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<p><a id="X87EAF8E578D95793" name="X87EAF8E578D95793"></a></p>
<div class="ChapSects"><a href="chap2.html#X87EAF8E578D95793">2 <span class="Heading">The User Interface of the <strong class="pkg">AtlasRep</strong> Package
</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X87D26B13819A8209">2.1 <span class="Heading">Accessing vs. Constructing Representations</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X81BF52FC7B8C08D4">2.2 <span class="Heading">Group Names Used in the <strong class="pkg">AtlasRep</strong> Package</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X795DB7E486E0817D">2.3 <span class="Heading">Standard Generators Used in the <strong class="pkg">AtlasRep</strong> Package
</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X861CD545803B97E8">2.4 <span class="Heading">Class Names Used in the <strong class="pkg">AtlasRep</strong> Package</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X850EEDEE831EE039">2.4-1 <span class="Heading">Definition of <strong class="pkg">ATLAS</strong> Class Names</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X78166D1D7D18EFBF">2.4-2 AtlasClassNames</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X806ED7BB83FD848B">2.5 <span class="Heading">Accessing Data of the <strong class="pkg">AtlasRep</strong> Package</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X79DACFFA7E2D1A99">2.5-1 DisplayAtlasInfo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7D1CCCF8852DFF39">2.5-2 AtlasGenerators</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X801F2E657C8A79ED">2.5-3 AtlasProgram</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X841478AB7CD06D44">2.5-4 OneAtlasGeneratingSetInfo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X84C2D76482E60E42">2.5-5 AllAtlasGeneratingSetInfos</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X80AABEE783363B70">2.5-6 AtlasGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7A3E460C82B3D9A3">2.5-7 AtlasSubgroup</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X87ACE06E82B68589">2.6 <span class="Heading">Examples of Using the <strong class="pkg">AtlasRep</strong> Package</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8563D96878AC685C">2.6-1 <span class="Heading">Example: Class Representatives</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X81C9233778A3A817">2.6-2 <span class="Heading">Example: Permutation and Matrix Representations</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8284D7E87D38889C">2.6-3 <span class="Heading">Example: Outer Automorphisms</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X794D669E7A507310">2.6-4 <span class="Heading">Example: Using Semi-presentations and Black Box Programs</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7CE7C2068017525C">2.6-5 <span class="Heading">Example: Using the <strong class="pkg">GAP</strong> Library of Tables of Marks</span></a>
</span>
</div>
</div>
<h3>2 <span class="Heading">The User Interface of the <strong class="pkg">AtlasRep</strong> Package
</span></h3>
<p>The <em>user interface</em> is the part of the <strong class="pkg">GAP</strong> interface that allows one to display information about the current contents of the database and to access individual data (perhaps from a remote server, see Section <a href="chap1.html#X7E1934D2780D108F"><b>1.7-1</b></a>). The corresponding functions are described in this chapter. See Section <a href="chap2.html#X87ACE06E82B68589"><b>2.6</b></a> for some small examples how to use the functions of the interface.</p>
<p>Extensions of the <strong class="pkg">AtlasRep</strong> package are regarded as another part of the <strong class="pkg">GAP</strong> interface, they are described in Chapter <a href="chap3.html#X7DF711157F9EFA28"><b>3</b></a>. Finally, the low level part of the interface are described in Chapter <a href="chap5.html#X7F77634D817156B3"><b>5</b></a>.</p>
<p>As stated in Section <a href="chap1.html#X8033B61682EE6A23"><b>1.2</b></a>, the user interface is preliminary. It will be extended when the <strong class="pkg">GAP</strong> version of the <strong class="pkg">ATLAS</strong> of Group Representations is connected to other <strong class="pkg">GAP</strong> databases such as the libraries of character tables and tables of marks.</p>
<p>For some of the examples in this chapter, the <strong class="pkg">GAP</strong> packages <strong class="pkg">CTblLib</strong> <a href="chapBib.html#biBCTblLib1.1.3">[Bre04]</a> and <strong class="pkg">TomLib</strong> are needed.</p>
<table class="example">
<tr><td><pre>
gap> LoadPackage( "ctbllib" );
true
gap> LoadPackage( "tomlib" );
true
</pre></td></tr></table>
<p><a id="X87D26B13819A8209" name="X87D26B13819A8209"></a></p>
<h4>2.1 <span class="Heading">Accessing vs. Constructing Representations</span></h4>
<p>Note that <em>accessing</em> the data means in particular that it is <em>not</em> the aim of this package to <em>construct</em> representations from known ones. For example, if at least one permutation representation for a group G is stored but no matrix representation in a positive characteristic p, say, then <code class="func">OneAtlasGeneratingSetInfo</code> (<a href="chap2.html#X841478AB7CD06D44"><b>2.5-4</b></a>) returns <code class="keyw">fail</code> when it is asked for a description of an available set of matrix generators for G in characteristic p, although such a representation can be obtained by reduction modulo p of an integral matrix representation, which in turn can be constructed from any permutation representation.</p>
