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<title> Multiplicity-Free Permutation Characters in GAP, part 2</title>
<h1 align="center">Multiplicity-Free Permutation Characters in GAP, part 2 </h1>
<body bgcolor="FFFFFF">
<div class="p"><!----></div>
<h3 align="center"> T<font size="-2">HOMAS</font> B<font size="-2">REUER</font> <br />
<i>Lehrstuhl D für Mathematik</i> <br />
<i>RWTH, 52056 Aachen, Germany</i> </h3>
<div class="p"><!----></div>
<h3 align="center">July 21th, 2003 </h3>
<div class="p"><!----></div>
<div class="p"><!----></div>
We complete the classification of the multiplicity-free permutation
actions of nearly simple groups that involve a sporadic simple group,
which had been started in [<a href="#BL96" name="CITEBL96">BL96</a>] and [<a href="#LM03" name="CITELM03">LM</a>].
<div class="p"><!----></div>
<div class="p"><!----></div>
<h1>Contents </h1><a href="#tth_sEc1"
>1 Introduction</a><br />
<a href="#tth_sEc2"
>2 The Approach</a><br />
<a href="#tth_sEc2.1"
>2.1 Computing Possible Permutation Characters</a><br />
<a href="#tth_sEc2.2"
>2.2 Verifying the Candidates</a><br />
<a href="#tth_sEc2.3"
>2.3 Isoclinic Groups</a><br />
<a href="#tth_sEc2.4"
>2.4 Tests for <font face="helvetica">GAP</font></a><br />
<a href="#tth_sEc3"
>3 The Groups</a><br />
<a href="#tth_sEc3.1"
>3.1 G = 2.M<sub>12</sub></a><br />
<a href="#tth_sEc3.2"
>3.2 G = 2.M<sub>12</sub>.2</a><br />
<a href="#tth_sEc3.3"
>3.3 G = 2.M<sub>22</sub></a><br />
<a href="#tth_sEc3.4"
>3.4 G = 2.M<sub>22</sub>.2</a><br />
<a href="#tth_sEc3.5"
>3.5 G = 3.M<sub>22</sub></a><br />
<a href="#tth_sEc3.6"
>3.6 G = 3.M<sub>22</sub>.2</a><br />
<a href="#tth_sEc3.7"
>3.7 G = 4.M<sub>22</sub> and G = 12.M<sub>22</sub></a><br />
<a href="#tth_sEc3.8"
>3.8 G = 4.M<sub>22</sub>.2 and G = 12.M<sub>22</sub>.2</a><br />
<a href="#tth_sEc3.9"
>3.9 G = 6.M<sub>22</sub></a><br />
<a href="#tth_sEc3.10"
>3.10 G = 6.M<sub>22</sub>.2</a><br />
<a href="#tth_sEc3.11"
>3.11 G = 2.J<sub>2</sub></a><br />
<a href="#tth_sEc3.12"
>3.12 G = 2.J<sub>2</sub>.2</a><br />
<a href="#tth_sEc3.13"
>3.13 G = 2.HS</a><br />
<a href="#tth_sEc3.14"
>3.14 G = 2.HS.2</a><br />
<a href="#tth_sEc3.15"
>3.15 G = 3.J<sub>3</sub></a><br />
<a href="#tth_sEc3.16"
>3.16 G = 3.J<sub>3</sub>.2</a><br />
<a href="#tth_sEc3.17"
>3.17 G = 3.McL</a><br />
<a href="#tth_sEc3.18"
>3.18 G = 3.McL.2</a><br />
<a href="#tth_sEc3.19"
>3.19 G = 2.Ru</a><br />
<a href="#tth_sEc3.20"
>3.20 G = 2.Suz</a><br />
<a href="#tth_sEc3.21"
>3.21 G = 2.Suz.2</a><br />
<a href="#tth_sEc3.22"
>3.22 G = 3.Suz</a><br />
<a href="#tth_sEc3.23"
>3.23 G = 3.Suz.2</a><br />
<a href="#tth_sEc3.24"
>3.24 G = 6.Suz</a><br />
<a href="#tth_sEc3.25"
>3.25 G = 6.Suz.2</a><br />
<a href="#tth_sEc3.26"
>3.26 G = 3.ON</a><br />
<a href="#tth_sEc3.27"
>3.27 G = 3.ON.2</a><br />
<a href="#tth_sEc3.28"
>3.28 G = 2.Fi<sub>22</sub></a><br />
<a href="#tth_sEc3.29"
>3.29 G = 2.Fi<sub>22</sub>.2</a><br />
<a href="#tth_sEc3.30"
>3.30 G = 3.Fi<sub>22</sub></a><br />
<a href="#tth_sEc3.31"
>3.31 G = 3.Fi<sub>22</sub>.2</a><br />
<a href="#tth_sEc3.32"
>3.32 G = 6.Fi<sub>22</sub></a><br />
<a href="#tth_sEc3.33"
>3.33 G = 6.Fi<sub>22</sub>.2</a><br />
<a href="#tth_sEc3.34"
>3.34 G = 2.Co<sub>1</sub></a><br />
<a href="#tth_sEc3.35"
>3.35 G = 3.F<sub>3+</sub></a><br />
<a href="#tth_sEc3.36"
>3.36 G = 3.F<sub>3+</sub>.2</a><br />
<a href="#tth_sEc3.37"
>3.37 G = 2.B</a><br />
<a href="#tth_sEc4"
>4 Appendix: Explicit Computations with Groups</a><br />
<a href="#tth_sEc4.1"
>4.1 2<sup>4</sup>:A<sub>6</sub> type subgroups in 2.M<sub>22</sub></a><br />
<a href="#tth_sEc4.2"
>4.2 2<sup>4</sup>:S<sub>5</sub> type subgroups in M<sub>22</sub>.2</a><br />
<div class="p"><!----></div>
<div class="p"><!----></div>
<h2><a name="tth_sEc1">
1</a> Introduction</h2>
<div class="p"><!----></div>
In [<a href="#BL96" name="CITEBL96">BL96</a>], the multiplicity-free permutation characters of the sporadic
simple groups and their automorphism groups were classified.
Based on this list,
the multiplicity-free permutation characters of the central extensions of the
sporadic simple groups were classified in [<a href="#LM03" name="CITELM03">LM</a>].
<div class="p"><!----></div>
The purpose of this writeup is to show how the multiplicity-free
permutation characters of the automorphic extensions of the central
extensions of the sporadic simple groups can be computed,
to verify the calculations in [<a href="#LM03" name="CITELM03">LM</a>] (and to correct an error,
see Section <a href="#LMerror">3.32</a>),
and to provide a testfile for the <font face="helvetica">GAP</font> functions and the database.
<div class="p"><!----></div>
The database has been extended in the sense that also most of the character
tables of the multiplicity-free permutation modules of the sporadic simple
groups and their automorphic and central extensions have been computed,
see [<a href="#Hoe01" name="CITEHoe01">Höh01</a>,<a href="#Mue03" name="CITEMue03">Mül03</a>] for details.
<div class="p"><!----></div>
<h2><a name="tth_sEc2">
2</a> The Approach</h2>
<div class="p"><!----></div>
Suppose that a group G contains a normal subgroup N.
If π is a faithful multiplicity-free permutation character of G
then π = 1<sub>U</sub><sup>G</sup> for a subgroup U of G that intersects N trivially,
so π contains a constituent 1<sub>UN</sub><sup>G</sup> of degree π(1) / |N|,
which can be viewed as a multiplicity-free permutation character of the
factor group G / N.
Moreover, no constituent of the difference π− 1<sub>UN</sub><sup>G</sup> has N in its
kernel.
<div class="p"><!----></div>
So if we know all multiplicity-free permutation characters of the factor group
G / N then we can compute all candidates for multiplicity-free permutation
characters of G by "filling up" each such character
[π] with a linear combination of characters not containing N
in their kernels, of total degree (|N|−1) ·π(1), and such that the
sum is a possible permutation character of G.
For this situation, <font face="helvetica">GAP</font> provides a special variant of the function
<tt>PermChars</tt>.
In a second step, the candidates are inspected whether the required point
stabilizers (and if yes, how many conjugacy classes of them) exist.
Finally, the permutation characters are verified by explicit induction from
the character tables of the point stabilizers.
<div class="p"><!----></div>
The multiplicity-free permutation actions of the sporadic simple groups
and their automorphism groups are known by [<a href="#BL96" name="CITEBL96">BL96</a>],
so this approach is suitable for these groups.
<div class="p"><!----></div>
For central extensions of sporadic simple groups, the multiplicity-free
permutation characters have been classified in [<a href="#LM03" name="CITELM03">LM</a>];
this note describes a slightly different approach,
so we will give an independent confirmation of their results.
<div class="p"><!----></div>
First we load the Character Table Library [<a href="#CTblLib" name="CITECTblLib">Bre04b</a>]
of the <font face="helvetica">GAP</font> system [<a href="#GAP4" name="CITEGAP4">GAP04</a>],
read the file with <font face="helvetica">GAP</font> functions for computing multiplicity-free
permutation characters,
and the known data for the sporadic simple groups and their automorphism
groups.
<div class="p"><!----></div>
<pre>
gap> LoadPackage( "ctbllib" );
true
gap> RereadPackage( "ctbllib", "tst/multfree.g" );
true
gap> RereadPackage( "ctbllib", "tst/multfree.dat" );
true
</pre>
<div class="p"><!----></div>
In order to compare the results computed below with the database that is
already available, we load also this database.
<div class="p"><!----></div>
<pre>
gap> RereadPackage( "ctbllib", "tst/multfre2.dat" );
true
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc2.1">
2.1</a> Computing Possible Permutation Characters</h3>
<div class="p"><!----></div>
Then we define the <font face="helvetica">GAP</font> functions that are needed in the following.
<div class="p"><!----></div>
<tt>FaithfulCandidates</tt> takes the character table <tt>tbl</tt> of a group G
and the name <tt>factname</tt> of a factor group F of G for which the
multiplicity-free permutation characters are known,
and returns a list of lists, the entry at the i-th position being
the list of possible permutation characters of G that are multiplicity-free
and such that the sum of all constituents that are characters of F is the
i-th multiplicity-free permutation character of F.
As a side-effect, if the i-th entry is nonempty then information is printed
about the structure of the point-stabilizer in F and the number of
candidates found.
<div class="p"><!----></div>
<pre>
gap> FaithfulCandidates:= function( tbl, factname )
> local factinfo, factchars, facttbl, fus, sizeN, faith, i;
>
> # Fetch the data for the factor group.
> factinfo:= MultFreePermChars( factname );
> factchars:= List( factinfo, x -> x.character );
> facttbl:= UnderlyingCharacterTable( factchars[1] );
> fus:= GetFusionMap( tbl, facttbl );
> sizeN:= Size( tbl ) / Size( facttbl );
>
> # Compute faithful possible permutation characters.
> faith:= List( factchars, pi -> PermChars( tbl,
> rec( torso:= [ sizeN * pi[1] ],
> normalsubgroup:= ClassPositionsOfKernel( fus ),
> nonfaithful:= pi{ fus } ) ) );
>
> # Take only the multiplicity-free ones.
> faith:= List( faith, x -> Filtered( x, pi -> ForAll( Irr( tbl ),
> chi -> ScalarProduct( tbl, pi, chi ) < 2 ) ) );
>
> # Print info about the candidates.
> for i in [ 1 .. Length( faith ) ] do
> if not IsEmpty( faith[i] ) then
> Print( i, ": subgroup ", factinfo[i].subgroup,
> ", degree ", faith[i][1][1],
> " (", Length( faith[i] ), " cand.)\n" );
> fi;
> od;
>
> # Return the candidates.
> return faith;
> end;;
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc2.2">
2.2</a> Verifying the Candidates</h3>
<div class="p"><!----></div>
In the verification step, we check which of the given candidates of G
are induced from a given subgroup S.
For that, we use the following function.
Its arguments are the character table <tt>s</tt> of S,
the character tables <tt>tbl2</tt> and <tt>tbl</tt> of G and its derived subgroup
G<sup>′</sup> of index 2
(if G is perfect then <tt>0</tt> must be entered for <tt>tbl2</tt>),
the list <tt>candidates</tt> of characters of G,
and one of the strings <tt>"all"</tt>, <tt>"extending"</tt>, which means that we consider
either all possible class fusions of <tt>s</tt> into <tt>tbl2</tt> or only those whose
image does not lie in G<sup>′</sup>.
Note that the table of the derived subgroup of G is needed because
we want to express the decomposition of the permutation characters
relative to G<sup>′</sup>.
<div class="p"><!----></div>
The idea is that we know that n different permutation characters arise
from subgroups isomorphic with S (with the additional property that the
image of the embedding of S into G is not contained in G<sup>′</sup>
if the last argument is <tt>"extending"</tt>), and that <tt>candidates</tt> is a set
of possible permutation characters, of length n.
If the possible fusions between the character tables <tt>s</tt> and <tt>tbl2</tt>
lead to exactly to the given n permutation characters then we have proved
that they are in fact the permutation characters of G in question.
In this case, <tt>VerifyCandidates</tt> prints information about the decomposition
of the permutation characters.
If none of <tt>candidates</tt> arises from the possible embeddings of S into G
then the function prints that S does not occur.
In all other cases, the function signals an error (so this will not happen
in the calls to this function below).
<div class="p"><!----></div>
<pre>
gap> VerifyCandidates:= function( s, tbl, tbl2, candidates, admissible )
> local fus, der, pi;
>
> if tbl2 = 0 then
> tbl2:= tbl;
> fi;
>
> # Compute the possible class fusions, and induce the trivial character.
> fus:= PossibleClassFusions( s, tbl2 );
> if admissible = "extending" then
> der:= Set( GetFusionMap( tbl, tbl2 ) );
> fus:= Filtered( fus, map -> not IsSubset( der, map ) );
> fi;
> pi:= Set( List( fus, map -> Induced( s, tbl2,
> [ TrivialCharacter( s ) ], map )[1] ) );
>
> # Compare the two lists.
> if pi = SortedList( candidates ) then
> Print( "G = ", Identifier( tbl2 ), ": point stabilizer ",
> Identifier( s ), ", ranks ",
> List( pi, x -> Length( ConstituentsOfCharacter(x) ) ), "\n" );
> if Size( tbl ) = Size( tbl2 ) then
> Print( PermCharInfo( tbl, pi ).ATLAS, "\n" );
> else
> Print( PermCharInfoRelative( tbl, tbl2, pi ).ATLAS, "\n" );
> fi;
> elif IsEmpty( Intersection( pi, candidates ) ) then
> Print( "G = ", Identifier( tbl2 ), ": no ", Identifier( s ), "\n" );
> else
> Error( "problem with verify" );
> fi;
> end;;
</pre>
<div class="p"><!----></div>
Since in most cases the character tables of possible point stabilizers
are contained in the <font face="helvetica">GAP</font> Character Table Library,
the above function provides an easy test.
Alternatively, we could compute <em>all</em> faithful possible permutation
characters (not only the multiplicity-free ones)
of the degree in question;
if there are as many different such characters as are known to be induced
from point stabilizers <em>and</em> if no other subgroups of this index
exist then the characters are indeed permutation characters,
and we can compare them with the multiplicity-free characters computed
before.
<div class="p"><!----></div>
In the verification of the candidates, the following situations occur.
<div class="p"><!----></div>
<b>Lemma 1</b> <em><a name="situationI">
</a>
Let Φ\colon [^G] → G be a group epimorphism,
with K = ker(Φ) cyclic of order m,
and let H be a subgroup of G such that m is coprime to the order
of the commutator factor group of H.
If it is known that Φ<sup>−1</sup>(H) is a direct product of H with K
then the preimages under Φ of the G-conjugates of H
in [^G] contain one [^G]-class of subgroups
that are isomorphic with H and intersect trivially with K.
(This holds for example if the order of the Schur multiplier of H
is coprime to m.)
<div class="p"><!----></div>
</em>
<b>Lemma 2</b> <em><a name="situationII">
</a>
Let Φ\colon [^G] → G be a group epimorphism,
with K = ker(Φ) of order 3, such that the derived subgroup
G<sup>′</sup> of G has index 2 in G
and such that K is not central in G.
Consider a subgroup H of G with a subgroup H<sub>0</sub> = H ∩G<sup>′</sup>
of index 2 in H, and such that it is known that the preimage
Φ<sup>−1</sup>(H<sub>0</sub>) is a direct product of H<sub>0</sub> with K.
(This holds for example if the order of the Schur multiplier of H<sub>0</sub>
is coprime to 3.)
Then each complement of K in Φ<sup>−1</sup>(H<sub>0</sub>)
extends in Φ<sup>−1</sup>(H) to exactly one complement of K
that is isomorphic with H.
<div class="p"><!----></div>
</em>
<b>Lemma 3</b> <em><a name="situationIII">
</a>
Let Φ\colon [^G] → G be a group epimorphism,
with K = ker(Φ) of order 2.
Consider a subgroup H of G, with derived subgroup H<sup>′</sup>
of index 2 in H and such that
Φ<sup>−1</sup>(H<sup>′</sup>) has the structure 2 ×H<sup>′</sup>.
<ul>
<br />(i)
Suppose that there is an element h ∈ H \H<sup>′</sup>
such that the squares of the preimages of h in [^G] lie in
the unique subgroup of index 2 in Φ<sup>−1</sup>(H<sup>′</sup>).
(This holds for example if the preimages of h are involutions.)
Then Φ<sup>−1</sup>(H) has the type K ×H.
<br />(ii)
If Φ<sup>−1</sup>(H) has the type K ×H then
this group contains exactly two subgroups that are isomorphic with H.
<br />(iii)
Suppose that case (ii) applies and that there is
h ∈ H \H<sup>′</sup> whose two preimages are not conjugate
in [^G] and such that none of the two subgroups of the type H in
Φ<sup>−1</sup>(H) contains elements in the two conjugacy classes
that contain the preimages of h.
Then the two subgroups of the type H induce different permutation
characters of [^G], in particular exactly two conjugacy classes of
subgroups of the type H in [^G] arise from the conjugates of H
in G.</ul>
<div class="p"><!----></div>
</em>With character theoretic methods, we can check a weaker form of
Lemma <a href="#situationIII">2.3</a> (i).
Namely, the conditions are clearly satisfied if there is a conjugacy class
C of elements in H that is not contained in H<sup>′</sup>
and such that the class of [^G] that
contains the squares of the preimages of C is <em>not</em> contained
in the images of the classes of 2 ×H<sup>′</sup> that lie outside
H<sup>′</sup>.
<div class="p"><!----></div>
The function <tt>CheckConditionsForLemma3</tt> tests this, and prints a message
if Lemma <a href="#situationIII">2.3</a> (i) applies becaue of this situation.
More precisely, the arguments are (in this order) the character tables of
H<sup>′</sup>, H, G, and [^G], and one of the strings <tt>"all"</tt>,
<tt>"extending"</tt>; the last argument means that either all embeddings of H
into G are considered or only those which do not lie inside the
derived subgroup of G.