<p><a id="X81BF52FC7B8C08D4" name="X81BF52FC7B8C08D4"></a></p>
<h4>2.2 <span class="Heading">Group Names Used in the <strong class="pkg">AtlasRep</strong> Package</span></h4>
<p><a id="sect:groupnames"/> The <strong class="pkg">AtlasRep</strong> package refers to data of the <strong class="pkg">ATLAS</strong> of Group Representations by the <em>name</em> of the group in question plus additional information. Thus it is essential to know this name, which is called <em>the <strong class="pkg">GAP</strong> name</em> of the group in the following.</p>
<p>For an almost simple group, the <strong class="pkg">GAP</strong> name is equal to the <code class="func">Identifier</code> (<a href="../../../doc/htm/ref/CHAP069.htm#SECT008"><b>Reference: Identifier!for character tables</b></a>) value of the character table of this group in the <strong class="pkg">GAP</strong> library (see <code class="func">Access to Library Character Tables</code> (<a href="../../../pkg/ctbllib/htm/CHAP002.htm#SECT002"><b>CTblLib: Access to Library Character Tables</b></a>)); this name is usually very similar to the name used in the <strong class="pkg">ATLAS</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCNPW85]</a>. For example, <code class="code">"M22"</code> is the <strong class="pkg">GAP</strong> name of the Mathieu group M_22, and <code class="code">"12_1.U4(3).2_1"</code> is the <strong class="pkg">GAP</strong> name of 12_1.U_4(3).2_1.</p>
<p>Internally, for example as part of filenames (see Section <a href="chap5.html#X7D16BE31788F0E8A"><b>5.6</b></a>), the package uses names that may differ from the <strong class="pkg">GAP</strong> names; these names are called <em><strong class="pkg">ATLAS</strong>-file names</em>. For example, <code class="code">A5</code>, <code class="code">TE62</code>, and <code class="code">F24</code> are possible values for <var class="Arg">groupname</var>. Of these, only <code class="code">A5</code> is also a <strong class="pkg">GAP</strong> name, but the other two are not; the corresponding <strong class="pkg">GAP</strong> names are <code class="code">2E6(2)</code> and <code class="code">Fi24'</code>, respectively.</p>
<p><a id="X795DB7E486E0817D" name="X795DB7E486E0817D"></a></p>
<h4>2.3 <span class="Heading">Standard Generators Used in the <strong class="pkg">AtlasRep</strong> Package
</span></h4>
<p>For the general definition of <em>standard generators</em> of a group, see Section <a href="../../../doc/htm/ref/CHAP068.htm#SECT010"><b>Reference: Standard Generators of Groups</b></a>; details can be found in <a href="chapBib.html#biBWil96">[Wil96]</a>.</p>
<p>Several <em>different</em> standard generators may be defined for a group, the definitions can be found at</p>
<p><span class="URL"><a href="http://brauer.maths.qmul.ac.uk/Atlas">http://brauer.maths.qmul.ac.uk/Atlas</a></span></p>
<p>When one specifies the standardization, the i-th set of standard generators is denoted by the number i. Note that when more than one set of standard generators is defined for a group, one must be careful to use <em>compatible standardization</em>. For example, the straight line programs, straight line decisions and black box programs in the database refer to a specific standardization of their inputs. That is, a straight line program for computing generators of a certain subgroup of a group G is defined only for a specific set of standard generators of G, and applying the program to matrix or permutation generators of G but w.r.t. a different standardization may yield unpredictable results. Therefore the results returned by the functions described in this chapter contain information about the standardizations they refer to.</p>
<p><a id="X861CD545803B97E8" name="X861CD545803B97E8"></a></p>
<h4>2.4 <span class="Heading">Class Names Used in the <strong class="pkg">AtlasRep</strong> Package</span></h4>
<p>For each straight line program (see <code class="func">AtlasProgram</code> (<a href="chap2.html#X801F2E657C8A79ED"><b>2.5-3</b></a>)) that is used to compute lists of class representatives, it is essential to describe the classes in which these elements lie. Therefore, in these cases the records returned by the function <code class="func">AtlasProgram</code> (<a href="chap2.html#X801F2E657C8A79ED"><b>2.5-3</b></a>) contain a component <code class="code">outputs</code> with value a list of <em>class names</em>.</p>
<p>Currently we define these class names only for simple groups and automorphic extensions and central extensions of simple groups, see Section <a href="chap2.html#X850EEDEE831EE039"><b>2.4-1</b></a>. The function <code class="func">AtlasClassNames</code> (<a href="chap2.html#X78166D1D7D18EFBF"><b>2.4-2</b></a>) can be used to compute the list of class names from the character table in the <strong class="pkg">GAP</strong> Library.</p>
<p><a id="X850EEDEE831EE039" name="X850EEDEE831EE039"></a></p>
<h5>2.4-1 <span class="Heading">Definition of <strong class="pkg">ATLAS</strong> Class Names</span></h5>