<div class="p"><!----></div>
The function <em>assumes</em> that <tt>s0</tt> is the character table of the derived
subgroup of <tt>s</tt>, and that H<sup>′</sup> lifts to a direct product in [^G].
<div class="p"><!----></div>
<pre>
gap> CheckConditionsForLemma3:= function( s0, s, fact, tbl, admissible )
> local s0fuss, sfusfact, der, outerins, outerinfact, preim, squares, dp,
> dpfustbl, s0indp, other, goodclasses;
>
> if Size( s ) <> 2 * Size( s0 ) then
> Error( "<s> must be twice as large as <s0>" );
> fi;
>
> s0fuss:= GetFusionMap( s0, s );
> sfusfact:= PossibleClassFusions( s, fact );
> if admissible = "extending" then
> der:= ClassPositionsOfDerivedSubgroup( fact );
> sfusfact:= Filtered( sfusfact, map -> not IsSubset( der, map ) );
> fi;
> outerins:= Difference( [ 1 .. NrConjugacyClasses( s ) ], s0fuss );
> outerinfact:= Set( List( sfusfact, map -> Set( map{ outerins } ) ) );
> if Length( outerinfact ) <> 1 then
> Error( "classes of `", s, "' inside `", fact, "' not determined" );
> fi;
>
> preim:= Flat( InverseMap( GetFusionMap( tbl, fact ) ){ outerinfact[1] } );
> squares:= Set( PowerMap( tbl, 2 ){ preim } );
> dp:= s0 * CharacterTable( "Cyclic", 2 );
> dpfustbl:= PossibleClassFusions( dp, tbl );
> s0indp:= GetFusionMap( s0, dp );
> other:= Difference( [ 1 .. NrConjugacyClasses( dp ) ], s0indp );
> goodclasses:= List( dpfustbl, map -> Intersection( squares,
> Difference( map{ s0indp }, map{ other } ) ) );
> if not IsEmpty( Intersection( goodclasses ) ) then
> Print( Identifier( tbl ), ": ", Identifier( s ),
> " lifts to a direct product,\n",
> "proved by squares in ", Intersection( goodclasses ), ".\n" );
> elif ForAll( goodclasses, IsEmpty ) then
> Print( Identifier( tbl ), ": ", Identifier( s ),
> " lifts to a nonsplit extension.\n" );
> else
> Print( "sorry, no proof of the splitting!\n" );
> fi;
> end;;
</pre>
<div class="p"><!----></div>
Let us assume we are in the situation of Lemma <a href="#situationIII">2.3</a>.
Then Φ<sup>−1</sup>(H) is a direct product
〈z 〉×H, where z is an involution.
The derived subgroup of Φ<sup>−1</sup>(H) is H<sub>0</sub> ≡ H<sup>′</sup>,
and Φ<sup>−1</sup>(H) contains two subgroups H<sub>1</sub>, H<sub>2</sub>
which are isomorphic with H,
and such that H<sub>2</sub> = H<sub>0</sub> ∪{ h z; h ∈ H<sub>1</sub> \H<sub>0</sub> }.
If the embedding of H<sub>1</sub>, say, into [^G] has the properties
that an element outside H<sub>0</sub> is mapped into a class C of [^G]
that is different from z C and such that no element of H<sub>1</sub> lies in z C
then z C contains element of H<sub>2</sub> but C does not.
In particular, the permutation characters of the two actions of [^G]
on the cosets of H<sub>1</sub> and H<sub>2</sub>, respectively, are necessarily different.
<div class="p"><!----></div>
We check this with the following function.
Its arguments are one class fusion from the character table of H<sub>1</sub> to that
of [^G], the factor fusion from the character table of [^G] to
that of G,
and the list of positions of the classes of H<sub>0</sub> in the character table
of H<sub>1</sub>.
The return value is <tt>true</tt> if there are two different permutation characters,
and <tt>false</tt> if this cannot be proved using the criterion.
<div class="p"><!----></div>
<pre>
gap> NecessarilyDifferentPermChars:= function( fusion, factfus, inner )
> local outer, inv;
>
> outer:= Difference( [ 1 .. Length( fusion ) ], inner );
> fusion:= fusion{ outer };
> inv:= Filtered( InverseMap( factfus ), IsList );
> return ForAny( inv, pair -> Length( Intersection( pair, fusion ) ) = 1 );
> end;;
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc2.3">
2.3</a> Isoclinic Groups</h3>
<div class="p"><!----></div>
For dealing with the character tables of groups of the type 2.G.2 that are
isoclinic to those whose tables are printed in the A<font size="-2">TLAS</font> ([<a href="#CCN85" name="CITECCN85">CCN<sup>+</sup>85</a>]),
it is necessary to store explicitly the factor fusion from 2.G.2 onto G.2
and the subgroup fusion from 2.G into 2.G.2,
in order to make the above functions work.
Note that these maps coincide for the two isoclinism types.
<div class="p"><!----></div>
<pre>
gap> IsoclinicTable:= function( tbl, tbl2, facttbl )
> local subfus, factfus;
>
> subfus:= GetFusionMap( tbl, tbl2 );
> factfus:= GetFusionMap( tbl2, facttbl );
> tbl2:= CharacterTableIsoclinic( tbl2 );
> StoreFusion( tbl, subfus, tbl2 );
> StoreFusion( tbl2, factfus, facttbl );
> return tbl2;
> end;;
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc2.4">
2.4</a> Tests for <font face="helvetica">GAP</font></h3>
<div class="p"><!----></div>
With the following function, we check whether the characters computed here
coincide with the lists computed in [<a href="#LM03" name="CITELM03">LM</a>].
<div class="p"><!----></div>
<pre>
gap> CompareWithDatabase:= function( name, chars )
> local info;
>
> info:= MultFreePermChars( name );
> info:= List( info, x -> x.character );;
> if SortedList( info ) <> SortedList( Concatenation( chars ) ) then
> Error( "contradiction 1 for ", name );
> fi;
> end;;
</pre>
<div class="p"><!----></div>
If the character tables of all maximal subgroups of G are known then
we could use alternatively the same method (and in fact the same <font face="helvetica">GAP</font>
functions) as in the classification in [<a href="#BL96" name="CITEBL96">BL96</a>].
This is shown in the following sections where applicable,
using the following function.
<div class="p"><!----></div>
<pre>
gap> CompareWithCandidatesByMaxes:= function( name, faith )
> local tbl, poss;
>
> tbl:= CharacterTable( name );
> if not HasMaxes( tbl ) then
> Error( "no maxes stored for ", name );
> fi;
> poss:= PossiblePermutationCharactersWithBoundedMultiplicity( tbl, 1 );
> poss:= List( poss.permcand, l -> Filtered( l,
> pi -> ClassPositionsOfKernel( pi ) = [ 1 ] ) );
> if SortedList( Concatenation( poss ) )
> <> SortedList( Concatenation( faith ) ) then
> Error( "contradiction 2 for ", name );
> fi;
> end;;
</pre>
<div class="p"><!----></div>
<h2><a name="tth_sEc3">
3</a> The Groups</h2>
<div class="p"><!----></div>
In the following,
we use A<font size="-2">TLAS</font> notation (see [<a href="#CCN85" name="CITECCN85">CCN<sup>+</sup>85</a>]) for the names of the groups.
In particular, 2 ×G and G ×2 denote the direct product
of the group G with a cyclic group of order 2,
and G.2 and 2.G denote an upward and downward extension, respectively,
of G by a cyclic group of order 2, such that these groups are <em>not</em>
direct products.
<div class="p"><!----></div>
For groups of the structure 2.G.2 where the character table of G is
contained in the A<font size="-2">TLAS</font>, we use the name 2.G.2 for the isoclinism type
whose character table is printed in the A<font size="-2">TLAS</font>,
and (2.G.2)<sup>∗</sup> for the other isoclinism type.
<div class="p"><!----></div>
Most of the computations that are shown in the following use only information
from the <font face="helvetica">GAP</font> Character Table Library.
The (few) explicit computations with groups are collected in
Section <a href="#explicit">4</a>.
<div class="p"><!----></div>
<h3><a name="tth_sEc3.1">
3.1</a> G = 2.M<sub>12</sub></h3>
<div class="p"><!----></div>
The group 2.M<sub>12</sub> has ten faithful multiplicity-free permutation actions,
with point stabilizers of the types M<sub>11</sub> (twice),
A<sub>6</sub>.2<sub>1</sub> (twice), 3<sup>2</sup>.2.S<sub>4</sub> (four classes), and 3<sup>2</sup>:2.A<sub>4</sub> (twice).
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "2.M12" );;
gap> faith:= FaithfulCandidates( tbl, "M12" );;
1: subgroup $M_{11}$, degree 24 (1 cand.)
2: subgroup $M_{11}$, degree 24 (1 cand.)
5: subgroup $A_6.2_1 \leq A_6.2^2$, degree 264 (1 cand.)
8: subgroup $A_6.2_1 \leq A_6.2^2$, degree 264 (1 cand.)
11: subgroup $3^2.2.S_4$, degree 440 (2 cand.)
12: subgroup $3^2:2.A_4 \leq 3^2.2.S_4$, degree 880 (1 cand.)
13: subgroup $3^2.2.S_4$, degree 440 (2 cand.)
14: subgroup $3^2:2.A_4 \leq 3^2.2.S_4$, degree 880 (1 cand.)
</pre>
<div class="p"><!----></div>
There are two classes of M<sub>11</sub> subgroups in M<sub>12</sub> as well as in
2.M<sub>12</sub>, so we apply Lemma <a href="#situationI">2.1</a>.
<div class="p"><!----></div>
<pre>
gap> VerifyCandidates( CharacterTable( "M11" ), tbl, 0,
> Concatenation( faith[1], faith[2] ), "all" );
G = 2.M12: point stabilizer M11, ranks [ 3, 3 ]
[ "1a+11a+12a", "1a+11b+12a" ]
</pre>
<div class="p"><!----></div>
According to the list of maximal subgroups of 2.M<sub>12</sub>,
any A<sub>6</sub>.2<sup>2</sup> subgroup in M<sub>12</sub> lifts to a group of the structure
A<sub>6</sub>.D<sub>8</sub> in M<sub>12</sub>, which contains two conjugate subgroups of the type
A<sub>6</sub>.2<sub>1</sub>; so we get two classes of such subgroups, with the same permutation
character.
<div class="p"><!----></div>
<pre>
gap> Maxes( tbl );
[ "2xM11", "2.M12M2", "A6.D8", "2.M12M4", "2.L2(11)", "2x3^2.2.S4",
"2.M12M7", "2.M12M8", "2.M12M9", "2.M12M10", "2.A4xS3" ]
gap> faith[5] = faith[8];
true
gap> VerifyCandidates( CharacterTable( "A6.2_1" ), tbl, 0, faith[5], "all" );
G = 2.M12: point stabilizer A6.2_1, ranks [ 7 ]
[ "1a+11ab+12a+54a+55a+120b" ]
</pre>
<div class="p"><!----></div>
The 3<sup>2</sup>.2.S<sub>4</sub> type subgroups of M<sub>12</sub> lift to direct products with
the centre of 2.M<sub>12</sub>, each such group contains two subgroups of the type
3<sup>2</sup>.2.S<sub>4</sub> which induce different permutation characters,
for example because the involutions in 3<sup>2</sup>.2.S<sub>4</sub> \3<sup>2</sup>.2.A<sub>4</sub>
lie in the two preimages of the class <tt>2B</tt> of M<sub>12</sub>.
<div class="p"><!----></div>
<pre>
gap> s:= CharacterTable( "3^2.2.S4" );;
gap> derpos:= ClassPositionsOfDerivedSubgroup( s );;
gap> facttbl:= CharacterTable( "M12" );;
gap> factfus:= GetFusionMap( tbl, facttbl );;
gap> ForAll( PossibleClassFusions( s, tbl ),
> map -> NecessarilyDifferentPermChars( map, factfus, derpos ) );
true
gap> VerifyCandidates( s, tbl, 0, Concatenation( faith[11], faith[13] ), "all" );
G = 2.M12: point stabilizer 3^2.2.S4, ranks [ 7, 7, 9, 9 ]
[ "1a+11a+54a+55a+99a+110ab", "1a+11b+54a+55a+99a+110ab",
"1a+11a+12a+44ab+54a+55a+99a+120b", "1a+11b+12a+44ab+54a+55a+99a+120b" ]
</pre>
<div class="p"><!----></div>
Each 3<sup>2</sup>.2.S<sub>4</sub> type group contains a unique subgroup of the type
3<sup>2</sup>.2.A<sub>4</sub>, we get two classes of such subgroups, with
different permutation characters because already the corresponding characters
for M<sub>12</sub> are different; we verify the candidates by inducing the degree
two permutation characters of the 3<sup>2</sup>.2.S<sub>4</sub> type groups to 2.M<sub>12</sub>.
<div class="p"><!----></div>
<pre>
gap> fus:= PossibleClassFusions( s, tbl );;
gap> deg2:= PermChars( s, 2 );
[ Character( CharacterTable( "3^2.2.S4" ), [ 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0
] ) ]
gap> pi:= Set( List( fus, map -> Induced( s, tbl, deg2, map )[1] ) );;
gap> pi = SortedList( Concatenation( faith[12], faith[14] ) );
true
gap> PermCharInfo( tbl, pi ).ATLAS;
[ "1a+11a+12a+44ab+45a+54a+55ac+99a+110ab+120ab",
"1a+11b+12a+44ab+45a+54a+55ab+99a+110ab+120ab" ]
gap> CompareWithDatabase( "2.M12", faith );
gap> CompareWithCandidatesByMaxes( "2.M12", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.2">
3.2</a> G = 2.M<sub>12</sub>.2</h3>
<div class="p"><!----></div>
The group 2.M<sub>12</sub>.2 that is printed in the A<font size="-2">TLAS</font> has three faithful
multiplicity-free permutation actions,
with point stabilizers of the types M<sub>11</sub> and L<sub>2</sub>(11).2 (twice),
respectively.
<div class="p"><!----></div>
<pre>
gap> tbl2:= CharacterTable( "2.M12.2" );;
gap> faith:= FaithfulCandidates( tbl2, "M12.2" );;
1: subgroup $M_{11}$, degree 48 (1 cand.)
2: subgroup $L_2(11).2$, degree 288 (2 cand.)
</pre>
<div class="p"><!----></div>
The two classes of subgroups of the type M<sub>11</sub> in 2.M<sub>12</sub> are fused in
2.M<sub>12</sub>.2, so we get one class of these subgroups.
<div class="p"><!----></div>
<pre>
gap> VerifyCandidates( CharacterTable( "M11" ), tbl, tbl2, faith[1], "all" );
G = 2.M12.2: point stabilizer M11, ranks [ 5 ]
[ "1a^{\\pm}+11ab+12a^{\\pm}" ]
</pre>
<div class="p"><!----></div>
The outer involutions in the maximal subgroups of the type L<sub>2</sub>(11).2
in M<sub>12</sub>.2 lift to involutions in 2.M<sub>12</sub>.2;
moreover, those subgroups of the type L<sub>2</sub>(11).2 that are novelties
(so the intersection with M<sub>12</sub> lies in M<sub>11</sub> subgroups)
contain <tt>2B</tt> elements which lift to involutions in 2.M<sub>12</sub>.2,
so the L<sub>2</sub>(11) subgroup lifts to a group of the type 2 ×L<sub>2</sub>(11),
and Lemma <a href="#situationIII">2.3</a> yields two classes of subgroups.
The permutation characters are different, for example because
each each of the two candidates contains elements in one of the
two preimages of the class <tt>2B</tt>.
<div class="p"><!----></div>
(The function <tt>CheckConditionsForLemma3</tt> fails here,
because of the two classes of maximal subgroups L<sub>2</sub>(11).2 in M<sub>12</sub>.2.
One of them contains <tt>2A</tt> elements, the other contains <tt>2B</tt> elements.
Only the latter type of subgroups, whose intersection with M<sub>12</sub> is not
maximal in M<sub>12</sub>, lifts to subgroups of 2.M<sub>12</sub>.2 that contain
L<sub>2</sub>(11).2 subgroups.)
<div class="p"><!----></div>
<pre>
gap> s:= CharacterTable( "L2(11).2" );;
gap> derpos:= ClassPositionsOfDerivedSubgroup( s );;
gap> facttbl:= CharacterTable( "M12.2" );;
gap> factfus:= GetFusionMap( tbl2, facttbl );;
gap> ForAll( PossibleClassFusions( s, tbl2 ),
> map -> NecessarilyDifferentPermChars( map, factfus, derpos ) );
true
gap> VerifyCandidates( s, tbl, tbl2, faith[2], "all" );
G = 2.M12.2: point stabilizer L2(11).2, ranks [ 7, 7 ]
[ "1a^++11ab+12a^{\\pm}+55a^++66a^++120b^-",
"1a^++11ab+12a^{\\pm}+55a^++66a^++120b^+" ]
gap> CompareWithDatabase( "2.M12.2", faith );
</pre>
<div class="p"><!----></div>
The group (2.M<sub>12</sub>.2)<sup>∗</sup> of the isoclinism type that is not printed
in the A<font size="-2">TLAS</font> has one faithful multiplicity-free permutation action,
with point stabilizer of the type M<sub>11</sub>;
as this subgroup lies inside 2.M<sub>12</sub>, its existence is clear,
and the permutation character in both groups of the type 2.M<sub>12</sub>.2
is the same.
<div class="p"><!----></div>
<pre>
gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );;
gap> faith:= FaithfulCandidates( tbl2, "M12.2" );;
1: subgroup $M_{11}$, degree 48 (1 cand.)
gap> CompareWithDatabase( "Isoclinic(2.M12.2)", faith );
</pre>
<div class="p"><!----></div>
Note that in (2.M<sub>12</sub>.2)<sup>∗</sup>,
the subgroup of the type (2 ×L<sub>2</sub>(11)).2 is a nonsplit extension,
so the unique index 2 subgroup in this group contains the centre of
2.M<sub>12</sub>.2, in particular there is no subgroup of the type L<sub>2</sub>(11).2.
<div class="p"><!----></div>
<pre>
gap> s:= CharacterTable( "L2(11).2" );;
gap> PossibleClassFusions( s, tbl2 );
[ ]
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.3">
3.3</a> G = 2.M<sub>22</sub></h3><a name="libtbl">
</a>
<div class="p"><!----></div>
The group 2.M<sub>22</sub> has four faithful multiplicity-free permutation actions,
with point stabilizers of the types 2<sup>4</sup>:A<sub>5</sub>, A<sub>7</sub> (twice),
and 2<sup>3</sup>:L<sub>3</sub>(2), by Lemma <a href="#situationI">2.1</a>.