<p>For the definition of class names of an almost simple group, we assume that the ordinary character tables of all nontrivial normal subgroups are shown in the <strong class="pkg">ATLAS</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCNPW85]</a>.</p>
<p>Each class name is a string consisting of the element order of the class in question followed by a combination of capital letters, digits, and the characters <code class="code">'</code> and <code class="code">-</code> (starting with a capital letter). For example, <code class="code">1A</code>, <code class="code">12A1</code>, and <code class="code">3B'</code> denote the class that contains the identity element, a class of element order 12, and a class of element order 3, respectively.</p>
<ol>
<li><p>For the table of a <em>simple</em> group, the class names are the same as returned by the two argument version of the <strong class="pkg">GAP</strong> function <code class="func">ClassNames</code> (<a href="../../../doc/htm/ref/CHAP069.htm#SECT008"><b>Reference: ClassNames</b></a>), cf. <a href="chapBib.html#biBCCN85">[CCNPW85, Chapter 7, Section 5]</a>: The classes are arranged w.r.t. increasing element order and for each element order w.r.t. decreasing centralizer order, the conjugacy classes that contain elements of order n are named n<code class="code">A</code>, n<code class="code">B</code>, n<code class="code">C</code>, ...; the alphabet used here is potentially infinite, and reads <code class="code">A</code>, <code class="code">B</code>, <code class="code">C</code>, ..., <code class="code">Z</code>, <code class="code">A1</code>, <code class="code">B1</code>, ..., <code class="code">A2</code>, <code class="code">B2</code>, ....</p>
<p>For example, the classes of the alternating group A_5 have the names <code class="code">1A</code>, <code class="code">2A</code>, <code class="code">3A</code>, <code class="code">5A</code>, and <code class="code">5B</code>.</p>
</li>
<li><p>Next we consider the case of an <em>upward extension</em> G.A of a simple group G by a <em>cyclic</em> group of order A. The <strong class="pkg">ATLAS</strong> defines class names for each element g of G.A only w.r.t. the group G.a, say, that is generated by G and g; namely, there is a power of g (with the exponent coprime to the order of g) for which the class has a name of the same form as the class names for simple groups, and the name of the class of g w.r.t. G.a is then obtained from this name by appending a suitable number of dashes <code class="code">'</code>. So dashed class names refer exactly to those classes that are not printed in the <strong class="pkg">ATLAS</strong>.</p>
<p>For example, those classes of the symmetric group S_5 that do not lie in A_5 have the names <code class="code">2B</code>, <code class="code">4A</code>, and <code class="code">6A</code>. The outer classes of the group L_2(8).3 have the names <code class="code">3B</code>, <code class="code">6A</code>, <code class="code">9D</code>, and <code class="code">3B'</code>, <code class="code">6A'</code>, <code class="code">9D'</code>. The outer elements of order 5 in the group Sz(32).5 lie in the classes with names <code class="code">5B</code>, <code class="code">5B'</code>, <code class="code">5B''</code>, and <code class="code">5B'''</code>.</p>
<p>In the group G.A, the class of g may fuse with other classes. The name of the class of g in G.A is obtained from the names of the involved classes of G.a by concatenating their names after removing the element order part from all of them except the first one.</p>
<p>For example, the elements of order 9 in the group L_2(27).6 are contained in the subgroup L_2(27).3 but not in L_2(27). In L_2(27).3, they lie in the classes <code class="code">9A</code>, <code class="code">9A'</code>, <code class="code">9B</code>, and <code class="code">9B'</code>; in L_2(27).6, these classes fuse to <code class="code">9AB</code> and <code class="code">9A'B'</code>.</p>
</li>
<li><p>Now we define class names for <em>general upward extensions</em> G.A of a simple group G. Each element g of such a group lies in an upward extension G.a by a cyclic group, and the class names w.r.t. G.a are already defined. The name of the class of g in G.A is obtained by concatenating the names of the classes in the orbit of G.A on the classes of cyclic upward extensions of G, after ordering the names lexicographically and removing the element order part from all of them except the first one. An <em>exception</em> is the situation where dashed and non-dashed class names appear in an orbit; in this case, the dashed names are omitted.</p>
<p>For example, the classes <code class="code">21A</code> and <code class="code">21B</code> of the group U_3(5).3 fuse in U_3(5).S_3 to the class <code class="code">21AB</code>, and the class <code class="code">2B</code> of U_3(5).2 fuses with the involution classes <code class="code">2B'</code>, <code class="code">2B''</code> in the groups U_3(5).2^' and U_3(5).2^{''} to the class <code class="code">2B</code> of U_3(5).S_3.</p>