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "2.M22" );;
gap> faith:= FaithfulCandidates( tbl, "M22" );;
3: subgroup $2^4:A_5 \leq 2^4:A_6$, degree 924 (1 cand.)
4: subgroup $A_7$, degree 352 (1 cand.)
5: subgroup $A_7$, degree 352 (1 cand.)
7: subgroup $2^3:L_3(2)$, degree 660 (1 cand.)
</pre>
<div class="p"><!----></div>
Note that one class of subgroups of the type 2<sup>4</sup>:A<sub>5</sub> in the maximal subgroup
of the type 2<sup>4</sup>:A<sub>6</sub> as well as the A<sub>7</sub> and 2<sup>3</sup>:L<sub>3</sub>(2) subgroups
lift to direct products in 2.M<sub>22</sub>.
A proof for 2<sup>4</sup>:A<sub>5</sub> using explicit computations with the group can be found
in Subsection <a href="#explicit1">4.1</a>.
<div class="p"><!----></div>
<pre>
gap> Maxes( tbl );
[ "2.L3(4)", "2.M22M2", "2xA7", "2xA7", "2.M22M5", "2x2^3:L3(2)",
"(2xA6).2_3", "2xL2(11)" ]
gap> s:= CharacterTable( "P1/G1/L1/V1/ext2" );;
gap> VerifyCandidates( s, tbl, 0, faith[3], "all" );
G = 2.M22: point stabilizer P1/G1/L1/V1/ext2, ranks [ 8 ]
[ "1a+21a+55a+126ab+154a+210b+231a" ]
gap> faith[4] = faith[5];
true
gap> VerifyCandidates( CharacterTable( "A7" ), tbl, 0, faith[4], "all" );
G = 2.M22: point stabilizer A7, ranks [ 5 ]
[ "1a+21a+56a+120a+154a" ]
gap> VerifyCandidates( CharacterTable( "M22M6" ), tbl, 0, faith[7], "all" );
G = 2.M22: point stabilizer 2^3:sl(3,2), ranks [ 7 ]
[ "1a+21a+55a+99a+120a+154a+210b" ]
gap> CompareWithDatabase( "2.M22", faith );
gap> CompareWithCandidatesByMaxes( "2.M22", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.4">
3.4</a> G = 2.M<sub>22</sub>.2</h3><a name="2.M22.2">
</a>
<div class="p"><!----></div>
The group 2.M<sub>22</sub>.2 that is printed in the A<font size="-2">TLAS</font> has eight faithful
multiplicity-free permutation actions,
with point stabilizers of the types 2<sup>4</sup>:S<sub>5</sub> (twice), A<sub>7</sub>,
2<sup>3</sup>:L<sub>3</sub>(2) ×2 (twice), 2<sup>3</sup>:L<sub>3</sub>(2), and L<sub>2</sub>(11).2 (twice).
<div class="p"><!----></div>
<pre>
gap> tbl2:= CharacterTable( "2.M22.2" );;
gap> faith:= FaithfulCandidates( tbl2, "M22.2" );;
6: subgroup $2^4:S_5 \leq 2^4:S_6$, degree 924 (2 cand.)
7: subgroup $A_7$, degree 704 (1 cand.)
11: subgroup $2^3:L_3(2) \times 2$, degree 660 (2 cand.)
12: subgroup $2^3:L_3(2) \leq 2^3:L_3(2) \times 2$, degree 1320 (2 cand.)
16: subgroup $L_2(11).2$, degree 1344 (2 cand.)
</pre>
<div class="p"><!----></div>
The character table of the 2<sup>4</sup>:S<sub>5</sub> type subgroup is contained in the <font face="helvetica">GAP</font>
Character Table Library,
with identifier <tt>w(d5)</tt> (cf. Subsection <a href="#explicit2">4.2</a>).
<div class="p"><!----></div>
<pre>
gap> s:= CharacterTable( "w(d5)" );;
gap> derpos:= ClassPositionsOfDerivedSubgroup( s );;
gap> facttbl:= CharacterTable( "M22.2" );;
gap> factfus:= GetFusionMap( tbl2, facttbl );;
gap> ForAll( PossibleClassFusions( s, tbl2 ),
> map -> NecessarilyDifferentPermChars( map, factfus, derpos ) );
true
gap> VerifyCandidates( s, tbl, tbl2, faith[6], "all" );
G = 2.M22.2: point stabilizer w(d5), ranks [ 7, 7 ]
[ "1a^++21a^++55a^++126ab+154a^++210b^-+231a^-",
"1a^++21a^++55a^++126ab+154a^++210b^++231a^-" ]
</pre>
<div class="p"><!----></div>
The two classes of the type A<sub>7</sub> subgroups in 2.M<sub>22</sub> are fused
in 2.M<sub>22</sub>.2.
<div class="p"><!----></div>
<pre>
gap> VerifyCandidates( CharacterTable( "A7" ), tbl, tbl2, faith[7], "all" );
G = 2.M22.2: point stabilizer A7, ranks [ 10 ]
[ "1a^{\\pm}+21a^{\\pm}+56a^{\\pm}+120a^{\\pm}+154a^{\\pm}" ]
</pre>
<div class="p"><!----></div>
The preimages of the 2<sup>3</sup>:L<sub>3</sub>(2) ×2 type subgroups of M<sub>22</sub>.2
in 2.M<sub>22</sub>.2 are direct products, by the discussion of 2.M<sub>22</sub>
and Lemma <a href="#situationIII">2.3</a> (i).
So Lemma <a href="#situationIII">2.3</a> (iii) yields two classes,
with different permutation characters.
<div class="p"><!----></div>
<pre>
gap> s:= CharacterTable( "2x2^3:L3(2)" );;
gap> s0:= CharacterTable( "2^3:sl(3,2)" );;
gap> s0fuss:= PossibleClassFusions( s0, s );;
gap> StoreFusion( s0, s0fuss[1], s );
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "extending" );
2.M22.2: 2x2^3:L3(2) lifts to a direct product,
proved by squares in [ 1, 5, 14, 16 ].
gap> derpos:= ClassPositionsOfDerivedSubgroup( s );;
gap> ForAll( PossibleClassFusions( s, tbl2 ),
> map -> NecessarilyDifferentPermChars( map, factfus, derpos ) );
true
gap> VerifyCandidates( s, tbl, tbl2, faith[11], "extending" );
G = 2.M22.2: point stabilizer 2x2^3:L3(2), ranks [ 7, 7 ]
[ "1a^++21a^++55a^++99a^++120a^-+154a^++210b^-",
"1a^++21a^++55a^++99a^++120a^++154a^++210b^+" ]
</pre>
<div class="p"><!----></div>
There is one class of subgroups of the type 2<sup>3</sup>:L<sub>3</sub>(2) in 2.M<sub>22</sub>.
One of the two candidates of degree 1 320 is excluded because it does not
arise from a possible class fusion.
<div class="p"><!----></div>
<pre>
gap> s:= CharacterTable( "M22M6" );;
gap> fus:= PossibleClassFusions( s, tbl );;
gap> pi1320:= Set( List( fus, x -> Induced( s, tbl2,
> [ TrivialCharacter( s ) ], x )[1] ) );;
gap> IsSubset( faith[12], pi1320 );
true
gap> faith[12]:= pi1320;;
gap> VerifyCandidates( s, tbl, tbl2, faith[12], "all" );
G = 2.M22.2: point stabilizer 2^3:sl(3,2), ranks [ 14 ]
[ "1a^{\\pm}+21a^{\\pm}+55a^{\\pm}+99a^{\\pm}+120a^{\\pm}+154a^{\\pm}+210b^{\\\
pm}" ]
</pre>
<div class="p"><!----></div>
The preimages of the L<sub>2</sub>(11).2 type subgroups of M<sub>22</sub>.2 in 2.M<sub>22</sub>.2
are direct products by Lemma <a href="#situationIII">2.3</a> (i),
so Lemma <a href="#situationIII">2.3</a> (iii) yields two classes,
with different permutation characters.
<div class="p"><!----></div>
<pre>
gap> s:= CharacterTable( "L2(11).2" );;
gap> s0:= CharacterTable( "L2(11)" );;
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "all" );
2.M22.2: L2(11).2 lifts to a direct product,
proved by squares in [ 1, 4, 10, 13 ].
gap> derpos:= ClassPositionsOfDerivedSubgroup( s );;
gap> ForAll( PossibleClassFusions( s, tbl2 ),
> map -> NecessarilyDifferentPermChars( map, factfus, derpos ) );
true
gap> VerifyCandidates( CharacterTable( "L2(11).2" ), tbl, tbl2, faith[16], "all" );
G = 2.M22.2: point stabilizer L2(11).2, ranks [ 10, 10 ]
[ "1a^++21a^-+55a^++56a^{\\pm}+120a^-+154a^++210a^-+231a^-+440a^+",
"1a^++21a^-+55a^++56a^{\\pm}+120a^++154a^++210a^-+231a^-+440a^-" ]
gap> CompareWithDatabase( "2.M22.2", faith );
</pre>
<div class="p"><!----></div>
The group (2.M<sub>22</sub>.2)<sup>∗</sup> of the isoclinism type that is not printed
in the A<font size="-2">TLAS</font> has two faithful multiplicity-free permutation actions,
with point stabilizers of the types A<sub>7</sub> and 2<sup>3</sup>:L<sub>3</sub>(2).
<div class="p"><!----></div>
<pre>
gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );;
gap> faith:= FaithfulCandidates( tbl2, "M22.2" );;
7: subgroup $A_7$, degree 704 (1 cand.)
12: subgroup $2^3:L_3(2) \leq 2^3:L_3(2) \times 2$, degree 1320 (2 cand.)
gap> faith[12]:= Filtered( faith[12], chi -> chi in pi1320 );;
gap> CompareWithDatabase( "Isoclinic(2.M22.2)", faith );
</pre>
<div class="p"><!----></div>
The two classes of subgroups lie inside 2.M<sub>22</sub>,
so their existence has been discussed already above.
<div class="p"><!----></div>
<h3><a name="tth_sEc3.5">
3.5</a> G = 3.M<sub>22</sub></h3>
<div class="p"><!----></div>
The group 3.M<sub>22</sub> has four faithful multiplicity-free permutation actions,
with point stabilizers of the types 2<sup>4</sup>:A<sub>5</sub>, 2<sup>4</sup>:S<sub>5</sub>, 2<sup>3</sup>:L<sub>3</sub>(2),
and L<sub>2</sub>(11).
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "3.M22" );;
gap> faith:= FaithfulCandidates( tbl, "M22" );;
3: subgroup $2^4:A_5 \leq 2^4:A_6$, degree 1386 (1 cand.)
6: subgroup $2^4:S_5$, degree 693 (1 cand.)
7: subgroup $2^3:L_3(2)$, degree 990 (1 cand.)
9: subgroup $L_2(11)$, degree 2016 (1 cand.)
</pre>
<div class="p"><!----></div>
The existence of one class of each of these types follows from
Lemma <a href="#situationI">2.1</a>.
<div class="p"><!----></div>
<pre>
gap> VerifyCandidates( CharacterTable( "P1/G1/L1/V1/ext2" ), tbl, 0, faith[3], "all" );
G = 3.M22: point stabilizer P1/G1/L1/V1/ext2, ranks [ 13 ]
[ "1a+21abc+55a+105abcd+154a+231abc" ]
gap> VerifyCandidates( CharacterTable( "M22M5" ), tbl, 0, faith[6], "all" );
G = 3.M22: point stabilizer 2^4:s5, ranks [ 10 ]
[ "1a+21abc+55a+105abcd+154a" ]
gap> VerifyCandidates( CharacterTable( "M22M6" ), tbl, 0, faith[7], "all" );
G = 3.M22: point stabilizer 2^3:sl(3,2), ranks [ 13 ]
[ "1a+21abc+55a+99abc+105abcd+154a" ]
gap> VerifyCandidates( CharacterTable( "M22M8" ), tbl, 0, faith[9], "all" );
G = 3.M22: point stabilizer L2(11), ranks [ 16 ]
[ "1a+21abc+55a+105abcd+154a+210abc+231abc" ]
gap> CompareWithDatabase( "3.M22", faith );
gap> CompareWithCandidatesByMaxes( "3.M22", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.6">
3.6</a> G = 3.M<sub>22</sub>.2</h3>
<div class="p"><!----></div>
The group 3.M<sub>22</sub>.2 has five faithful multiplicity-free permutation
actions, with point stabilizers of the types 2<sup>4</sup>:S<sub>5</sub>, 2<sup>5</sup>:S<sub>5</sub>,
2<sup>4</sup>:(A<sub>5</sub> ×2), 2<sup>3</sup>:L<sub>3</sub>(2) ×2, and L<sub>2</sub>(11).2.
<div class="p"><!----></div>
<pre>
gap> tbl2:= CharacterTable( "3.M22.2" );;
gap> faith:= FaithfulCandidates( tbl2, "M22.2" );;
6: subgroup $2^4:S_5 \leq 2^4:S_6$, degree 1386 (1 cand.)
8: subgroup $2^5:S_5$, degree 693 (1 cand.)
10: subgroup $2^4:(A_5 \times 2) \leq 2^5:S_5$, degree 1386 (1 cand.)
11: subgroup $2^3:L_3(2) \times 2$, degree 990 (1 cand.)
16: subgroup $L_2(11).2$, degree 2016 (1 cand.)
</pre>
<div class="p"><!----></div>
Subgroups of these types exist by Lemma <a href="#situationII">2.2</a>.
The verification is straightforward in all cases
except that of 2<sup>4</sup>:(A<sub>5</sub> ×2).
<div class="p"><!----></div>
<pre>
gap> VerifyCandidates( CharacterTable( "w(d5)" ), tbl, tbl2, faith[6], "all" );
G = 3.M22.2: point stabilizer w(d5), ranks [ 9 ]
[ "1a^++21a^+bc+55a^++105adbc+154a^++231a^-bc" ]
gap> VerifyCandidates( CharacterTable( "M22.2M4" ), tbl, tbl2, faith[8], "all" );
G = 3.M22.2: point stabilizer M22.2M4, ranks [ 7 ]
[ "1a^++21a^+bc+55a^++105adbc+154a^+" ]
gap> VerifyCandidates( CharacterTable( "2x2^3:L3(2)" ), tbl, tbl2, faith[11], "all" );
G = 3.M22.2: point stabilizer 2x2^3:L3(2), ranks [ 9 ]
[ "1a^++21a^+bc+55a^++99a^+bc+105adbc+154a^+" ]
gap> VerifyCandidates( CharacterTable( "L2(11).2" ), tbl, tbl2, faith[16], "all" );
G = 3.M22.2: point stabilizer L2(11).2, ranks [ 11 ]
[ "1a^++21a^-bc+55a^++105adbc+154a^++210a^-bc+231a^-bc" ]
</pre>
<div class="p"><!----></div>
In the remaining case, we note that the 2<sup>4</sup>:(A<sub>5</sub> ×2) type subgroup
has index 2 in the maximal subgroup of the type 2<sup>5</sup>:S<sub>5</sub>,
whose character table is available via the identifier <tt>M22.2M4</tt>.
It is sufficient to show that exactly one of the three index 2
subgroups in this group induces a multiplicity-free permutation character
of 3.M<sub>22</sub>.2,
and this can be done by inducing the degree 2 permutation characters
of 2<sup>5</sup>:S<sub>5</sub> to 3.M<sub>22</sub>.2.
<div class="p"><!----></div>
<pre>
gap> s:= CharacterTable( "M22.2M4" );;
gap> lin:= LinearCharacters( s );
[ Character( CharacterTable( "M22.2M4" ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ),
Character( CharacterTable( "M22.2M4" ), [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 ] ),
Character( CharacterTable( "M22.2M4" ), [ 1, 1, 1, 1, 1, 1, 1, -1, -1, -1,
-1, -1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1 ] ),
Character( CharacterTable( "M22.2M4" ), [ 1, 1, 1, 1, 1, 1, 1, -1, -1, -1,
-1, -1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1 ] ) ]
gap> perms:= List( lin{ [ 2 .. 4 ] }, chi -> chi + lin[1] );;
gap> sfustbl2:= PossibleClassFusions( s, tbl2 );;
gap> Length( sfustbl2 );
2
gap> ind1:= Induced( s, tbl2, perms, sfustbl2[1] );;
gap> ind2:= Induced( s, tbl2, perms, sfustbl2[2] );;
gap> PermCharInfo( tbl2, ind1 ).ATLAS;
[ "1ab+21ab+42aa+55ab+154ab+210ccdd", "1a+21ab+42a+55a+154a+210bcd+462a",
"1a+21aa+42a+55a+154a+210acd+462a" ]
gap> PermCharInfo( tbl2, ind2 ).ATLAS;
[ "1a+21aa+42a+55a+154a+210acd+462a", "1a+21ab+42a+55a+154a+210bcd+462a",
"1ab+21ab+42aa+55ab+154ab+210ccdd" ]
gap> ind1[2] = ind2[2];
true
gap> [ ind1[2] ] = faith[10];
true
gap> CompareWithDatabase( "3.M22.2", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.7">
3.7</a> G = 4.M<sub>22</sub> and G = 12.M<sub>22</sub></h3>
<div class="p"><!----></div>
The group 4.M<sub>22</sub> and hence also the group 12.M<sub>22</sub> has no
faithful multiplicity-free permutation action.
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "4.M22" );;
gap> faith:= FaithfulCandidates( tbl, "2.M22" );;
gap> CompareWithDatabase( "4.M22", faith );
gap> CompareWithCandidatesByMaxes( "4.M22", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.8">
3.8</a> G = 4.M<sub>22</sub>.2 and G = 12.M<sub>22</sub>.2</h3>
<div class="p"><!----></div>
The two isoclinism types of groups of the type 4.M<sub>22</sub>.2 and hence also all
groups of the type 12.M<sub>22</sub>.2 have no faithful multiplicity-free
permutation actions.
<div class="p"><!----></div>
<pre>
gap> tbl2:= CharacterTable( "4.M22.2" );;
gap> faith:= FaithfulCandidates( tbl2, "M22.2" );;
gap> CompareWithDatabase( "4.M22.2", faith );
gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );;
gap> faith:= FaithfulCandidates( tbl2, "M22.2" );;
gap> CompareWithDatabase( "Isoclinic(4.M22.2)", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.9">
3.9</a> G = 6.M<sub>22</sub></h3>
<div class="p"><!----></div>
The group 6.M<sub>22</sub> has two faithful multiplicity-free permutation actions,
with point stabilizers of the types 2<sup>4</sup>:A<sub>5</sub> and 2<sup>3</sup>:L<sub>3</sub>(2).
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "6.M22" );;
gap> faith:= FaithfulCandidates( tbl, "3.M22" );;
1: subgroup $2^4:A_5 \rightarrow (M_{22},3)$, degree 2772 (1 cand.)
3: subgroup $2^3:L_3(2) \rightarrow (M_{22},7)$, degree 1980 (1 cand.)