<p>It may happen that some names in the <code class="code">outputs</code> component of a record returned by <code class="func">AtlasProgram</code> (<a href="chap2.html#X801F2E657C8A79ED"><b>2.5-3</b></a>) do not uniquely determine the classes of the corresponding elements. For example, the (algebraically conjugate) classes <code class="code">39A</code> and <code class="code">39B</code> of the group Co_1 have not been distinguished yet. In such cases, the names used contain a minus sign <code class="code">-</code>, and mean "one of the classes in the range described by the name before and the name after the minus sign"; the element order part of the name does not appear after the minus sign. So the name <code class="code">39A-B</code> for the group Co_1 means <code class="code">39A</code> or <code class="code">39B</code>, and the name <code class="code">20A-B'''</code> for the group Sz(32).5 means one of the classes of element order 20 in this group (these classes lie outside the simple group Sz).</p>
</li>
<li><p>For a central <em>downward extension</em> m.G of a simple group G by a cyclic group of order m, let pi denote the natural epimorphism from m.G onto G. Each class name of m.G has the form <code class="code">nX_0</code>, <code class="code">nX_1</code> etc., where <code class="code">nX</code> is the class name of the image under pi, and the indices <code class="code">0</code>, <code class="code">1</code> etc. are chosen according to the position of the class in the lifting order rows for G, see <a href="chapBib.html#biBCCN85">[CCNPW85, Chapter 7, Section 7, and the example in Section 8]</a>).</p>
<p>For example, if m = 6 then <code class="code">1A_1</code> and <code class="code">1A_5</code> denote the classes containing the generators of the kernel of pi, that is, central elements of order 6.</p>
</li>
</ol>
<p><a id="X78166D1D7D18EFBF" name="X78166D1D7D18EFBF"></a></p>
<h5>2.4-2 AtlasClassNames</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AtlasClassNames</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b>a list of class names.</p>
<p>Let <var class="Arg">tbl</var> be the ordinary character table of a group G that is simple or an automorphic or a central extension of a simple group and such that <var class="Arg">tbl</var> is an <strong class="pkg">ATLAS</strong> table from the <strong class="pkg">GAP</strong> Character Table Library, according to its <code class="func">InfoText</code> (<a href="../../../doc/htm/ref/CHAP069.htm#SECT008"><b>Reference: InfoText</b></a>) value. Then <code class="func">AtlasClassNames</code> returns the list of class names for G, as defined in Section <a href="chap2.html#X850EEDEE831EE039"><b>2.4-1</b></a>. The ordering of class names is the same as the ordering of the columns of <var class="Arg">tbl</var>.</p>
<p>(The function may work also for character tables that are not <strong class="pkg">ATLAS</strong> tables, but then clearly the class names returned are somewhat arbitrary.)</p>
<table class="example">
<tr><td><pre>
gap> AtlasClassNames( CharacterTable( "L3(4).3" ) );
[ "1A", "2A", "3A", "4ABC", "5A", "5B", "7A", "7B", "3B", "3B'", "3C", "3C'",
"6B", "6B'", "15A", "15A'", "15B", "15B'", "21A", "21A'", "21B", "21B'" ]
gap> AtlasClassNames( CharacterTable( "U3(5).2" ) );
[ "1A", "2A", "3A", "4A", "5A", "5B", "5CD", "6A", "7AB", "8AB", "10A", "2B",
"4B", "6D", "8C", "10B", "12B", "20A", "20B" ]
gap> AtlasClassNames( CharacterTable( "L2(27).6" ) );
[ "1A", "2A", "3AB", "7ABC", "13ABC", "13DEF", "14ABC", "2B", "4A", "26ABC",
"26DEF", "28ABC", "28DEF", "3C", "3C'", "6A", "6A'", "9AB", "9A'B'", "6B",
"6B'", "12A", "12A'" ]
gap> AtlasClassNames( CharacterTable( "L3(4).3.2_2" ) );
[ "1A", "2A", "3A", "4ABC", "5AB", "7A", "7B", "3B", "3C", "6B", "15A",
"15B", "21A", "21B", "2C", "4E", "6E", "8D", "14A", "14B" ]
gap> AtlasClassNames( CharacterTable( "3.A6" ) );
[ "1A_0", "1A_1", "1A_2", "2A_0", "2A_1", "2A_2", "3A_0", "3B_0", "4A_0",
"4A_1", "4A_2", "5A_0", "5A_1", "5A_2", "5B_0", "5B_1", "5B_2" ]
</pre></td></tr></table>
<p><a id="X806ED7BB83FD848B" name="X806ED7BB83FD848B"></a></p>
<h4>2.5 <span class="Heading">Accessing Data of the <strong class="pkg">AtlasRep</strong> Package</span></h4>
<p>(Note that the output of the examples in this section refers to a perhaps outdated table of contents; the current version of the database may contain more information than is shown here.)</p>
<p><a id="X79DACFFA7E2D1A99" name="X79DACFFA7E2D1A99"></a></p>
<h5>2.5-1 DisplayAtlasInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> DisplayAtlasInfo</code>( <var class="Arg"></var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> DisplayAtlasInfo</code>( <var class="Arg">listofnames</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> DisplayAtlasInfo</code>( <var class="Arg">gapname[, std][, ...]</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function lists the information available via the <strong class="pkg">AtlasRep</strong> package, for the given input. Depending on whether remote access to data is enabled (see Section <a href="chap1.html#X7E1934D2780D108F"><b>1.7-1</b></a>), all the data provided by the <strong class="pkg">ATLAS</strong> of Group Representations or only those in the local installation are considered.</p>
<p>(An interactive alternative to <code class="func">DisplayAtlasInfo</code> is the function <code class="func">BrowseAtlasInfo</code> (<a href="../../../pkg/Browse/doc/chap6.html#X8411AF928194C5AB"><b>Browse: BrowseAtlasInfo</b></a>), see <a href="chapBib.html#biBBrowse1.2">[BL08]</a>; this function provides also the functionality of <code class="func">AtlasGenerators</code> (<a href="chap2.html#X7D1CCCF8852DFF39"><b>2.5-2</b></a>).)</p>