</pre>
<div class="p"><!----></div>
The existence of one class of each of these subgroups follows from the
treatment of 2.M<sub>22</sub> and 3.M<sub>22</sub>.
<div class="p"><!----></div>
<pre>
gap> VerifyCandidates( CharacterTable( "P1/G1/L1/V1/ext2" ), tbl, 0, faith[1], "all" );
G = 6.M22: point stabilizer P1/G1/L1/V1/ext2, ranks [ 22 ]
[ "1a+21abc+55a+105abcd+126abcdef+154a+210bef+231abc" ]
gap> VerifyCandidates( CharacterTable( "M22M6" ), tbl, 0, faith[3], "all" );
G = 6.M22: point stabilizer 2^3:sl(3,2), ranks [ 17 ]
[ "1a+21abc+55a+99abc+105abcd+120a+154a+210b+330de" ]
gap> CompareWithDatabase( "6.M22", faith );
gap> CompareWithCandidatesByMaxes( "6.M22", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.10">
3.10</a> G = 6.M<sub>22</sub>.2</h3>
<div class="p"><!----></div>
The group 6.M<sub>22</sub>.2 that is printed in the A<font size="-2">TLAS</font> has six faithful
multiplicity-free permutation actions,
with point stabilizers of the types 2<sup>4</sup>:S<sub>5</sub> (twice),
2<sup>3</sup>:L<sub>3</sub>(2) ×2 (twice), and L<sub>2</sub>(11).2 (twice).
<div class="p"><!----></div>
<pre>
gap> tbl2:= CharacterTable( "6.M22.2" );;
gap> faith:= FaithfulCandidates( tbl2, "M22.2" );;
6: subgroup $2^4:S_5 \leq 2^4:S_6$, degree 2772 (2 cand.)
11: subgroup $2^3:L_3(2) \times 2$, degree 1980 (2 cand.)
16: subgroup $L_2(11).2$, degree 4032 (2 cand.)
</pre>
<div class="p"><!----></div>
We know that 2.M<sub>22</sub>.2 contains two classes of subgroups isomorphic with
each of the required point stabilizers, so we apply Lemma <a href="#situationII">2.2</a>.
<div class="p"><!----></div>
<pre>
gap> s:= CharacterTable( "w(d5)" );;
gap> VerifyCandidates( s, tbl, tbl2, faith[6], "all" );
G = 6.M22.2: point stabilizer w(d5), ranks [ 14, 14 ]
[ "1a^++21a^+bc+55a^++105adbc+126abcfde+154a^++210b^-ef+231a^-bc",
"1a^++21a^+bc+55a^++105adbc+126abcfde+154a^++210b^+ef+231a^-bc" ]
</pre>
<div class="p"><!----></div>
(Since 6.M<sub>22</sub> contains subgroups of the type 2<sup>3</sup>:L<sub>3</sub>(2) ×2
in which we are not interested,
we must use <tt>"extending"</tt> as the last argument of <tt>VerifyCandidates</tt>
for this case.)
<div class="p"><!----></div>
<pre>
gap> s:= CharacterTable( "2x2^3:L3(2)" );;
gap> VerifyCandidates( s, tbl, tbl2, faith[11], "extending" );
G = 6.M22.2: point stabilizer 2x2^3:L3(2), ranks [ 12, 12 ]
[ "1a^++21a^+bc+55a^++99a^+bc+105adbc+120a^-+154a^++210b^-+330de",
"1a^++21a^+bc+55a^++99a^+bc+105adbc+120a^++154a^++210b^++330de" ]
gap> VerifyCandidates( CharacterTable( "L2(11).2" ), tbl, tbl2, faith[16], "all" );
G = 6.M22.2: point stabilizer L2(11).2, ranks [ 20, 20 ]
[ "1a^++21a^-bc+55a^++56a^{\\pm}+66abcd+105adbc+120a^-bc+154a^++210a^-cdgjhi+2\
31a^-bc+440a^+",
"1a^++21a^-bc+55a^++56a^{\\pm}+66abcd+105adbc+120a^+bc+154a^++210a^-cdgjhi+2\
31a^-bc+440a^-" ]
gap> CompareWithDatabase( "6.M22.2", faith );
</pre>
<div class="p"><!----></div>
The group (6.M<sub>22</sub>.2)<sup>∗</sup> of the isoclinism type that is not printed
in the A<font size="-2">TLAS</font> has no faithful multiplicity-free permutation action.
<div class="p"><!----></div>
<pre>
gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );;
gap> faith:= FaithfulCandidates( tbl2, "M22.2" );;
gap> CompareWithDatabase( "Isoclinic(6.M22.2)", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.11">
3.11</a> G = 2.J<sub>2</sub></h3>
<div class="p"><!----></div>
The group 2.J<sub>2</sub> has one faithful multiplicity-free permutation action,
with point stabilizer of the type U<sub>3</sub>(3), by Lemma <a href="#situationI">2.1</a>.
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "2.J2" );;
gap> faith:= FaithfulCandidates( tbl, "J2" );;
1: subgroup $U_3(3)$, degree 200 (1 cand.)
gap> VerifyCandidates( CharacterTable( "U3(3)" ), tbl, 0, faith[1], "all" );
G = 2.J2: point stabilizer U3(3), ranks [ 5 ]
[ "1a+36a+50ab+63a" ]
gap> CompareWithDatabase( "2.J2", faith );
gap> CompareWithCandidatesByMaxes( "2.J2", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.12">
3.12</a> G = 2.J<sub>2</sub>.2</h3>
<div class="p"><!----></div>
The group 2.J<sub>2</sub>.2 that is printed in the A<font size="-2">TLAS</font> has no faithful
multiplicity-free permutation action.
<div class="p"><!----></div>
<pre>
gap> tbl2:= CharacterTable( "2.J2.2" );;
gap> faith:= FaithfulCandidates( tbl2, "J2.2" );;
gap> CompareWithDatabase( "2.J2.2", faith );
</pre>
<div class="p"><!----></div>
The group (2.J<sub>2</sub>.2)<sup>∗</sup> of the isoclinism type that is not printed
in the A<font size="-2">TLAS</font> has four faithful multiplicity-free permutation actions,
with point stabilizers of the types U<sub>3</sub>(3).2 (twice) and
3.A<sub>6</sub>.2<sub>3</sub> (twice).
<div class="p"><!----></div>
<pre>
gap> facttbl:= CharacterTable( "J2.2" );;
gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );;
gap> faith:= FaithfulCandidates( tbl2, "J2.2" );;
1: subgroup $U_3(3).2$, degree 200 (1 cand.)
5: subgroup $3.A_6.2_3 \leq 3.A_6.2^2$, degree 1120 (1 cand.)
</pre>
<div class="p"><!----></div>
The existence of two classes of each of these subgroups follows from
Lemma <a href="#situationIII">2.3</a>.
(Note that the Schur multiplier of U<sub>3</sub>(3) is trivial and 6.A<sub>6</sub>
does not admit an automorphic extension that has a factor group A<sub>6</sub>.2<sub>3</sub>.)
<div class="p"><!----></div>
<pre>
gap> s0:= CharacterTable( "U3(3)" );;
gap> s:= CharacterTable( "U3(3).2" );;
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "all" );
Isoclinic(2.J2.2): U3(3).2 lifts to a direct product,
proved by squares in [ 1, 3, 8, 16 ].
gap> VerifyCandidates( s, tbl, tbl2, faith[1], "all" );
G = Isoclinic(2.J2.2): point stabilizer U3(3).2, ranks [ 4 ]
[ "1a^++36a^++50ab+63a^+" ]
gap> s0:= CharacterTable( "3.A6" );;
gap> s:= CharacterTable( "3.A6.2_3" );;
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "all" );
Isoclinic(2.J2.2): 3.A6.2_3 lifts to a direct product,
proved by squares in [ 3, 10, 16, 25 ].
gap> VerifyCandidates( s, tbl, tbl2, faith[5], "all" );
G = Isoclinic(2.J2.2): point stabilizer 3.A6.2_3, ranks [ 12 ]
[ "1a^++14c^{\\pm}+21ab+50ab+63a^{\\pm}+90a^++126a^++175a^-+216a^{\\pm}" ]
gap> faith[1]:= faith[1]{ [ 1, 1 ] };;
gap> faith[5]:= faith[5]{ [ 1, 1 ] };;
gap> CompareWithDatabase( "Isoclinic(2.J2.2)", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.13">
3.13</a> G = 2.HS</h3>
<div class="p"><!----></div>
The group 2.HS has five faithful multiplicity-free permutation actions,
with point stabilizers of the types U<sub>3</sub>(5) (twice), A<sub>8</sub>,
and M<sub>11</sub> (twice).
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "2.HS" );;
gap> faith:= FaithfulCandidates( tbl, "HS" );;
3: subgroup $U_3(5) \leq U_3(5).2$, degree 704 (1 cand.)
5: subgroup $U_3(5) \leq U_3(5).2$, degree 704 (1 cand.)
8: subgroup $A_8 \leq A_8.2$, degree 4400 (1 cand.)
10: subgroup $M_{11}$, degree 11200 (1 cand.)
11: subgroup $M_{11}$, degree 11200 (1 cand.)
</pre>
<div class="p"><!----></div>
Lemma <a href="#situationI">2.1</a> applies in all cases; note that 2.HS does not admit
an embedding of 2.A<sub>8</sub>.
<div class="p"><!----></div>
<pre>
gap> VerifyCandidates( CharacterTable( "U3(5)" ), tbl, 0,
> Concatenation( faith[3], faith[5] ), "all" );
G = 2.HS: point stabilizer U3(5), ranks [ 6, 6 ]
[ "1a+22a+154c+175a+176ab", "1a+22a+154b+175a+176ab" ]
gap> PossibleClassFusions( CharacterTable( "2.A8" ), tbl );
[ ]
gap> VerifyCandidates( CharacterTable( "A8" ), tbl, 0, faith[8], "all" );
G = 2.HS: point stabilizer A8, ranks [ 13 ]
[ "1a+22a+77a+154abc+175a+176ab+693a+770a+924ab" ]
gap> VerifyCandidates( CharacterTable( "M11" ), tbl, 0,
> Concatenation( faith[10], faith[11] ), "all" );
G = 2.HS: point stabilizer M11, ranks [ 16, 16 ]
[ "1a+22a+56a+77a+154c+175a+176ab+616ab+770a+825a+1056a+1980ab+2520a",
"1a+22a+56a+77a+154b+175a+176ab+616ab+770a+825a+1056a+1980ab+2520a" ]
gap> CompareWithDatabase( "2.HS", faith );
gap> CompareWithCandidatesByMaxes( "2.HS", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.14">
3.14</a> G = 2.HS.2</h3>
<div class="p"><!----></div>
The group 2.HS.2 that is printed in the A<font size="-2">TLAS</font> has four faithful
multiplicity-free permutation actions,
with point stabilizers of the types A<sub>8</sub> ×2 (twice)
and A<sub>8</sub>.2 (twice).
<div class="p"><!----></div>
<pre>
gap> tbl2:= CharacterTable("2.HS.2");;
gap> faith:= FaithfulCandidates( tbl2, "HS.2" );;
10: subgroup $A_8 \times 2 \leq A_8.2 \times 2$, degree 4400 (1 cand.)
11: subgroup $A_8.2 \leq A_8.2 \times 2$, degree 4400 (1 cand.)
</pre>
<div class="p"><!----></div>
The existence of two classes of subgroups for each candidate follows from
Lemma <a href="#situationIII">2.3</a>.
(Since there are A<sub>8</sub> ×2 type subgroups inside 2.HS in which we are
not interested,
we must use <tt>"extending"</tt> as the last argument of <tt>VerifyCandidates</tt>.)
<div class="p"><!----></div>
<pre>
gap> facttbl:= CharacterTable( "HS.2" );;
gap> factfus:= GetFusionMap( tbl2, facttbl );;
gap> s0:= CharacterTable( "A8");;
gap> s:= s0 * CharacterTable( "Cyclic", 2 );
CharacterTable( "A8xC2" )
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "all" );
2.HS.2: A8xC2 lifts to a direct product,
proved by squares in [ 1, 6, 13, 20, 30 ].
gap> VerifyCandidates( s, tbl, tbl2, faith[10], "extending" );
G = 2.HS.2: point stabilizer A8xC2, ranks [ 10 ]
[ "1a^++22a^++77a^++154a^+bc+175a^++176ab+693a^++770a^++924ab" ]
gap> s:= CharacterTable( "A8.2" );;
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "extending" );
2.HS.2: A8.2 lifts to a direct product,
proved by squares in [ 1, 6, 13 ].
gap> VerifyCandidates( s, tbl, tbl2, faith[11], "all" );
G = 2.HS.2: point stabilizer A8.2, ranks [ 10 ]
[ "1a^++22a^-+77a^++154a^+bc+175a^++176ab+693a^++770a^-+924ab" ]
gap> faith[10]:= faith[10]{ [ 1, 1 ] };;
gap> faith[11]:= faith[11]{ [ 1, 1 ] };;
gap> CompareWithDatabase( "2.HS.2", faith );
</pre>
<div class="p"><!----></div>
The group (2.HS.2)<sup>∗</sup> of the isoclinism type that is not printed
in the A<font size="-2">TLAS</font> has no faithful multiplicity-free permutation action.
<div class="p"><!----></div>
<pre>
gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );;
gap> faith:= FaithfulCandidates( tbl2, "HS.2" );;
gap> CompareWithDatabase( "Isoclinic(2.HS.2)", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.15">
3.15</a> G = 3.J<sub>3</sub></h3>
<div class="p"><!----></div>
The group 3.J<sub>3</sub> has no faithful multiplicity-free permutation action.
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "3.J3" );;
gap> faith:= FaithfulCandidates( tbl, "J3" );;
gap> CompareWithDatabase( "3.J3", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.16">
3.16</a> G = 3.J<sub>3</sub>.2</h3>
<div class="p"><!----></div>
The group 3.J<sub>3</sub>.2 has no faithful multiplicity-free permutation action.
<div class="p"><!----></div>
<pre>
gap> tbl2:= CharacterTable( "3.J3.2" );;
gap> faith:= FaithfulCandidates( tbl2, "J3.2" );;
gap> CompareWithDatabase( "3.J3.2", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.17">
3.17</a> G = 3.McL</h3>
<div class="p"><!----></div>
The group 3.McL has one faithful multiplicity-free permutation action,
with point stabilizer of the type 2.A<sub>8</sub>, by Lemma <a href="#situationI">2.1</a>.
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "3.McL" );;
gap> faith:= FaithfulCandidates( tbl, "McL" );;
6: subgroup $2.A_8$, degree 66825 (1 cand.)
gap> VerifyCandidates( CharacterTable( "2.A8" ), tbl, 0, faith[6], "all" );
G = 3.McL: point stabilizer 2.A8, ranks [ 14 ]
[ "1a+252a+1750a+2772ab+5103abc+5544a+6336ab+8064ab+9625a" ]
gap> CompareWithDatabase( "3.McL", faith );
gap> CompareWithCandidatesByMaxes( "3.McL", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.18">
3.18</a> G = 3.McL.2</h3>
<div class="p"><!----></div>
The group 3.McL.2 has one faithful multiplicity-free permutation action,
with point stabilizer of the type (2.A<sub>8</sub>.2)<sup>∗</sup>,
by Lemma <a href="#situationII">2.2</a>.
<div class="p"><!----></div>
<pre>
gap> tbl2:= CharacterTable( "3.McL.2" );;
gap> faith:= FaithfulCandidates( tbl2, "McL.2" );;
9: subgroup $2.S_8$, degree 66825 (1 cand.)
gap> s:= CharacterTable( "Isoclinic(2.A8.2)" );;
gap> VerifyCandidates( s, tbl, tbl2, faith[9], "all" );
G = 3.McL.2: point stabilizer Isoclinic(2.A8.2), ranks [ 10 ]
[ "1a^++252a^++1750a^++2772ab+5103a^+bc+5544a^++6336ab+8064ab+9625a^+" ]
gap> CompareWithDatabase( "3.McL.2", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.19">
3.19</a> G = 2.Ru</h3>
<div class="p"><!----></div>
The group 2.Ru has one faithful multiplicity-free permutation action,
with point stabilizer of the type <sup>2</sup>F<sub>4</sub>(2)<sup>′</sup>,
by Lemma <a href="#situationI">2.1</a>.
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "2.Ru" );;
gap> faith:= FaithfulCandidates( tbl, "Ru" );;
2: subgroup ${^2F_4(2)^{\prime}} \leq {^2F_4(2)^{\prime}}.2$, degree 16240 (
1 cand.)
gap> VerifyCandidates( CharacterTable( "2F4(2)'" ), tbl, 0, faith[2], "all" );
G = 2.Ru: point stabilizer 2F4(2)', ranks [ 9 ]
[ "1a+28ab+406a+783a+3276a+3654a+4032ab" ]
gap> CompareWithDatabase( "2.Ru", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.20">
3.20</a> G = 2.Suz</h3>
<div class="p"><!----></div>
The group 2.Suz has one faithful multiplicity-free permutation action,
with point stabilizer of the type U<sub>5</sub>(2), by Lemma <a href="#situationI">2.1</a>.
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "2.Suz" );;
gap> faith:= FaithfulCandidates( tbl, "Suz" );;
4: subgroup $U_5(2)$, degree 65520 (1 cand.)
gap> VerifyCandidates( CharacterTable( "U5(2)" ), tbl, 0, faith[4], "all" );
G = 2.Suz: point stabilizer U5(2), ranks [ 10 ]
[ "1a+143a+364abc+5940a+12012a+14300a+16016ab" ]
gap> CompareWithDatabase( "2.Suz", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.21">
3.21</a> G = 2.Suz.2</h3>
<div class="p"><!----></div>
The group 2.Suz.2 that is printed in the A<font size="-2">TLAS</font> has four faithful
multiplicity-free permutation actions,
with point stabilizers of the types U<sub>5</sub>(2).2 (twice)
and 3<sup>5</sup>:(M<sub>11</sub> ×2) (twice), respectively.
<div class="p"><!----></div>
<pre>
gap> tbl2:= CharacterTable( "2.Suz.2" );;
gap> faith:= FaithfulCandidates( tbl2, "Suz.2" );;
8: subgroup $U_5(2).2$, degree 65520 (1 cand.)
12: subgroup $3^5:(M_{11} \times 2)$, degree 465920 (1 cand.)
</pre>
<div class="p"><!----></div>
We verify the conditions of Lemma <a href="#situationIII">2.3</a>.