<p>Called without arguments, <code class="func">DisplayAtlasInfo</code> prints an overview what information the <strong class="pkg">ATLAS</strong> of Group Representations provides. One line is printed for each group G, with the following columns.</p>
<dl>
<dt><strong class="Mark"><code class="code">group</code></strong></dt>
<dd><p>the <strong class="pkg">GAP</strong> name of G (see Section <a href="chap2.html#X81BF52FC7B8C08D4"><b>2.2</b></a>),</p>
</dd>
<dt><strong class="Mark"><code class="code">#</code></strong></dt>
<dd><p>the number of representations stored for G,</p>
</dd>
<dt><strong class="Mark"><code class="code">maxes</code></strong></dt>
<dd><p>the available straight line programs for computing generators of maximal subgroups of G,</p>
</dd>
<dt><strong class="Mark"><code class="code">cl</code></strong></dt>
<dd><p>a <code class="code">+</code> sign if at least one program for computing representatives of conjugacy classes of elements of G is stored, and a <code class="code">-</code> sign otherwise,</p>
</dd>
<dt><strong class="Mark"><code class="code">cyc</code></strong></dt>
<dd><p>a <code class="code">+</code> sign if at least one program for computing representatives of classes of maximally cyclic subgroups of G is stored, and a <code class="code">-</code> sign otherwise,</p>
</dd>
<dt><strong class="Mark"><code class="code">out</code></strong></dt>
<dd><p>descriptions of outer automorphisms of G for which at least one program is stored,</p>
</dd>
<dt><strong class="Mark"><code class="code">check</code></strong></dt>
<dd><p>a <code class="code">+</code> sign if at least one program is available for checking whether a set of generators is a set of standard generators, and a <code class="code">-</code> sign otherwise,</p>
</dd>
<dt><strong class="Mark"><code class="code">pres</code></strong></dt>
<dd><p>a <code class="code">+</code> sign if at least one program is available that encodes a presentation, and a <code class="code">-</code> sign otherwise,</p>
</dd>
<dt><strong class="Mark"><code class="code">find</code></strong></dt>
<dd><p>a <code class="code">+</code> sign if at least one program is available for finding standard generators, and a <code class="code">-</code> sign otherwise,</p>
</dd>
</dl>
<p>Called with a list <var class="Arg">listofnames</var> of strings that are <strong class="pkg">GAP</strong> names for a group from the <strong class="pkg">ATLAS</strong> of Group Representations, <code class="func">DisplayAtlasInfo</code> prints the overview described above but restricted to the groups in this list.</p>
<p>Called with a string <var class="Arg">gapname</var> that is a <strong class="pkg">GAP</strong> name for a group from the <strong class="pkg">ATLAS</strong> of Group Representations, <code class="func">DisplayAtlasInfo</code> prints an overview of the information that is available for this group. One line is printed for each representation, showing the number of this representation (which can be used in calls of <code class="func">AtlasGenerators</code> (<a href="chap2.html#X7D1CCCF8852DFF39"><b>2.5-2</b></a>)), and a string of one of the following forms; in both cases, <var class="Arg">id</var> is a (possibly empty) string.</p>
<dl>
<dt><strong class="Mark"><code class="code">G <= Sym(<var class="Arg">n</var><var class="Arg">id</var>)</code></strong></dt>
<dd><p>denotes a permutation representation of degree <var class="Arg">n</var>, for example <code class="code">G <= Sym(40a)</code> and <code class="code">G <= Sym(40b)</code> denote two (nonequivalent) representations of degree 40.</p>
</dd>
<dt><strong class="Mark"><code class="code">G <= GL(<var class="Arg">n</var><var class="Arg">id</var>,<var class="Arg">descr</var>)</code></strong></dt>
<dd><p>denotes a matrix representation of dimension <var class="Arg">n</var> over a coefficient ring described by <var class="Arg">descr</var>, which can be a prime power, <code class="code">Z</code> (denoting the ring of integers), a description of an algebraic extension field, <code class="code">C</code> (denoting an unspecified algebraic extension field), or <code class="code">Z/<var class="Arg">m</var>Z</code> for an integer <var class="Arg">m</var> (denoting the ring of residues mod <var class="Arg">m</var>); for example, <code class="code">G <= GL(2a,4)</code> and <code class="code">G <= GL(2b,4)</code> denote two (nonequivalent) representations of dimension 2 over the field with four elements.</p>
</dd>
</dl>
<p>After the representations, the programs available for <var class="Arg">gapname</var> are listed.</p>
<p>If the first argument is a string <var class="Arg">gapname</var>, the following optional arguments can be used to restrict the overview.</p>
<dl>
<dt><strong class="Mark"><var class="Arg">std</var></strong></dt>
<dd><p>must be a positive integer or a list of positive integers; if it is given then only those representations are considered that refer to the <var class="Arg">std</var>-th set of standard generators or the i-th set of standard generators, for i in <var class="Arg">std</var> (see Section <a href="chap2.html#X795DB7E486E0817D"><b>2.3</b></a>),</p>
</dd>
<dt><strong class="Mark"><code class="code">IsPermGroup</code> and <code class="keyw">true</code></strong></dt>