<div class="p"><!----></div>
<pre>
gap> s0:= CharacterTable( "U5(2)" );;
gap> s:= CharacterTable( "U5(2).2" );;
gap> facttbl:= CharacterTable( "Suz.2" );;
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "all" );
2.Suz.2: U5(2).2 lifts to a direct product,
proved by squares in [ 1, 8, 13, 19, 31, 44 ].
gap> VerifyCandidates( s, tbl, tbl2, faith[8], "all" );
G = 2.Suz.2: point stabilizer U5(2).2, ranks [ 8 ]
[ "1a^++143a^-+364a^+bc+5940a^++12012a^-+14300a^-+16016ab" ]
gap> s0:= CharacterTable( "SuzM5" );
CharacterTable( "3^5:M11" )
gap> s:= CharacterTable( "Suz.2M6" );
CharacterTable( "3^5:(M11x2)" )
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "all" );
2.Suz.2: 3^5:(M11x2) lifts to a direct product,
proved by squares in [ 1, 4, 8, 10, 19, 22, 26, 39 ].
gap> VerifyCandidates( s, tbl, tbl2, faith[12], "all" );
G = 2.Suz.2: point stabilizer 3^5:(M11x2), ranks [ 14 ]
[ "1a^++364a^{\\pm}bc+5940a^++12012a^-+14300a^-+15015ab+15795a^++16016ab+54054\
a^++100100a^-b^{\\pm}" ]
gap> faith[8]:= faith[8]{ [ 1, 1 ] };;
gap> faith[12]:= faith[12]{ [ 1, 1 ] };;
gap> CompareWithDatabase( "2.Suz.2", faith );
</pre>
<div class="p"><!----></div>
The group (2.Suz.2)<sup>∗</sup> of the isoclinism type that is not printed
in the A<font size="-2">TLAS</font> has no faithful multiplicity-free permutation action.
<div class="p"><!----></div>
<pre>
gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );;
gap> faith:= FaithfulCandidates( tbl2, "Suz.2" );;
gap> CompareWithDatabase( "Isoclinic(2.Suz.2)", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.22">
3.22</a> G = 3.Suz</h3>
<div class="p"><!----></div>
The group 3.Suz has four faithful multiplicity-free permutation actions,
with point stabilizers of the types G<sub>2</sub>(4), U<sub>5</sub>(2),
2<sup>1+6</sup><sub>−</sub>.U<sub>4</sub>(2), and 2<sup>4+6</sup>:3A<sub>6</sub>, respectively,
by Lemma <a href="#situationI">2.1</a>.
<div class="p"><!----></div>
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "3.Suz" );;
gap> faith:= FaithfulCandidates( tbl, "Suz" );;
1: subgroup $G_2(4)$, degree 5346 (1 cand.)
4: subgroup $U_5(2)$, degree 98280 (1 cand.)
5: subgroup $2^{1+6}_-.U_4(2)$, degree 405405 (1 cand.)
6: subgroup $2^{4+6}:3A_6$, degree 1216215 (1 cand.)
gap> Maxes( tbl );
[ "3xG2(4)", "3^2.U4(3).2_3'", "3xU5(2)", "3x2^(1+6)_-.U4(2)", "3^6.M11",
"3xJ2.2", "3x2^(4+6).3A6", "(A4x3.L3(4)).2", "3x2^(2+8):(A5xS3)",
"3xM12.2", "3.3^(2+4):2(A4x2^2).2", "(3.A6xA5):2", "(3^(1+2):4xA6).2",
"3xL3(3).2", "3xL3(3).2", "3xL2(25)", "3.A7" ]
gap> VerifyCandidates( CharacterTable( "G2(4)" ), tbl, 0, faith[1], "all" );
G = 3.Suz: point stabilizer G2(4), ranks [ 7 ]
[ "1a+66ab+780a+1001a+1716ab" ]
gap> VerifyCandidates( CharacterTable( "U5(2)" ), tbl, 0, faith[4], "all" );
G = 3.Suz: point stabilizer U5(2), ranks [ 14 ]
[ "1a+78ab+143a+364a+1365ab+4290ab+5940a+12012a+14300a+27027ab" ]
gap> VerifyCandidates( CharacterTable( "SuzM4" ), tbl, 0, faith[5], "all" );
G = 3.Suz: point stabilizer 2^1+6.u4q2, ranks [ 23 ]
[ "1a+66ab+143a+429ab+780a+1716ab+3432a+5940a+6720ab+14300a+18954abc+25025a+42\
900ab+64350cd+66560a" ]
gap> VerifyCandidates( CharacterTable( "SuzM7" ), tbl, 0, faith[6], "all" );
G = 3.Suz: point stabilizer 2^4+6:3a6, ranks [ 27 ]
[ "1a+364a+780a+1001a+1365ab+4290ab+5940a+12012a+14300a+15795a+25025a+27027ab+\
42900ab+66560a+75075a+85800ab+88452a+100100a+104247ab+139776ab" ]
gap> CompareWithDatabase( "3.Suz", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.23">
3.23</a> G = 3.Suz.2</h3>
<div class="p"><!----></div>
The group 3.Suz.2 has four faithful multiplicity-free permutation actions,
with point stabilizers of the types G<sub>2</sub>(4).2, U<sub>5</sub>(2).2,
2<sup>1+6</sup><sub>−</sub>.U<sub>4</sub>(2).2, and 2<sup>4+6</sup>:3S<sub>6</sub>, respectively.
We know from the treatment of 3.Suz that we can apply
Lemma <a href="#situationII">2.2</a>.
<div class="p"><!----></div>
<pre>
gap> tbl2:= CharacterTable( "3.Suz.2" );;
gap> faith:= FaithfulCandidates( tbl2, "Suz.2" );;
1: subgroup $G_2(4).2$, degree 5346 (1 cand.)
8: subgroup $U_5(2).2$, degree 98280 (1 cand.)
10: subgroup $2^{1+6}_-.U_4(2).2$, degree 405405 (1 cand.)
13: subgroup $2^{4+6}:3S_6$, degree 1216215 (1 cand.)
gap> Maxes( CharacterTable( "Suz.2" ) );
[ "Suz", "G2(4).2", "3_2.U4(3).(2^2)_{133}", "U5(2).2", "2^(1+6)_-.U4(2).2",
"3^5:(M11x2)", "J2.2x2", "2^(4+6):3S6", "(A4xL3(4):2_3):2",
"2^(2+8):(S5xS3)", "M12.2x2", "3^(2+4):2(S4xD8)", "(A6:2_2xA5).2",
"(3^2:8xA6).2", "L2(25).2_2", "A7.2" ]
gap> VerifyCandidates( CharacterTable( "G2(4).2" ), tbl, tbl2, faith[1], "all" );
G = 3.Suz.2: point stabilizer G2(4).2, ranks [ 5 ]
[ "1a^++66ab+780a^++1001a^++1716ab" ]
gap> VerifyCandidates( CharacterTable( "U5(2).2" ), tbl, tbl2, faith[8], "all" );
G = 3.Suz.2: point stabilizer U5(2).2, ranks [ 10 ]
[ "1a^++78ab+143a^-+364a^++1365ab+4290ab+5940a^++12012a^-+14300a^-+27027ab" ]
gap> VerifyCandidates( CharacterTable( "Suz.2M5" ), tbl, tbl2, faith[10], "all" );
G = 3.Suz.2: point stabilizer 2^(1+6)_-.U4(2).2, ranks [ 16 ]
[ "1a^++66ab+143a^-+429ab+780a^++1716ab+3432a^++5940a^++6720ab+14300a^-+18954a\
^-bc+25025a^++42900ab+64350cd+66560a^+" ]
gap> VerifyCandidates( CharacterTable( "Suz.2M8" ), tbl, tbl2, faith[13], "all" );
G = 3.Suz.2: point stabilizer 2^(4+6):3S6, ranks [ 20 ]
[ "1a^++364a^++780a^++1001a^++1365ab+4290ab+5940a^++12012a^-+14300a^-+15795a^+\
+25025a^++27027ab+42900ab+66560a^++75075a^++85800ab+88452a^++100100a^++104247a\
b+139776ab" ]
gap> CompareWithDatabase( "3.Suz.2", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.24">
3.24</a> G = 6.Suz</h3>
<div class="p"><!----></div>
The group 6.Suz has one faithful multiplicity-free permutation action,
with point stabilizer of the type U<sub>5</sub>(2), by Lemma <a href="#situationI">2.1</a>.
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "6.Suz" );;
gap> faith:= FaithfulCandidates( tbl, "2.Suz" );;
1: subgroup $U_5(2) \rightarrow (Suz,4)$, degree 196560 (1 cand.)
gap> VerifyCandidates( CharacterTable( "U5(2)" ), tbl, 0, faith[1], "all" );
G = 6.Suz: point stabilizer U5(2), ranks [ 26 ]
[ "1a+12ab+78ab+143a+364abc+924ab+1365ab+4290ab+4368ab+5940a+12012a+14300a+160\
16ab+27027ab+27456ab" ]
gap> CompareWithDatabase( "6.Suz", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.25">
3.25</a> G = 6.Suz.2</h3>
<div class="p"><!----></div>
The group 6.Suz.2 that is printed in the A<font size="-2">TLAS</font> has two faithful
multiplicity-free permutation actions,
with point stabilizers of the type U<sub>5</sub>(2).2 (twice).
We know from the treatment of 6.Suz that we can apply
Lemma <a href="#situationII">2.2</a>.
<div class="p"><!----></div>
<pre>
gap> tbl2:= CharacterTable( "6.Suz.2" );;
gap> faith:= FaithfulCandidates( tbl2, "Suz.2" );;
8: subgroup $U_5(2).2$, degree 196560 (1 cand.)
gap> VerifyCandidates( CharacterTable( "U5(2).2" ), tbl, tbl2, faith[8], "all" );
G = 6.Suz.2: point stabilizer U5(2).2, ranks [ 16 ]
[ "1a^++12ab+78ab+143a^-+364a^+bc+924ab+1365ab+4290ab+4368ab+5940a^++12012a^-+\
14300a^-+16016ab+27027ab+27456ab" ]
gap> faith[8]:= faith[8]{ [ 1, 1 ] };;
gap> CompareWithDatabase( "6.Suz.2", faith );
</pre>
<div class="p"><!----></div>
It follows from the treatment of 2.Suz.2 that the group (6.Suz.2)<sup>∗</sup>
of the isoclinism type that is not printed in the A<font size="-2">TLAS</font> does not have a
faithful multiplicity-free permutation action.
<div class="p"><!----></div>
<h3><a name="tth_sEc3.26">
3.26</a> G = 3.ON</h3>
<div class="p"><!----></div>
The group 3.ON has four faithful multiplicity-free permutation actions,
with point stabilizers of the types L<sub>3</sub>(7).2 (twice) and L<sub>3</sub>(7) (twice).
(The Schur multiplier of L<sub>3</sub>(7).2 is trivial, so the L<sub>3</sub>(7) type
subgroups lift to direct products with the centre of 3.ON, that is,
we can apply Lemma <a href="#situationI">2.1</a>.)
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "3.ON" );;
gap> faith:= FaithfulCandidates( tbl, "ON" );;
1: subgroup $L_3(7).2$, degree 368280 (1 cand.)
2: subgroup $L_3(7) \leq L_3(7).2$, degree 736560 (1 cand.)
3: subgroup $L_3(7).2$, degree 368280 (1 cand.)
4: subgroup $L_3(7) \leq L_3(7).2$, degree 736560 (1 cand.)
gap> VerifyCandidates( CharacterTable( "L3(7).2" ), tbl, 0,
> Concatenation( faith[1], faith[3] ), "all" );
G = 3.ON: point stabilizer L3(7).2, ranks [ 11, 11 ]
[ "1a+495ab+10944a+26752a+32395b+52668a+58653bc+63612ab",
"1a+495cd+10944a+26752a+32395a+52668a+58653bc+63612ab" ]
gap> VerifyCandidates( CharacterTable( "L3(7)" ), tbl, 0,
> Concatenation( faith[2], faith[4] ), "all" );
G = 3.ON: point stabilizer L3(7), ranks [ 15, 15 ]
[ "1a+495ab+10944a+26752a+32395b+37696a+52668a+58653bc+63612ab+85064a+122760ab\
",
"1a+495cd+10944a+26752a+32395a+37696a+52668a+58653bc+63612ab+85064a+122760ab\
" ]
gap> CompareWithDatabase( "3.ON", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.27">
3.27</a> G = 3.ON.2</h3>
<div class="p"><!----></div>
The group 3.ON.2 has no faithful multiplicity-free permutation action.
<div class="p"><!----></div>
<pre>
gap> tbl2:= CharacterTable( "3.ON.2" );;
gap> faith:= FaithfulCandidates( tbl2, "ON.2" );;
gap> CompareWithDatabase( "3.ON.2", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.28">
3.28</a> G = 2.Fi<sub>22</sub></h3>
<div class="p"><!----></div>
The group 2.Fi<sub>22</sub> has seven faithful multiplicity-free permutation
actions, with point stabilizers of the types O<sub>7</sub>(3) (twice), O<sub>8</sub><sup>+</sup>(2):S<sub>3</sub>
(twice), O<sub>8</sub><sup>+</sup>(2):3, and O<sub>8</sub><sup>+</sup>(2):2 (twice).
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "2.Fi22" );;
gap> faith:= FaithfulCandidates( tbl, "Fi22" );;
2: subgroup $O_7(3)$, degree 28160 (2 cand.)
3: subgroup $O_7(3)$, degree 28160 (2 cand.)
4: subgroup $O_8^+(2).3.2$, degree 123552 (2 cand.)
5: subgroup $O_8^+(2).3 \leq O_8^+(2).3.2$, degree 247104 (1 cand.)
6: subgroup $O_8^+(2).2 \leq O_8^+(2).3.2$, degree 370656 (2 cand.)
</pre>
<div class="p"><!----></div>
The two classes of maximal subgroups of the type O<sub>7</sub>(3) in Fi<sub>22</sub> induce
the same permutation character and lift to two classes of the type
2 ×O<sub>7</sub>(3) in 2.Fi<sub>22</sub>.
We get the same two candidates for these two classes.
One of them belongs to the first class of O<sub>7</sub>(3) subgroups in 2.Fi<sub>22</sub>,
the other candidate belongs to the second class;
this can be seen from the fact that the outer automorphism of Fi<sub>22</sub>
swaps the two classes of O<sub>7</sub>(3) subgroups, and the lift of this automorphism
to 2.Fi<sub>22</sub> interchanges the candidates
-this action can be read off from the embedding of 2.Fi<sub>22</sub> into any group
of the type 2.Fi<sub>22</sub>.2.
<div class="p"><!----></div>
<pre>
gap> faith[2] = faith[3];
true
gap> tbl2:= CharacterTable("2.Fi22.2");;
gap> embed:= GetFusionMap( tbl, tbl2 );;
gap> swapped:= Filtered( InverseMap( embed ), IsList );
[ [ 3, 4 ], [ 17, 18 ], [ 25, 26 ], [ 27, 28 ], [ 33, 34 ], [ 36, 37 ],
[ 42, 43 ], [ 51, 52 ], [ 59, 60 ], [ 63, 65 ], [ 64, 66 ], [ 71, 72 ],
[ 73, 75 ], [ 74, 76 ], [ 81, 82 ], [ 85, 87 ], [ 86, 88 ], [ 89, 90 ],
[ 93, 94 ], [ 95, 98 ], [ 96, 97 ], [ 99, 100 ], [ 103, 104 ],
[ 107, 110 ], [ 108, 109 ], [ 113, 114 ] ]
gap> perm:= Product( List( swapped, pair -> ( pair[1], pair[2] ) ) );;
gap> Permuted( faith[2][1], perm ) = faith[2][2];
true
gap> VerifyCandidates( CharacterTable( "O7(3)" ), tbl, 0, faith[2], "all" );
G = 2.Fi22: point stabilizer O7(3), ranks [ 5, 5 ]
[ "1a+352a+429a+13650a+13728b", "1a+352a+429a+13650a+13728a" ]
gap> faith[2]:= [ faith[2][1] ];;
gap> faith[3]:= [ faith[3][2] ];;
</pre>
<div class="p"><!----></div>
All involutions in Fi<sub>22</sub> lift to involutions in 2.Fi<sub>22</sub>,
so the preimages of the maximal subgroups of the type O<sub>8</sub><sup>+</sup>(2).S<sub>3</sub>
in Fi<sub>22</sub> have the type 2 ×O<sub>8</sub><sup>+</sup>(2).S<sub>3</sub>.
We apply Lemma <a href="#situationIII">2.3</a>, using that the two subgroups of the type
O<sub>8</sub><sup>+</sup>(2).S<sub>3</sub> contain involutions outside O<sub>8</sub><sup>+</sup>(2) which lie in the two
nonconjugate preimages of the class <tt>2A</tt> of Fi<sub>22</sub>;
this proves the existence of the two candidates of degree 123 552.
<div class="p"><!----></div>
<pre>
gap> s:= CharacterTable( "O8+(2).S3" );;
gap> s0:= CharacterTable( "O8+(2).3" );;
gap> facttbl:= CharacterTable( "Fi22" );;
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl, "all" );
2.Fi22: O8+(2).3.2 lifts to a direct product,
proved by squares in [ 1, 8, 10, 12, 20, 23, 30, 46, 55, 61, 91 ].
gap> derpos:= ClassPositionsOfDerivedSubgroup( s );;
gap> factfus:= GetFusionMap( tbl, facttbl );;
gap> ForAll( PossibleClassFusions( s, tbl ),
> map -> NecessarilyDifferentPermChars( map, factfus, derpos ) );
true
gap> VerifyCandidates( CharacterTable( "O8+(2).S3" ), tbl, 0, faith[4], "all" );
G = 2.Fi22: point stabilizer O8+(2).3.2, ranks [ 6, 6 ]
[ "1a+3080a+13650a+13728b+45045a+48048c",
"1a+3080a+13650a+13728a+45045a+48048b" ]
</pre>
<div class="p"><!----></div>
The existence of one class of O<sub>8</sub><sup>+</sup>(2).3 subgroups follows from
Lemma <a href="#situationI">2.1</a>, and the proof for O<sub>8</sub><sup>+</sup>(2).S<sub>3</sub> also establishes
two classes of O<sub>8</sub><sup>+</sup>(2).2 subgroups, with different permutation characters,
<div class="p"><!----></div>
<pre>
gap> VerifyCandidates( CharacterTable( "O8+(2).3" ), tbl, 0, faith[5], "all" );
G = 2.Fi22: point stabilizer O8+(2).3, ranks [ 11 ]
[ "1a+1001a+3080a+10725a+13650a+13728ab+45045a+48048bc+50050a" ]
gap> VerifyCandidates( CharacterTable( "O8+(2).2" ), tbl, 0, faith[6], "all" );
G = 2.Fi22: point stabilizer O8+(2).2, ranks [ 11, 11 ]
[ "1a+352a+429a+3080a+13650a+13728b+45045a+48048ac+75075a+123200a",
"1a+352a+429a+3080a+13650a+13728a+45045a+48048ab+75075a+123200a" ]
gap> CompareWithDatabase( "2.Fi22", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.29">
3.29</a> G = 2.Fi<sub>22</sub>.2</h3>
<div class="p"><!----></div>
The group 2.Fi<sub>22</sub>.2 that is printed in the A<font size="-2">TLAS</font> has seven faithful
multiplicity-free permutation actions,
with point stabilizers of the types O<sub>7</sub>(3), O<sub>8</sub><sup>+</sup>(2):S<sub>3</sub>,
O<sub>8</sub><sup>+</sup>(2):3 ×2 (twice), O<sub>8</sub><sup>+</sup>(2):2, and <sup>2</sup>F<sub>4</sub>(2) (twice).