<dd><p>restrict to permutation representations,</p>
</dd>
<dt><strong class="Mark"><code class="code">NrMovedPoints</code> and <var class="Arg">n</var></strong></dt>
<dd><p>for a positive integer, a list of positive integers, or a property <var class="Arg">n</var>, restrict to permutation representations of degree equal to <var class="Arg">n</var>, or in the list <var class="Arg">n</var>, or satisfying the function <var class="Arg">n</var>,</p>
</dd>
<dt><strong class="Mark"><code class="code">NrMovedPoints</code> and the string <code class="code">"minimal"</code></strong></dt>
<dd><p>restrict to faithful permutation representations of minimal degree (if this information is available),</p>
</dd>
<dt><strong class="Mark"><code class="code">IsMatrixGroup</code> and <code class="keyw">true</code></strong></dt>
<dd><p>restrict to matrix representations,</p>
</dd>
<dt><strong class="Mark"><code class="code">Characteristic</code> and <var class="Arg">p</var></strong></dt>
<dd><p>for a prime integer, a list of prime integers, or a property <var class="Arg">p</var>, restrict to matrix representations over fields of characteristic equal to <var class="Arg">p</var>, or in the list <var class="Arg">p</var>, or satisfying the function <var class="Arg">p</var> (representations over residue class rings that are not fields can be addressed by entering <code class="keyw">fail</code> as the value of <var class="Arg">p</var>),</p>
</dd>
<dt><strong class="Mark"><code class="code">Dimension</code> and <var class="Arg">n</var></strong></dt>
<dd><p>for a positive integer, a list of positive integers, or a property <var class="Arg">n</var>, restrict to matrix representations of dimension equal to <var class="Arg">n</var>, or in the list <var class="Arg">n</var>, or satisfying the function <var class="Arg">n</var>,</p>
</dd>
<dt><strong class="Mark"><code class="code">Characteristic</code>, <var class="Arg">p</var>, <code class="code">Dimension</code>,
and the string <code class="code">"minimal"</code></strong></dt>
<dd><p>for a prime integer <var class="Arg">p</var>, restrict to faithful matrix representations over fields of characteristic <var class="Arg">p</var> that have minimal dimension (if this information is available),</p>
</dd>
<dt><strong class="Mark"><code class="code">Ring</code> and <var class="Arg">R</var></strong></dt>
<dd><p>for a ring or a property <var class="Arg">R</var>, restrict to matrix representations over this ring or satisfying this function (note that the representation might be defined over a proper subring of <var class="Arg">R</var>), and</p>
</dd>
<dt><strong class="Mark"><code class="code">Ring</code>, <var class="Arg">R</var>, <code class="code">Dimension</code>,
and the string <code class="code">"minimal"</code></strong></dt>
<dd><p>for a ring <var class="Arg">R</var>, restrict to faithful matrix representations over this ring that have minimal dimension (if this information is available),</p>
</dd>
<dt><strong class="Mark"><code class="code">IsStraightLineProgram</code></strong></dt>
<dd><p>restricts to straight line programs, straight line decisions (see Section <a href="chap4.html#X8121E9567A7137C9"><b>4.1</b></a>), and black box programs (see Section <a href="chap4.html#X7BE856BC785A9E8F"><b>4.2</b></a>).</p>
</dd>
</dl>
<p>If "minimality" information is requested and no available representation matches this condition then either no minimal representation is available or the information about the minimality is missing. See <code class="func">MinimalRepresentationInfo</code> (<a href="chap4.html#X7DC66D8282B2BB7F"><b>4.3-1</b></a>) for checking whether the minimality information is available for the group in question. Note that in the cases where the string <code class="code">"minimal"</code> occurs as an argument, <code class="func">MinimalRepresentationInfo</code> (<a href="chap4.html#X7DC66D8282B2BB7F"><b>4.3-1</b></a>) is called with third argument <code class="code">"lookup"</code>; this is because the stored information was computed just for the groups in the <strong class="pkg">ATLAS</strong> of Group Representations, so trying to compute non-stored minimality information (using other available databases) will hardly be successful.</p>
<p>The representations are ordered as follows. Permutation representations come first (ordered according to their degrees), followed by matrix representations over finite fields (ordered first according to the field size and second according to the dimension), matrix representations over the integers, and then matrix representations over algebraic extension fields (both kinds ordered according to the dimension), the last representations are matrix representations over residue class rings (ordered first according to the modulus and second according to the dimension).</p>
<p>The maximal subgroups are ordered according to decreasing group order. For an extension G.p of a simple group G by an outer automorphism of prime order p, this means that G is the first maximal subgroup and then come the extensions of the maximal subgroups of G and the novelties; so the n-th maximal subgroup of G and the n-th maximal subgroup of G.p are in general not related. (This coincides with the numbering used for the <code class="func">Maxes</code> (<a href="../../../pkg/ctbllib/htm/CHAP002.htm#SECT002"><b>CTblLib: Maxes</b></a>) attribute for character tables.)</p>