<div class="p"><!----></div>
<pre>
gap> tbl2:= CharacterTable( "2.Fi22.2" );;
gap> faith:= FaithfulCandidates( tbl2, "Fi22.2" );;
3: subgroup $O_7(3)$, degree 56320 (1 cand.)
5: subgroup $O_8^+(2).3.2 \leq O_8^+(2).3.2 \times 2$, degree 247104 (
1 cand.)
6: subgroup $O_8^+(2).3 \times 2 \leq O_8^+(2).3.2 \times 2$, degree 247104 (
1 cand.)
10: subgroup $O_8^+(2).2 \leq O_8^+(2).3.2 \times 2$, degree 741312 (1 cand.)
16: subgroup ${^2F_4(2)}$, degree 7185024 (1 cand.)
</pre>
<div class="p"><!----></div>
The third, fifth, and tenth multiplicity-free permutation character of
Fi<sub>22</sub>.2 are induced from subgroups of the types O<sub>7</sub>(3), O<sub>8</sub><sup>+</sup>(2).S<sub>3</sub>,
and O<sub>8</sub><sup>+</sup>(2).2 that lie inside Fi<sub>22</sub>, and we have discussed above that
these groups lift to direct products in 2.Fi<sub>22</sub>.
In fact all such subgroups of 2.Fi<sub>22</sub>.2 lie inside 2.Fi<sub>22</sub>,
and the two classes of such subgroups in 2.Fi<sub>22</sub> are fused in
2.Fi<sub>22</sub>.2, hence we get only one class of these subgroups.
<div class="p"><!----></div>
<div class="p"><!----></div>
<pre>
gap> VerifyCandidates( CharacterTable( "O7(3)" ), tbl, tbl2, faith[3], "all" );
G = 2.Fi22.2: point stabilizer O7(3), ranks [ 9 ]
[ "1a^{\\pm}+352a^{\\pm}+429a^{\\pm}+13650a^{\\pm}+13728ab" ]
gap> VerifyCandidates( CharacterTable( "O8+(2).S3" ), tbl, tbl2, faith[5], "all" );
G = 2.Fi22.2: point stabilizer O8+(2).3.2, ranks [ 10 ]
[ "1a^{\\pm}+3080a^{\\pm}+13650a^{\\pm}+13728ab+45045a^{\\pm}+48048bc" ]
gap> VerifyCandidates( CharacterTable( "O8+(2).2" ), tbl, tbl2, faith[10], "all" );
G = 2.Fi22.2: point stabilizer O8+(2).2, ranks [ 20 ]
[ "1a^{\\pm}+352a^{\\pm}+429a^{\\pm}+3080a^{\\pm}+13650a^{\\pm}+13728ab+45045a\
^{\\pm}+48048a^{\\pm}bc+75075a^{\\pm}+123200a^{\\pm}" ]
</pre>
<div class="p"><!----></div>
The sixth multiplicity-free permutation character of Fi<sub>22</sub>.2
is induced from a subgroup of the type O<sub>8</sub><sup>+</sup>(2).3 ×2 that does not lie
inside Fi<sub>22</sub>.
As we have discussed above, the O<sub>8</sub><sup>+</sup>(2).3 type subgroup of Fi<sub>22</sub>
lifts to a subgroup of the type 2 ×O<sub>8</sub><sup>+</sup>(2).3 in 2.Fi<sub>22</sub>,
and the outer involutions in the subgroup O<sub>8</sub><sup>+</sup>(2).3 ×2 of Fi<sub>22</sub>.2
lift to involutions in 2.Fi<sub>22</sub>.2, so there are two subgroups of the type
O<sub>8</sub><sup>+</sup>(2).3 ×2 not containing the centre of 2.Fi<sub>22</sub>.2,
which induce the same permutation character.
Since also 2.Fi<sub>22</sub> contains subgroups of the type O<sub>8</sub><sup>+</sup>(2).3 ×2,
we must use <tt>"extending"</tt> as the last argument of <tt>VerifyCandidates</tt>.
<div class="p"><!----></div>
<pre>
gap> s:= CharacterTable( "O8+(2).3" ) * CharacterTable( "Cyclic", 2 );;
gap> VerifyCandidates( s, tbl, tbl2, faith[6], "extending" );
G = 2.Fi22.2: point stabilizer O8+(2).3xC2, ranks [ 9 ]
[ "1a^++1001a^-+3080a^++10725a^++13650a^++13728ab+45045a^++48048bc+50050a^+" ]
gap> faith[6]:= faith[6]{ [ 1, 1 ] };;
</pre>
<div class="p"><!----></div>
By Lemma <a href="#situationIII">2.3</a>, the subgroup <sup>2</sup>F<sub>4</sub>(2) of Fi<sub>22</sub>.2 lifts
to 2 ×<sup>2</sup>F<sub>4</sub>(2) in 2.Fi<sub>22</sub>.2;
for that, note that the class <tt>4D</tt> of <sup>2</sup>F<sub>4</sub>(2) does not lie inside
<sup>2</sup>F<sub>4</sub>(2)<sup>′</sup> and the preimages in 2.Fi<sub>22</sub>.2 of the images in
Fi<sub>22</sub>.2 square into the subgroup <sup>2</sup>F<sub>4</sub>(2)<sup>′</sup> of the direct
product 2 ×<sup>2</sup>F<sub>4</sub>(2)<sup>′</sup>.
Since the group 2 ×<sup>2</sup>F<sub>4</sub>(2) contains two subgroups of the type
<sup>2</sup>F<sub>4</sub>(2), with normalizer 2 ×<sup>2</sup>F<sub>4</sub>(2), there are two classes
of such subgroups, which induce the same permutation character.
<div class="p"><!----></div>
<pre>
gap> facttbl:= CharacterTable( "Fi22.2" );;
gap> s0:= CharacterTable( "2F4(2)'" );;
gap> s:= CharacterTable( "2F4(2)" );;
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "all" );
2.Fi22.2: 2F4(2)'.2 lifts to a direct product,
proved by squares in [ 5, 38, 53 ].
gap> VerifyCandidates( s, tbl, tbl2, faith[16], "all" );
G = 2.Fi22.2: point stabilizer 2F4(2)'.2, ranks [ 13 ]
[ "1a^++1001a^++1430a^++13650a^++30030a^++133056a^{\\pm}+289575a^-+400400ab+57\
9150a^++675675a^-+1201200a^-+1663200ab" ]
gap> faith[16]:= faith[16]{ [ 1, 1 ] };;
gap> CompareWithDatabase( "2.Fi22.2", faith );
</pre>
<div class="p"><!----></div>
The group (2.Fi<sub>22</sub>.2)<sup>∗</sup> of the isoclinism type that is not printed
in the A<font size="-2">TLAS</font> has seven faithful multiplicity-free permutation actions,
with point stabilizers of the types O<sub>7</sub>(3), O<sub>8</sub><sup>+</sup>(2):S<sub>3</sub> (three times),
and O<sub>8</sub><sup>+</sup>(2):2 (three times).
<div class="p"><!----></div>
<pre>
gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );;
gap> faith:= FaithfulCandidates( tbl2, "Fi22.2" );;
3: subgroup $O_7(3)$, degree 56320 (1 cand.)
5: subgroup $O_8^+(2).3.2 \leq O_8^+(2).3.2 \times 2$, degree 247104 (
1 cand.)
7: subgroup $O_8^+(2).S_3 \leq O_8^+(2).3.2 \times 2$, degree 247104 (
1 cand.)
10: subgroup $O_8^+(2).2 \leq O_8^+(2).3.2 \times 2$, degree 741312 (1 cand.)
11: subgroup $O_8^+(2).2 \leq O_8^+(2).3.2 \times 2$, degree 741312 (1 cand.)
</pre>
<div class="p"><!----></div>
The characters arising from the third, fifth, and tenth multiplicity-free
permutation character of Fi<sub>22</sub>.2 are induced from subgroups of
2.Fi<sub>22</sub>, so these actions have been verified above.
<div class="p"><!----></div>
The seventh multiplicity-free permutation character of Fi<sub>22</sub>.2 is induced
from an O<sub>8</sub><sup>+</sup>(2):S<sub>3</sub> type subgroup that does not lie inside Fi<sub>22</sub>.
By Lemma <a href="#situationIII">2.3</a> (i), this subgroup lifts to a direct
product in (2.Fi<sub>22</sub>.2)<sup>∗</sup>;
this yields two actions
(because the O<sub>8</sub><sup>+</sup>(2):S<sub>3</sub> type subgroups have index 2 in their normalizer),
which induce the same permutation character.
Note that the involutions in O<sub>8</sub><sup>+</sup>(2):S<sub>3</sub> \O<sub>8</sub><sup>+</sup>(2) lie in the
class <tt>2F</tt> of Fi<sub>22</sub>.2, and these elements lift to involutions in
(2.Fi<sub>22</sub>.2)<sup>∗</sup>.
Since also 2.Fi<sub>22</sub> contains subgroups of the type O<sub>8</sub><sup>+</sup>(2):S<sub>3</sub>,
we must use <tt>"extending"</tt> as the last argument of <tt>VerifyCandidates</tt>.
<div class="p"><!----></div>
This argument also proves the existence of two classes of O<sub>8</sub><sup>+</sup>(2):2 type
subgroups that are not contained in 2.Fi<sub>22</sub>;
they arise from the 11th multiplicity-free permutation character
of Fi<sub>22</sub>.2.
<div class="p"><!----></div>
<pre>
gap> s0:= CharacterTable( "O8+(2).3" );;
gap> s:= CharacterTable( "O8+(2).S3" );;
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl2, "extending" );
Isoclinic(2.Fi22.2): O8+(2).3.2 lifts to a direct product,
proved by squares in [ 1, 7, 9, 11, 18, 21, 26, 39, 47, 52, 73 ].
gap> VerifyCandidates( s, tbl, tbl2, faith[7], "extending" );
G = Isoclinic(2.Fi22.2): point stabilizer O8+(2).3.2, ranks [ 9 ]
[ "1a^++1001a^++3080a^++10725a^-+13650a^++13728ab+45045a^++48048bc+50050a^-" ]
gap> faith[7]:= faith[7]{ [ 1, 1 ] };;
gap> s:= CharacterTable( "O8+(2).2" );;
gap> VerifyCandidates( s, tbl, tbl2, faith[11], "extending" );
G = Isoclinic(2.Fi22.2): point stabilizer O8+(2).2, ranks [ 19 ]
[ "1a^++352a^{\\pm}+429a^{\\pm}+1001a^++3080a^++10725a^-+13650a^++13728ab+4504\
5a^++48048a^{\\pm}bc+50050a^-+75075a^{\\pm}+123200a^{\\pm}" ]
gap> faith[11]:= faith[11]{ [ 1, 1 ] };;
gap> CompareWithDatabase( "Isoclinic(2.Fi22.2)", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.30">
3.30</a> G = 3.Fi<sub>22</sub></h3>
<div class="p"><!----></div>
The group 3.Fi<sub>22</sub> has six faithful multiplicity-free permutation actions,
with point stabilizers of the types O<sub>8</sub><sup>+</sup>(2):S<sub>3</sub>, O<sub>8</sub><sup>+</sup>(2):3 (twice),
O<sub>8</sub><sup>+</sup>(2):2, 2<sup>6</sup>:S<sub>6</sub>(2), and <sup>2</sup>F<sub>4</sub>(2).
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "3.Fi22" );;
gap> faith:= FaithfulCandidates( tbl, "Fi22" );;
4: subgroup $O_8^+(2).3.2$, degree 185328 (1 cand.)
5: subgroup $O_8^+(2).3 \leq O_8^+(2).3.2$, degree 370656 (2 cand.)
6: subgroup $O_8^+(2).2 \leq O_8^+(2).3.2$, degree 555984 (1 cand.)
8: subgroup $2^6:S_6(2)$, degree 2084940 (1 cand.)
9: subgroup ${^2F_4(2)^{\prime}}$, degree 10777536 (1 cand.)
</pre>
<div class="p"><!----></div>
The preimages of the maximal subgroups of the type O<sub>8</sub><sup>+</sup>(2).S<sub>3</sub> in Fi<sub>22</sub>
have the type 3 ×O<sub>8</sub><sup>+</sup>(2).S<sub>3</sub>,
because the Schur multiplier of O<sub>8</sub><sup>+</sup>(2) has order 4 and the only central
extension of S<sub>3</sub> by a group of order 3 is 3 ×S<sub>3</sub>.
Each such preimage contains one subgroup of the type O<sub>8</sub><sup>+</sup>(2).S<sub>3</sub>
with one subgroup of the type O<sub>8</sub><sup>+</sup>(2).3,
two conjugate O<sub>8</sub><sup>+</sup>(2).3 subgroups which are not contained in O<sub>8</sub><sup>+</sup>(2).S<sub>3</sub>,
and one class of O<sub>8</sub><sup>+</sup>(2).2 subgroups.
The two classes of O<sub>8</sub><sup>+</sup>(2).3 subgroups contain elements of order 3
outside O<sub>8</sub><sup>+</sup>(2) which lie in nonconjugate preimages of the class <tt>3A</tt>
of Fi<sub>22</sub>, so we get two classes of O<sub>8</sub><sup>+</sup>(2).3 subgroups in 3.Fi<sub>22</sub>
which induce different permutation characters.
<div class="p"><!----></div>
<pre>
gap> VerifyCandidates( CharacterTable( "O8+(2).S3" ), tbl, 0, faith[4], "all" );
G = 3.Fi22: point stabilizer O8+(2).3.2, ranks [ 10 ]
[ "1a+351ab+3080a+13650a+19305ab+42120ab+45045a" ]
gap> s:= CharacterTable( "O8+(2).3" );;
gap> fus:= PossibleClassFusions( s, tbl );;
gap> facttbl:= CharacterTable( "Fi22" );;
gap> factfus:= GetFusionMap( tbl, facttbl );;
gap> outer:= Difference( [ 1 .. NrConjugacyClasses( s ) ],
> ClassPositionsOfDerivedSubgroup( s ) );;
gap> outerfus:= List( fus, map -> map{ outer } );
[ [ 13, 13, 18, 18, 46, 46, 50, 50, 59, 59, 75, 75, 95, 95, 98, 98, 95, 95,
116, 116, 142, 142, 148, 148, 157, 157, 160, 160 ],
[ 14, 15, 18, 18, 47, 48, 51, 52, 59, 59, 76, 77, 96, 97, 99, 100, 96, 97,
116, 116, 143, 144, 149, 150, 158, 159, 161, 162 ],
[ 15, 14, 18, 18, 48, 47, 52, 51, 59, 59, 77, 76, 97, 96, 100, 99, 97, 96,
116, 116, 144, 143, 150, 149, 159, 158, 162, 161 ] ]
gap> preim:= InverseMap( factfus )[5];
[ 13, 14, 15 ]
gap> List( outerfus, x -> List( preim, i -> i in x ) );
[ [ true, false, false ], [ false, true, true ], [ false, true, true ] ]
gap> VerifyCandidates( s, tbl, 0, faith[5], "all" );
G = 3.Fi22: point stabilizer O8+(2).3, ranks [ 11, 17 ]
[ "1a+1001a+3080a+10725a+13650a+27027ab+45045a+50050a+96525ab",
"1a+351ab+1001a+3080a+7722ab+10725a+13650a+19305ab+42120ab+45045a+50050a+540\
54ab" ]
gap> VerifyCandidates( CharacterTable( "O8+(2).2" ), tbl, 0, faith[6], "all" );
G = 3.Fi22: point stabilizer O8+(2).2, ranks [ 17 ]
[ "1a+351ab+429a+3080a+13650a+19305ab+27027ab+42120ab+45045a+48048a+75075a+965\
25ab" ]
</pre>
<div class="p"><!----></div>
Lemma <a href="#situationI">2.1</a> applies to the maximal subgroups of the types
2<sup>6</sup>:S<sub>6</sub>(2) and <sup>2</sup>F<sub>4</sub>(2)<sup>′</sup> in Fi<sub>22</sub> and their preimages
in 3.Fi<sub>22</sub>.
<div class="p"><!----></div>
<pre>
gap> VerifyCandidates( CharacterTable( "2^6:s6f2" ), tbl, 0, faith[8], "all" );
G = 3.Fi22: point stabilizer 2^6:s6f2, ranks [ 24 ]
[ "1a+351ab+429a+1430a+3080a+13650a+19305ab+27027ab+30030a+42120ab+45045a+7507\
5a+96525ab+123552ab+205920a+320320a+386100ab" ]
gap> VerifyCandidates( CharacterTable( "2F4(2)'" ), tbl, 0, faith[9], "all" );
G = 3.Fi22: point stabilizer 2F4(2)', ranks [ 25 ]
[ "1a+1001a+1430a+13650a+19305ab+27027ab+30030a+51975ab+289575a+386100ab+40040\
0ab+405405ab+579150a+675675a+1201200a+1351350efgh" ]
gap> CompareWithDatabase( "3.Fi22", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.31">
3.31</a> G = 3.Fi<sub>22</sub>.2</h3>
<div class="p"><!----></div>
The group 3.Fi<sub>22</sub>.2 has seven faithful multiplicity-free permutation
actions,
with point stabilizers of the types O<sub>8</sub><sup>+</sup>(2):S<sub>3</sub> ×2,
O<sub>8</sub><sup>+</sup>(2):3 ×2, O<sub>8</sub><sup>+</sup>(2):S<sub>3</sub> (twice), O<sub>8</sub><sup>+</sup>(2):2 ×2,
2<sup>7</sup>:S<sub>6</sub>(2), and <sup>2</sup>F<sub>4</sub>(2).
<div class="p"><!----></div>
<pre>
gap> tbl2:= CharacterTable( "3.Fi22.2" );;
gap> faith:= FaithfulCandidates( tbl2, "Fi22.2" );;
4: subgroup $O_8^+(2).3.2 \times 2$, degree 185328 (1 cand.)
6: subgroup $O_8^+(2).3 \times 2 \leq O_8^+(2).3.2 \times 2$, degree 370656 (
1 cand.)
7: subgroup $O_8^+(2).S_3 \leq O_8^+(2).3.2 \times 2$, degree 370656 (
2 cand.)
8: subgroup $O_8^+(2).2 \times 2 \leq O_8^+(2).3.2 \times 2$, degree 555984 (
1 cand.)
9: subgroup $O_8^+(2).3 \leq O_8^+(2).3.2 \times 2$, degree 741312 (1 cand.)