<table class="example">
<tr><td><pre>
gap> DisplayAtlasInfo( [ "M11", "A5" ] );
group # maxes cl cyc out find check pres
---------------------------------------------------
M11 42 5 + + + + +
A5 18 3 - - - + +
</pre></td></tr></table>
<p>The above output means that the <strong class="pkg">ATLAS</strong> of Group Representations contains 42 representations of the Mathieu group M_11, straight line programs for computing generators of representatives of all five classes of maximal subgroups, for computing representatives of the conjugacy classes of elements and of generators of maximally cyclic subgroups, contains no straight line program for applying outer automorphisms (well, in fact M_11 admits no nontrivial outer automorphism), and contains a straight line decision that checks generators for being standard generators. Analogously, 18 representations of the alternating group A_5 are available, straight line programs for computing generators of representatives of all three classes of maximal subgroups, and no straight line programs for computing representatives of the conjugacy classes of elements, of generators of maximally cyclic subgroups, and no for computing images under outer automorphisms; a straight line decision for checking the standardization of generators is contained.</p>
<table class="example">
<tr><td><pre>
gap> DisplayAtlasInfo( "A5", IsPermGroup, true );
Representations for G = A5: (all refer to std. generators 1)
---------------------------
1: G <= Sym(5)
2: G <= Sym(6)
3: G <= Sym(10)
gap> DisplayAtlasInfo( "A5", NrMovedPoints, [ 4 .. 9 ] );
Representations for G = A5: (all refer to std. generators 1)
---------------------------
1: G <= Sym(5)
2: G <= Sym(6)
</pre></td></tr></table>
<p>The first three representations stored for A_5 are (in fact primitive) permutation representations.</p>
<table class="example">
<tr><td><pre>
gap> DisplayAtlasInfo( "A5", Dimension, [ 1 .. 3 ] );
Representations for G = A5: (all refer to std. generators 1)
---------------------------
8: G <= GL(2a,4)
9: G <= GL(2b,4)
10: G <= GL(3,5)
12: G <= GL(3a,9)
13: G <= GL(3b,9)
17: G <= GL(3a,Field([Sqrt(5)]))
18: G <= GL(3b,Field([Sqrt(5)]))
gap> DisplayAtlasInfo( "A5", Characteristic, 0 );
Representations for G = A5: (all refer to std. generators 1)
---------------------------
14: G <= GL(4,Z)
15: G <= GL(5,Z)
16: G <= GL(6,Z)
17: G <= GL(3a,Field([Sqrt(5)]))
18: G <= GL(3b,Field([Sqrt(5)]))
</pre></td></tr></table>
<p>The representations with number between 4 and 13 are (in fact irreducible) matrix representations over various finite fields, those with numbers 14 to 16 are integral matrix representations, and the last two are matrix representations over the field generated by sqrt{5} over the rational number field.</p>
<table class="example">
<tr><td><pre>
gap> DisplayAtlasInfo( "A5", NrMovedPoints, IsPrimeInt );
Representations for G = A5: (all refer to std. generators 1)
---------------------------
1: G <= Sym(5)
gap> DisplayAtlasInfo( "A5", Characteristic, IsOddInt );
Representations for G = A5: (all refer to std. generators 1)
---------------------------
6: G <= GL(4,3)
7: G <= GL(6,3)
10: G <= GL(3,5)
11: G <= GL(5,5)
12: G <= GL(3a,9)
13: G <= GL(3b,9)
gap> DisplayAtlasInfo( "A5", Dimension, IsPrimeInt );
Representations for G = A5: (all refer to std. generators 1)
---------------------------
8: G <= GL(2a,4)
9: G <= GL(2b,4)
10: G <= GL(3,5)
11: G <= GL(5,5)
12: G <= GL(3a,9)
13: G <= GL(3b,9)
15: G <= GL(5,Z)
17: G <= GL(3a,Field([Sqrt(5)]))
18: G <= GL(3b,Field([Sqrt(5)]))
gap> DisplayAtlasInfo( "A5", Ring, IsFinite and IsPrimeField );
Representations for G = A5: (all refer to std. generators 1)
---------------------------
4: G <= GL(4a,2)
5: G <= GL(4b,2)
6: G <= GL(4,3)
7: G <= GL(6,3)
10: G <= GL(3,5)
11: G <= GL(5,5)
</pre></td></tr></table>
<p>The above examples show how the output can be restricted using a property (a unary function that returns either <code class="keyw">true</code> or <code class="keyw">false</code>) that follows <code class="func">NrMovedPoints</code> (<a href="../../../doc/htm/ref/CHAP040.htm#SECT002"><b>Reference: NrMovedPoints</b></a>), <code class="func">Characteristic</code> (<a href="../../../doc/htm/ref/CHAP030.htm#SECT010"><b>Reference: Characteristic</b></a>), <code class="func">Dimension</code> (<a href="../../../doc/htm/ref/CHAP055.htm#SECT003"><b>Reference: Dimension</b></a>), or <code class="func">Ring</code> (<a href="../../../doc/htm/ref/CHAP054.htm#SECT001"><b>Reference: Ring</b></a>) in the argument list of <code class="func">DisplayAtlasInfo</code>.</p>
<table class="example">
<tr><td><pre>
gap> DisplayAtlasInfo( "A5", IsStraightLineProgram, true );
Programs for G = A5: (all refer to std. generators 1)
--------------------
available maxes of G: [ 1 .. 3 ] (all)
standard generators checker available
presentation available
</pre></td></tr></table>
<p>Straight line programs are available for computing generators of representatives of the three classes of maximal subgroups of A_5, and a straight line decision for checking whether given generators are in fact standard generators ia available as well as a presentation in terms of standard generators, see <code class="func">AtlasProgram</code> (<a href="chap2.html#X801F2E657C8A79ED"><b>2.5-3</b></a>).</p>