14: subgroup $2^7:S_6(2)$, degree 2084940 (1 cand.)
16: subgroup ${^2F_4(2)}$, degree 10777536 (1 cand.)
</pre>
<div class="p"><!----></div>
Let H be a subgroup of the type O<sub>8</sub><sup>+</sup>(2):S<sub>3</sub> ×2 in Fi<sub>22</sub>.2;
it induces the 4th multiplicity-free permutation character of Fi<sub>22</sub>.2.
The intersection of H with Fi<sub>22</sub> is of the type O<sub>8</sub><sup>+</sup>(2):S<sub>3</sub>;
it lifts to a direct product in 3.Fi<sub>22</sub>, which contains one subgroup
of the type O<sub>8</sub><sup>+</sup>(2):S<sub>3</sub> that is normal in the preimage of H.
By Lemma <a href="#situationII">2.2</a>, we get one class of subgroups of the type
O<sub>8</sub><sup>+</sup>(2):S<sub>3</sub> ×2 in 3.Fi<sub>22</sub>.2.
The same argument yields one class of each of the types O<sub>8</sub><sup>+</sup>(2):3 ×2
and O<sub>8</sub><sup>+</sup>(2):2 ×2,
which arise from the 6th and 8th multiplicity-free permutation character
of Fi<sub>22</sub>.2, respectively.
<div class="p"><!----></div>
<pre>
gap> s:= CharacterTable( "O8+(2).S3" ) * CharacterTable( "Cyclic", 2 );;
gap> VerifyCandidates( s, tbl, tbl2, faith[4], "all" );
G = 3.Fi22.2: point stabilizer O8+(2).3.2xC2, ranks [ 7 ]
[ "1a^++351ab+3080a^++13650a^++19305ab+42120ab+45045a^+" ]
gap> s:= CharacterTable( "O8+(2).3" ) * CharacterTable( "Cyclic", 2 );;
gap> VerifyCandidates( s, tbl, tbl2, faith[6], "all" );
G = 3.Fi22.2: point stabilizer O8+(2).3xC2, ranks [ 12 ]
[ "1a^++351ab+1001a^-+3080a^++7722ab+10725a^++13650a^++19305ab+42120ab+45045a^\
++50050a^++54054ab" ]
gap> s:= CharacterTable( "O8+(2).2" ) * CharacterTable( "Cyclic", 2 );;
gap> VerifyCandidates( s, tbl, tbl2, faith[8], "all" );
G = 3.Fi22.2: point stabilizer O8+(2).2xC2, ranks [ 12 ]
[ "1a^++351ab+429a^++3080a^++13650a^++19305ab+27027ab+42120ab+45045a^++48048a^\
++75075a^++96525ab" ]
</pre>
<div class="p"><!----></div>
Let H be a subgroup of the type O<sub>8</sub><sup>+</sup>(2):S<sub>3</sub> in Fi<sub>22</sub>.2 that is not
contained in Fi<sub>22</sub>; it induces the 7-th multiplicity-free
permutation character of Fi<sub>22</sub>.2.
The intersection of H with Fi<sub>22</sub> is of the type O<sub>8</sub><sup>+</sup>(2):3;
it lifts to a direct product in 3.Fi<sub>22</sub>, which contains four subgroups
of the type O<sub>8</sub><sup>+</sup>(2):3,
three of them not containing the centre of 3.Fi<sub>22</sub>.
By Lemma <a href="#situationII">2.2</a>, we get three subgroups of the type
O<sub>8</sub><sup>+</sup>(2):S<sub>3</sub> in 3.Fi<sub>22</sub>.2, two of which are conjugate;
they induce two different permutation characters, so we get two classes.
<div class="p"><!----></div>
(Since there are O<sub>8</sub><sup>+</sup>(2).S<sub>3</sub> type subgroups also inside 3.Fi<sub>22</sub>,
we must use <tt>"extending"</tt> as the last argument of <tt>VerifyCandidates</tt>.)
<div class="p"><!----></div>
<pre>
gap> s:= CharacterTable( "O8+(2).S3" );;
gap> derpos:= ClassPositionsOfDerivedSubgroup( s );;
gap> facttbl:= CharacterTable("Fi22.2");;
gap> sfustbl2:= PossibleClassFusions( s, tbl2,
> rec( permchar:= faith[7][1] ) );;
gap> ForAll( sfustbl2,
> map -> NecessarilyDifferentPermChars( map, factfus, derpos ) );
true
gap> VerifyCandidates( s, tbl, tbl2, faith[7], "extending" );
G = 3.Fi22.2: point stabilizer O8+(2).3.2, ranks [ 9, 12 ]
[ "1a^++1001a^++3080a^++10725a^-+13650a^++27027ab+45045a^++50050a^-+96525ab",
"1a^++351ab+1001a^++3080a^++7722ab+10725a^-+13650a^++19305ab+42120ab+45045a^\
++50050a^-+54054ab" ]
</pre>
<div class="p"><!----></div>
The nineth multiplicity-free permutation character of Fi<sub>22</sub>.2
is induced from a subgroup of the type O<sub>8</sub><sup>+</sup>(2).3 that lies inside Fi<sub>22</sub>
and is known to lift to s group of the type 3 ×O<sub>8</sub><sup>+</sup>(2).3
in 3.Fi<sub>22</sub>.
All subgroups of index three in this group either contain the centre of
3.Fi<sub>22</sub> or have the type O<sub>8</sub><sup>+</sup>(2).3, and it turns out that the
permutation characters of 3.Fi<sub>22</sub>.2 induced from these subgroups are
not multiplicity-free.
So the candidate can be excluded.
<div class="p"><!----></div>
<pre>
gap> VerifyCandidates( CharacterTable( "O8+(2).3" ), tbl, tbl2, faith[9], "all" );
G = 3.Fi22.2: no O8+(2).3
gap> faith[9]:= [];;
</pre>
<div class="p"><!----></div>
Lemma <a href="#situationII">2.2</a> guarantees the existence of one class of subgroups
of each of the types 2<sup>7</sup>:S<sub>6</sub>(2) and <sup>2</sup>F<sub>4</sub>(2).
<div class="p"><!----></div>
<pre>
gap> VerifyCandidates( CharacterTable( "2^7:S6(2)" ), tbl, tbl2, faith[14], "all" );
G = 3.Fi22.2: point stabilizer 2^7:S6(2), ranks [ 17 ]
[ "1a^++351ab+429a^++1430a^++3080a^++13650a^++19305ab+27027ab+30030a^++42120ab\
+45045a^++75075a^++96525ab+123552ab+205920a^++320320a^++386100ab" ]
gap> VerifyCandidates( CharacterTable( "2F4(2)" ), tbl, tbl2, faith[16], "all" );
G = 3.Fi22.2: point stabilizer 2F4(2)'.2, ranks [ 17 ]
[ "1a^++1001a^++1430a^++13650a^++19305ab+27027ab+30030a^++51975ab+289575a^-+38\
6100ab+400400ab+405405ab+579150a^++675675a^-+1201200a^-+1351350efgh" ]
gap> CompareWithDatabase( "3.Fi22.2", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.32">
3.32</a> G = 6.Fi<sub>22</sub></h3><a name="LMerror">
</a>
<div class="p"><!----></div>
The group 6.Fi<sub>22</sub> has six faithful multiplicity-free permutation actions,
with point stabilizers of the types O<sub>8</sub><sup>+</sup>(2):S<sub>3</sub> (twice),
O<sub>8</sub><sup>+</sup>(2):3 (twice), and O<sub>8</sub><sup>+</sup>(2):2 (twice).
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "6.Fi22" );;
gap> facttbl:= CharacterTable( "3.Fi22" );;
gap> faith:= FaithfulCandidates( tbl, "3.Fi22" );;
1: subgroup $O_8^+(2):S_3 \rightarrow (Fi_{22},4)$, degree 370656 (2 cand.)
2: subgroup $O_8^+(2):3 \rightarrow (Fi_{22},5)$, degree 741312 (1 cand.)
3: subgroup $O_8^+(2):3 \rightarrow (Fi_{22},5)$, degree 741312 (1 cand.)
4: subgroup $O_8^+(2):2 \rightarrow (Fi_{22},6)$, degree 1111968 (2 cand.)
</pre>
<div class="p"><!----></div>
From the discussion of the cases 2.Fi<sub>22</sub> and 3.Fi<sub>22</sub>,
we conclude that the maximal subgroups of the type O<sub>8</sub><sup>+</sup>(2).S<sub>3</sub> lift to
groups of the type 6 ×O<sub>8</sub><sup>+</sup>(2).S<sub>3</sub> in 6.Fi<sub>22</sub>.
So Lemma <a href="#situationIII">2.3</a> (iii) yields two classes of O<sub>8</sub><sup>+</sup>(2):S<sub>3</sub> type
subgroups, which induce different permutation characters.
<div class="p"><!----></div>
<pre>
gap> s:= CharacterTable( "O8+(2).S3" );;
gap> s0:= CharacterTable( "O8+(2).3" );;
gap> CheckConditionsForLemma3( s0, s, facttbl, tbl, "all" );
6.Fi22: O8+(2).3.2 lifts to a direct product,
proved by squares in [ 1, 22, 28, 30, 46, 55, 76, 104, 131, 141, 215 ].
gap> derpos:= ClassPositionsOfDerivedSubgroup( s );;
gap> factfus:= GetFusionMap( tbl, facttbl );;
gap> ForAll( PossibleClassFusions( s, tbl ),
> map -> NecessarilyDifferentPermChars( map, factfus, derpos ) );
true
gap> VerifyCandidates( s, tbl, 0, faith[1], "all" );
G = 6.Fi22: point stabilizer O8+(2).3.2, ranks [ 14, 14 ]
[ "1a+351ab+3080a+13650a+13728b+19305ab+42120ab+45045a+48048c+61776cd",
"1a+351ab+3080a+13650a+13728a+19305ab+42120ab+45045a+48048b+61776ab" ]
</pre>
<div class="p"><!----></div>
Each subgroup of the type O<sub>8</sub><sup>+</sup>(2):3 in 3.Fi<sub>22</sub> lifts to a direct product
in 6.Fi<sub>22</sub>, which yields one action;
as the two constituents that are permutation characters of 3.Fi<sub>22</sub>
are different,
we get two different permutation characters induced from O<sub>8</sub><sup>+</sup>(2):3.
<div class="p"><!----></div>
<pre>
gap> VerifyCandidates( CharacterTable( "O8+(2).3" ), tbl, 0,
> Concatenation( faith[2], faith[3] ), "all" );
G = 6.Fi22: point stabilizer O8+(2).3, ranks [ 17, 25 ]
[ "1a+1001a+3080a+10725a+13650a+13728ab+27027ab+45045a+48048bc+50050a+96525ab+\
123552cd",
"1a+351ab+1001a+3080a+7722ab+10725a+13650a+13728ab+19305ab+42120ab+45045a+48\
048bc+50050a+54054ab+61776abcd" ]
</pre>
<div class="p"><!----></div>
Each subgroup of the type O<sub>8</sub><sup>+</sup>(2):2 in 3.Fi<sub>22</sub> lifts to a direct product
in 6.Fi<sub>22</sub>, which yields two actions; the permutation characters are
different by the argument used for O<sub>8</sub><sup>+</sup>(2):S<sub>3</sub>.
<div class="p"><!----></div>
<pre>
gap> VerifyCandidates( CharacterTable( "O8+(2).2" ), tbl, 0, faith[4], "all" );
G = 6.Fi22: point stabilizer O8+(2).2, ranks [ 25, 25 ]
[ "1a+351ab+352a+429a+3080a+13650a+13728b+19305ab+27027ab+42120ab+45045a+48048\
ac+61776cd+75075a+96525ab+123200a+123552cd",
"1a+351ab+352a+429a+3080a+13650a+13728a+19305ab+27027ab+42120ab+45045a+48048\
ab+61776ab+75075a+96525ab+123200a+123552cd" ]
gap> CompareWithDatabase( "6.Fi22", faith );
</pre>
<div class="p"><!----></div>
(Note that the rank 17 permutation character above was missing in the first
version of [<a href="#LM03" name="CITELM03">LM</a>].)
<div class="p"><!----></div>
<h3><a name="tth_sEc3.33">
3.33</a> G = 6.Fi<sub>22</sub>.2</h3>
<div class="p"><!----></div>
The group 6.Fi<sub>22</sub>.2 that is printed in the A<font size="-2">TLAS</font> has four faithful
multiplicity-free permutation actions,
with point stabilizers of the types O<sub>8</sub><sup>+</sup>(2):3 ×2 (twice)
and <sup>2</sup>F<sub>4</sub>(2) (twice).
<div class="p"><!----></div>
<pre>
gap> tbl2:= CharacterTable( "6.Fi22.2" );;
gap> faith:= FaithfulCandidates( tbl2, "Fi22.2" );;
6: subgroup $O_8^+(2).3 \times 2 \leq O_8^+(2).3.2 \times 2$, degree 741312 (
1 cand.)
16: subgroup ${^2F_4(2)}$, degree 21555072 (1 cand.)
</pre>
<div class="p"><!----></div>
Each O<sub>8</sub><sup>+</sup>(2):3 ×2 type subgroup of 3.Fi<sub>22</sub>.2 gives rise to two
subgroups of the same type in 6.Fi<sub>22</sub>.2, so we get two classes inducing
the same permutation character.
(Since there are O<sub>8</sub><sup>+</sup>(2).3 ×2 type subgroups also inside 6.Fi<sub>22</sub>,
we must use <tt>"extending"</tt> as the last argument of <tt>VerifyCandidates</tt>.)
<div class="p"><!----></div>
<pre>
gap> s:= CharacterTable( "O8+(2).3" ) * CharacterTable( "Cyclic", 2 );;
gap> VerifyCandidates( s, tbl, tbl2, faith[6], "extending" );
G = 6.Fi22.2: point stabilizer O8+(2).3xC2, ranks [ 16 ]
[ "1a^++351ab+1001a^-+3080a^++7722ab+10725a^++13650a^++13728ab+19305ab+42120ab\
+45045a^++48048bc+50050a^++54054ab+61776adbc" ]
</pre>
<div class="p"><!----></div>
The subgroup of the type 6 ×<sup>2</sup>F<sub>4</sub>(2)<sup>′</sup> of 6.Fi<sub>22</sub> extends
to 6 ×<sup>2</sup>F<sub>4</sub>(2) in 6.Fi<sub>22</sub>.2, which contains two subgroups
of the type <sup>2</sup>F<sub>4</sub>(2), by Lemma <a href="#situationIII">2.3</a>;
so we get two classes of such subgroups,
which induce the same permutation character.
<div class="p"><!----></div>
<pre>
gap> VerifyCandidates( CharacterTable( "2F4(2)" ), tbl, tbl2, faith[16], "all" );
G = 6.Fi22.2: point stabilizer 2F4(2)'.2, ranks [ 22 ]
[ "1a^++1001a^++1430a^++13650a^++19305ab+27027ab+30030a^++51975ab+133056a^{\\p\
m}+289575a^-+386100ab+400400ab+405405ab+579150a^++675675a^-+1201200a^-+1351350\
efgh+1663200ab+1796256adbc" ]
gap> faith[6]:= faith[6]{ [ 1, 1 ] };;
gap> faith[16]:= faith[16]{ [ 1, 1 ] };;
gap> CompareWithDatabase( "6.Fi22.2", faith );
</pre>
<div class="p"><!----></div>
The group (6.Fi<sub>22</sub>.2)<sup>∗</sup> of the isoclinism type that is not printed
in the A<font size="-2">TLAS</font> has two faithful multiplicity-free permutation actions,
with point stabilizers of the type O<sub>8</sub><sup>+</sup>(2):S<sub>3</sub> (twice).
<div class="p"><!----></div>
<pre>
gap> facttbl:= CharacterTable( "Fi22.2" );;
gap> tbl2:= IsoclinicTable( tbl, tbl2, facttbl );;
gap> faith:= FaithfulCandidates( tbl2, "Fi22.2" );;
7: subgroup $O_8^+(2).S_3 \leq O_8^+(2).3.2 \times 2$, degree 741312 (
2 cand.)
</pre>
<div class="p"><!----></div>
The existence of O<sub>8</sub><sup>+</sup>(2):S<sub>3</sub> type subgroups (not contained in 6.Fi<sub>22</sub>)
follows from Lemma <a href="#situationII">2.2</a> and the existence of these subgroups in
(2.Fi<sub>22</sub>.2)<sup>∗</sup>; we get one class for each of the two classes in
(2.Fi<sub>22</sub>.2)<sup>∗</sup>, with different permutation characters.
<div class="p"><!----></div>
<pre>
gap> s:= CharacterTable( "O8+(2).S3" );;
gap> VerifyCandidates( s, tbl, tbl2, faith[7], "extending" );
G = Isoclinic(6.Fi22.2): point stabilizer O8+(2).3.2, ranks [ 12, 16 ]
[ "1a^++1001a^++3080a^++10725a^-+13650a^++13728ab+27027ab+45045a^++48048bc+500\
50a^-+96525ab+123552cd",
"1a^++351ab+1001a^++3080a^++7722ab+10725a^-+13650a^++13728ab+19305ab+42120ab\
+45045a^++48048bc+50050a^-+54054ab+61776adbc" ]
gap> CompareWithDatabase( "Isoclinic(6.Fi22.2)", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.34">
3.34</a> G = 2.Co<sub>1</sub></h3>
<div class="p"><!----></div>
The group 2.Co<sub>1</sub> has two faithful multiplicity-free permutation actions,
with point stabilizers of the types Co<sub>2</sub> and Co<sub>3</sub>,
respectively, by Lemma <a href="#situationI">2.1</a>.
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "2.Co1" );;
gap> faith:= FaithfulCandidates( tbl, "Co1" );;
1: subgroup $Co_2$, degree 196560 (1 cand.)
5: subgroup $Co_3$, degree 16773120 (1 cand.)
gap> VerifyCandidates( CharacterTable( "Co2" ), tbl, 0, faith[1], "all" );
G = 2.Co1: point stabilizer Co2, ranks [ 7 ]
[ "1a+24a+299a+2576a+17250a+80730a+95680a" ]
gap> VerifyCandidates( CharacterTable( "Co3" ), tbl, 0, faith[5], "all" );
G = 2.Co1: point stabilizer Co3, ranks [ 12 ]
[ "1a+24a+299a+2576a+17250a+80730a+95680a+376740a+1841840a+2417415a+5494125a+6\
446440a" ]
gap> CompareWithDatabase( "2.Co1", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.35">
3.35</a> G = 3.F<sub>3+</sub></h3>
<div class="p"><!----></div>
The group 3.F<sub>3+</sub> has two faithful multiplicity-free permutation actions,
with point stabilizers of the types Fi<sub>23</sub> and O<sub>10</sub><sup>−</sup>(2),
respectively, by Lemma <a href="#situationI">2.1</a>.
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "3.F3+" );;
gap> faith:= FaithfulCandidates( tbl, "F3+" );;
1: subgroup $Fi_{23}$, degree 920808 (1 cand.)
2: subgroup $O_{10}^-(2)$, degree 150532080426 (1 cand.)
gap> VerifyCandidates( CharacterTable( "Fi23" ), tbl, 0, faith[1], "all" );
G = 3.F3+: point stabilizer Fi23, ranks [ 7 ]
[ "1a+783ab+57477a+249458a+306153ab" ]
gap> VerifyCandidates( CharacterTable( "O10-(2)" ), tbl, 0, faith[2], "all" );
G = 3.F3+: point stabilizer O10-(2), ranks [ 43 ]
[ "1a+783ab+8671a+57477a+64584ab+249458a+306153ab+555611a+1666833a+6724809ab+1\
9034730ab+35873145a+43779879ab+48893768a+79452373a+195019461ab+203843871ab+415\
098112a+1050717096ab+1264015025a+1540153692a+1818548820ab+2346900864a+32086535\
25a+10169903744a+10726070355ab+13904165275a+15016498497ab+17161712568a+2109675\
1104ab" ]
gap> CompareWithDatabase( "3.F3+", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.36">
3.36</a> G = 3.F<sub>3+</sub>.2</h3>
<div class="p"><!----></div>
The group 3.F<sub>3+</sub>.2 has two faithful multiplicity-free permutation actions,
with point stabilizers of the types Fi<sub>23</sub> ×2 and O<sub>10</sub><sup>−</sup>(2).2,
respectively, by Lemma <a href="#situationII">2.2</a>.
<div class="p"><!----></div>
<pre>
gap> tbl2:= CharacterTable( "3.F3+.2" );;
gap> faith:= FaithfulCandidates( tbl2, "F3+.2" );;
1: subgroup $Fi_{23} \times 2$, degree 920808 (1 cand.)
3: subgroup $O_{10}^-(2).2$, degree 150532080426 (1 cand.)
gap> VerifyCandidates( CharacterTable( "2xFi23" ), tbl, tbl2, faith[1], "all" );
G = 3.F3+.2: point stabilizer 2xFi23, ranks [ 5 ]
[ "1a^++783ab+57477a^++249458a^++306153ab" ]
gap> VerifyCandidates( CharacterTable( "O10-(2).2" ), tbl, tbl2, faith[3], "all" );
G = 3.F3+.2: point stabilizer O10-(2).2, ranks [ 30 ]
[ "1a^++783ab+8671a^-+57477a^++64584ab+249458a^++306153ab+555611a^-+1666833a^+\
+6724809ab+19034730ab+35873145a^++43779879ab+48893768a^-+79452373a^++195019461\
ab+203843871ab+415098112a^-+1050717096ab+1264015025a^++1540153692a^++181854882\
0ab+2346900864a^-+3208653525a^++10169903744a^-+10726070355ab+13904165275a^++15\
016498497ab+17161712568a^++21096751104ab" ]
gap> CompareWithDatabase( "3.F3+.2", faith );
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc3.37">
3.37</a> G = 2.B</h3>
<div class="p"><!----></div>
The group 2.B has one faithful multiplicity-free permutation action,
with point stabilizer of the type Fi<sub>23</sub>, by Lemma <a href="#situationI">2.1</a>.
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "2.B" );;
gap> faith:= FaithfulCandidates( tbl, "B" );;
4: subgroup $Fi_{23}$, degree 2031941058560000 (1 cand.)
gap> VerifyCandidates( CharacterTable( "Fi23" ), tbl, 0, faith[4], "all" );
G = 2.B: point stabilizer Fi23, ranks [ 34 ]
[ "1a+4371a+96255a+96256a+9458750a+10506240a+63532485a+347643114a+356054375a+4\
10132480a+4221380670a+4275362520a+8844386304a+9287037474a+13508418144a+3665765\
3760a+108348770530a+309720864375a+635966233056a+864538761216a+1095935366250a+4\
322693806080a+6145833622500a+6619124890560a+10177847623680a+12927978301875a+38\
348970335820a+60780833777664a+89626740328125a+110949141022720a+211069033500000\
a+284415522641250b+364635285437500a+828829551513600a" ]
gap> CompareWithDatabase( "2.B", faith );
</pre>
<div class="p"><!----></div>
<h2><a name="tth_sEc4">
4</a> Appendix: Explicit Computations with Groups</h2><a name="explicit">
</a>
<div class="p"><!----></div>
Only in the proofs for the groups involving M<sub>22</sub>, explicit computations
with the groups were necessary.
These computations are collected in this appendix.
<div class="p"><!----></div>
<h3><a name="tth_sEc4.1">
4.1</a> 2<sup>4</sup>:A<sub>6</sub> type subgroups in 2.M<sub>22</sub></h3><a name="explicit1">
</a>
<div class="p"><!----></div>
We show that the preimage in 2.M<sub>22</sub> of each maximal subgroup of the type
2<sup>4</sup>:A<sub>6</sub> in M<sub>22</sub> contains one class of subgroups of the type
2 ×2<sup>4</sup>:A<sub>5</sub>.
For that, we first note that there are two classes of subgroups of the type
2<sup>4</sup>:A<sub>5</sub> inside 2<sup>4</sup>:A<sub>6</sub>, and that the A<sub>5</sub> subgroups lift to groups
of the type 2 ×A<sub>5</sub> because 2.M<sub>22</sub> does not admit an embedding of
2.A<sub>6</sub>.
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "2.M22" );;
gap> PossibleClassFusions( CharacterTable( "2.A6" ), tbl );
[ ]
</pre>
<div class="p"><!----></div>
Now we fetch a permutation representation of 2.M<sub>22</sub> on 352 points,
from the A<font size="-2">TLAS</font> of Group Representations (see [<a href="#AGR" name="CITEAGR">Wil</a>]),
via the <font face="helvetica">GAP</font> package AtlasRep (see [<a href="#AtlasRep" name="CITEAtlasRep">Bre04a</a>]),
and compute generators for the second class of maximal subgroups,
via the straight line program for M<sub>22</sub>.
<div class="p"><!----></div>
<pre>
gap> LoadPackage( "atlasrep" );
true
gap> gens:= OneAtlasGeneratingSet( "2.M22", NrMovedPoints, 352 );;
gap> slp:= AtlasStraightLineProgram( "M22", "maxes", 2 );;
gap> sgens:= ResultOfStraightLineProgram( slp.program, gens.generators );;
gap> s:= Group( sgens );; Size( s );
11520
gap> 2^5 * 360;
11520
</pre>
<div class="p"><!----></div>
The subgroup acts intransitively on the 352 points.
We switch to the representation on 192 points,
and compute the normal subgroup N of order 2<sup>5</sup>.
<div class="p"><!----></div>
<pre>
gap> orbs:= Orbits( s, MovedPoints( s ) );;
gap> List( orbs, Length );
[ 160, 192 ]
gap> s:= Action( s, orbs[2] );;
gap> Size( s );
11520
gap> syl2:= SylowSubgroup( s, 2 );;
gap> repeat
> x:= Random( syl2 );
> n:= NormalClosure( s, SubgroupNC( s, [ x ] ) );
> until Size( n ) = 32;
</pre>
<div class="p"><!----></div>
The point stabilizer S in this group has type A<sub>5</sub>,
and generates together with N one of the desired subgroups of the type
2<sup>5</sup>:A<sub>5</sub>.
However, S does not normalize a subgroup of order 2<sup>4</sup>,
and so there is no subgroup of the type 2<sup>4</sup>:A<sub>5</sub>.
<div class="p"><!----></div>
<pre>
gap> stab:= Stabilizer( s, 192 );;
gap> sub:= ClosureGroup( n, stab );;
gap> Size( sub );
1920
gap> Set( List( Elements( n ),
> x -> Size( NormalClosure( sub, SubgroupNC( sub, [ x ] ) ) ) ) );
[ 1, 2, 32 ]
</pre>
<div class="p"><!----></div>
A representative of the other class of A<sub>5</sub> type subgroups can be found
by taking an element x of order three that is not conjugate to one in S,
and to choose an element y of order five such that the product is an
involution.
<div class="p"><!----></div>
<pre>
gap> syl3:= SylowSubgroup( s, 3 );;
gap> repeat three:= Random( stab ); until Order( three ) = 3;
gap> repeat other:= Random( syl3 );
> until Order( other ) = 3 and not IsConjugate( s, three, other );
gap> syl5:= SylowSubgroup( s, 5 );;
gap> repeat y:= Random( syl5 )^Random( s ); until Order( other*y ) = 2;
gap> a5:= Group( other, y );;
gap> IsConjugate( s, a5, stab );
false
gap> sub:= ClosureGroup( n, a5 );;
gap> Size( sub );
1920
gap> Set( List( Elements( n ),
> x -> Size( NormalClosure( sub, SubgroupNC( sub, [ x ] ) ) ) ) );
[ 1, 2, 16, 32 ]
</pre>
<div class="p"><!----></div>
This proves the existence of one class of the desired subgroups.
Finally, we show that the character table of these groups is indeed
the one we used in Section <a href="#libtbl">3.3</a>.
<div class="p"><!----></div>
<pre>
gap> g:= First( Elements( n ),
> x -> Size( NormalClosure( sub, SubgroupNC( sub, [ x ] ) ) ) = 16 );;
gap> compl:= ClosureGroup( a5, g );;
gap> Size( compl );
960
gap> tbl:= CharacterTable( compl );;
gap> IsRecord( TransformingPermutationsCharacterTables( tbl,
> CharacterTable( "P1/G1/L1/V1/ext2" ) ) );
true
</pre>
<div class="p"><!----></div>
<h3><a name="tth_sEc4.2">
4.2</a> 2<sup>4</sup>:S<sub>5</sub> type subgroups in M<sub>22</sub>.2</h3><a name="explicit2">
</a>
<div class="p"><!----></div>
A maximal subgroup of the type 2<sup>4</sup>:S<sub>6</sub> in M<sub>22</sub>.2 is perhaps easiest
found as the point stabilizer in the degree 77 permutation representation.
In order to find its index 6 subgroups,
the degree 22 permutation representation of M<sub>22</sub>.2 is more suitable
because the restriction to the 2<sup>4</sup>:S<sub>6</sub> type subgroup has orbits of the
lengths 6 and 16, where the action of the orbit of length 6 is the
natural permutation action of S<sub>6</sub>.
<div class="p"><!----></div>
So we choose the sum of the two representations, of total degree 99.
For convenience, we find this representation as the point stabilizer in the
degree 100 representation of HS.2, which is contained in the A<font size="-2">TLAS</font>
of Group Representations (see [<a href="#AGR" name="CITEAGR">Wil</a>]).
<div class="p"><!----></div>
<pre>
gap> gens:= OneAtlasGeneratingSet( "HS.2", NrMovedPoints, 100 );;
gap> stab:= Stabilizer( Group( gens.generators ), 100 );;
gap> orbs:= Orbits( stab, MovedPoints( stab ) );;
gap> List( orbs, Length );
[ 77, 22 ]
gap> pnt:= First( orbs, x -> Length( x ) = 77 )[1];;
gap> m:= Stabilizer( stab, pnt );;
gap> Size( m );
11520
</pre>
<div class="p"><!----></div>
Now we find two nonconjugate subgroups of the type 2<sup>4</sup>:S<sub>5</sub> as the stabilizer
of a point and of a total in S<sub>6</sub>, respectively (cf. [<a href="#CCN85" name="CITECCN85">CCN<sup>+</sup>85</a>,p. 4]).
<div class="p"><!----></div>
<pre>
gap> orbs:= Orbits( m, MovedPoints( m ) );;
gap> List( orbs, Length );
[ 60, 16, 6, 16 ]
gap> six:= First( orbs, x -> Length( x ) = 6 );;
gap> p:= ( six[1], six[2] )( six[3], six[4] )( six[5], six[6] );;
gap> conj:= ( six[2], six[4], six[5], six[6], six[3] );;
gap> total:= List( [ 0 .. 4 ], i -> p^( conj^i ) );;
gap> stab1:= Stabilizer( m, six[1] );;
gap> stab2:= Stabilizer( m, Set( total ), OnSets );;
gap> IsConjugate( m, stab1, stab2 );
false
</pre>
<div class="p"><!----></div>
We identify the character tables of the two groups in the <font face="helvetica">GAP</font> Character
Table Library.
<div class="p"><!----></div>
<pre>
gap> s1:= CharacterTable( stab1 );;
gap> s2:= CharacterTable( stab2 );;
gap> NrConjugacyClasses( s1 ); NrConjugacyClasses( s2 );
12
18
gap> lib1:= CharacterTable( "2^4:s5" );;
gap> IsRecord( TransformingPermutationsCharacterTables( lib1, s1 ) );
true
gap> lib2:= CharacterTable( "w(d5)" );;
gap> IsRecord( TransformingPermutationsCharacterTables( lib2, s2 ) );
true
</pre>
<div class="p"><!----></div>
The first subgroup does not lead to multiplicity-free permutation characters
of 2.M<sub>22</sub>.2.
Note that there are two classes of subgroups of this type in M<sub>22</sub>.2,
one of them is contained in M<sub>22</sub> and the other is not.
The action on the cosets of the former is multiplicity-free,
but it does not lift to a multiplicity-free candidate of 2.M<sub>22</sub>.2;
and the action on the cosets of the latter is not multiplicity-free.
<div class="p"><!----></div>
<pre>
gap> tbl:= CharacterTable( "M22" );;
gap> tbl2:= CharacterTable( "M22.2" );;
gap> s1fustbl2:= PossibleClassFusions( s1, tbl2 );
[ [ 1, 2, 2, 4, 4, 2, 5, 3, 7, 5, 10, 6 ],
[ 1, 2, 12, 15, 4, 2, 5, 3, 16, 15, 17, 6 ] ]
gap> pi:= List( s1fustbl2, map -> Induced( s1, tbl2,
> [ TrivialCharacter( s1 ) ], map )[1] );
[ Character( CharacterTable( "M22.2" ), [ 462, 46, 12, 6, 6, 2, 4, 0, 0, 2,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ),
Character( CharacterTable( "M22.2" ), [ 462, 30, 12, 2, 2, 2, 0, 0, 0, 0,
0, 56, 0, 0, 12, 2, 2, 0, 0, 0, 0 ] ) ]
gap> PermCharInfoRelative( tbl, tbl2, pi ).ATLAS;
[ "1a^{\\pm}+21a^{\\pm}+55a^{\\pm}+154a^{\\pm}",
"1a^++21(a^+)^{2}+55a^++154a^++210a^+" ]
</pre>
<div class="p"><!----></div>
So only the second type of 2<sup>4</sup>:S<sub>5</sub> type subgroups can lift to the
multiplicity-free candidate in question,
and this situation is dealt with in Section <a href="#2.M22.2">3.4</a>.
<div class="p"><!----></div>
<pre>
gap> s2fustbl2:= PossibleClassFusions( s2, tbl2 );;
gap> pi:= List( s2fustbl2, map -> Induced( s2, tbl2,
> [ TrivialCharacter( s2 ) ], map )[1] );
[ Character( CharacterTable( "M22.2" ), [ 462, 30, 3, 2, 2, 2, 3, 0, 0, 0, 0,
28, 20, 4, 8, 1, 2, 0, 1, 0, 0 ] ) ]
gap> PermCharInfoRelative( tbl, tbl2, pi ).ATLAS;
[ "1a^++21a^++55a^++154a^++231a^-" ]
</pre>
<div class="p"><!----></div>
<h2>References</h2>
<dl compact="compact">
<dt><a href="#CITEBL96" name="BL96">[BL96]</a></dt><dd>
Thomas Breuer and Klaus Lux, <em>The multiplicity-free permutation characters
of the sporadic simple groups and their automorphism groups</em>, Comm. Alg.
<b>24</b> (1996), no. 7, 2293-2316.
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEAtlasRep" name="AtlasRep">[Bre04a]</a></dt><dd>
Thomas Breuer, <em>Manual for the <font face="helvetica">GAP</font> 4 Package AtlasRep, Version 1.2</em>,
Lehrstuhl D für Mathematik, Rheinisch
Westfälische Technische Hochschule, Aachen, Germany,
2004.
<div class="p"><!----></div>
</dd>
<dt><a href="#CITECTblLib" name="CTblLib">[Bre04b]</a></dt><dd>
Thomas Breuer, <em>Manual for the <font face="helvetica">GAP</font> Character Table Library, Version
1.1</em>, Lehrstuhl D für Mathematik, Rheinisch
Westfälische Technische Hochschule, Aachen, Germany,
2004.
<div class="p"><!----></div>
</dd>
<dt><a href="#CITECCN85" name="CCN85">[CCN<sup>+</sup>85]</a></dt><dd>
J[ohn] H. Conway, R[obert] T. Curtis, S[imon] P. Norton, R[ichard] A. Parker,
and R[obert] A. Wilson, <em>Atlas of finite groups</em>, Oxford University
Press, 1985.
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEGAP4" name="GAP4">[GAP04]</a></dt><dd>
The GAP Group, <em>GAP - Groups, Algorithms, and Programming, Version
4.4</em>, 2004, <a href="http://www.gap-system.org"><tt>http://www.gap-system.org</tt></a>.
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEHoe01" name="Hoe01">[Höh01]</a></dt><dd>
Ines Höhler, <em>Vielfachheitsfreie Permutationsdarstellungen und die
Invarianten zugehöriger Graphen</em>, Examensarbeit, Lehrstuhl D
für Mathematik, Rheinisch Westfälische
Technische Hochschule, Aachen, Germany, 2001.
<div class="p"><!----></div>
</dd>
<dt><a href="#CITELM03" name="LM03">[LM]</a></dt><dd>
S. A. Linton and Z. E. Mpono, <em>Multiplicity-free permutation characters of
covering groups of sporadic simple groups</em>.
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEMue03" name="Mue03">[Mül03]</a></dt><dd>
Jürgen Müller, <em>On endomorphism rings and character tables</em>,
Habilitationsschrift, Lehrstuhl D für Mathematik,
Rheinisch Westfälische Technische Hochschule,
Aachen, Germany, 2003.
<div class="p"><!----></div>
</dd>
<dt><a href="#CITEAGR" name="AGR">[Wil]</a></dt><dd>
Robert A. Wilson, <em>ATLAS of Finite Group Representations</em>,
<a href="http://www.mat.bham.ac.uk/atlas/"><tt>http://www.mat.bham.ac.uk/atlas/</tt></a>.</dd>
</dl>
<div class="p"><!----></div>
<div class="p"><!----></div>
<br /><br /><hr /><small>File translated from
T<sub><font size="-1">E</font></sub>X
by <a href="http://hutchinson.belmont.ma.us/tth/">
T<sub><font size="-1">T</font></sub>H</a>,
version 3.55.<br />On 31 Mar 2004, 10:54.</small>
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