<p><a id="X7D1CCCF8852DFF39" name="X7D1CCCF8852DFF39"></a></p>
<h5>2.5-2 AtlasGenerators</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AtlasGenerators</code>( <var class="Arg">gapname, repnr[, maxnr]</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AtlasGenerators</code>( <var class="Arg">identifier</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b>a record containing generators for a representation, or <code class="keyw">fail</code>.</p>
<p>In the first form, <var class="Arg">gapname</var> must be a string denoting a <strong class="pkg">GAP</strong> name (see Section <a href="chap2.html#X81BF52FC7B8C08D4"><b>2.2</b></a>) of a group, and <var class="Arg">repnr</var> a positive integer. If the <strong class="pkg">ATLAS</strong> of Group Representations contains at least <var class="Arg">repnr</var> representations for the group with <strong class="pkg">GAP</strong> name <var class="Arg">gapname</var> then <code class="func">AtlasGenerators</code>, when called with <var class="Arg">gapname</var> and <var class="Arg">repnr</var>, returns an immutable record describing the <var class="Arg">repnr</var>-th representation; otherwise <code class="keyw">fail</code> is returned. If a third argument <var class="Arg">maxnr</var>, a positive integer, is given then an immutable record describing the restriction of the <var class="Arg">repnr</var>-th representation to the <var class="Arg">maxnr</var>-th maximal subgroup is returned.</p>
<p>The result record has the following components.</p>
<dl>
<dt><strong class="Mark"><code class="code">groupname</code></strong></dt>
<dd><p>the <strong class="pkg">GAP</strong> name of the group (see Section <a href="chap2.html#X81BF52FC7B8C08D4"><b>2.2</b></a>),</p>
</dd>
<dt><strong class="Mark"><code class="code">generators</code></strong></dt>
<dd><p>a list of generators for the group,</p>
</dd>
<dt><strong class="Mark"><code class="code">standardization</code></strong></dt>
<dd><p>the positive integer denoting the underlying standard generators,</p>
</dd>
<dt><strong class="Mark"><code class="code">size</code> (only if known)</strong></dt>
<dd><p>the order of the group,</p>
</dd>
<dt><strong class="Mark"><code class="code">identifier</code></strong></dt>
<dd><p>a <strong class="pkg">GAP</strong> object (a list of filenames plus additional information) that uniquely determines the representation; the value can be used as <var class="Arg">identifier</var> argument of <code class="func">AtlasGenerators</code>.</p>
</dd>
<dt><strong class="Mark"><code class="code">repnr</code></strong></dt>
<dd><p>the number of the representation in the current session, equal to the argument <var class="Arg">repnr</var> if this is given.</p>
</dd>
</dl>
<p>Additionally, there are describing components dependent on the data type of the representation: For permutation representations, these are <code class="code">p</code> for the number of moved points and <code class="code">id</code> for the distinguishing string as described for <code class="func">DisplayAtlasInfo</code> (<a href="chap2.html#X79DACFFA7E2D1A99"><b>2.5-1</b></a>); for matrix representations, these are <code class="code">dim</code> for the dimension of the matrices, <code class="code">ring</code> (if known) for the ring generated by the matrix entries, and <code class="code">id</code> for the distinguishing string.</p>
<p>It should be noted that the number <var class="Arg">repnr</var> refers to the number shown by <code class="func">DisplayAtlasInfo</code> (<a href="chap2.html#X79DACFFA7E2D1A99"><b>2.5-1</b></a>) <em>in the current session</em>; it may be that after the addition of new representations, <var class="Arg">repnr</var> refers to another representation.</p>
<p>The alternative form of <code class="func">AtlasGenerators</code>, with only argument <var class="Arg">identifier</var>, can be used to fetch the result record with <code class="code">identifier</code> value equal to <var class="Arg">identifier</var>. The purpose of this variant is to access the <em>same</em> representation also in <em>different</em> <strong class="pkg">GAP</strong> sessions.</p>
<table class="example">
<tr><td><pre>
gap> gens1:= AtlasGenerators( "A5", 1 );
rec( generators := [ (1,2)(3,4), (1,3,5) ], groupname := "A5",
standardization := 1, repnr := 1,
identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ], p := 5,
id := "", size := 60 )
gap> gens8:= AtlasGenerators( "A5", 8 );
rec(
generators := [ [ [ Z(2)^0, 0*Z(2) ], [ Z(2^2), Z(2)^0 ] ], [ [ 0*Z(2), Z(2
)^0 ], [ Z(2)^0, Z(2)^0 ] ] ], groupname := "A5",
standardization := 1, repnr := 8,
identifier := [ "A5", [ "A5G1-f4r2aB0.m1", "A5G1-f4r2aB0.m2" ], 1, 4 ],
dim := 2, id := "a", ring := GF(2^2), size := 60 )
gap> gens17:= AtlasGenerators( "A5", 17 );
rec(
generators := [ [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ -E(5)-E(5)^4, -E(5)-E(5)^4,
1 ] ], [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ],
groupname := "A5", standardization := 1, repnr := 17,
identifier := [ "A5", "A5G1-Ar3aB0.g", 1, 3 ], dim := 3, id := "a",
ring := NF(5,[ 1, 4 ]), size := 60 )
</pre></td></tr></table>
<p>Each of the above pairs of elements generates a group isomorphic to A_5.</p>
<table class="example">
<tr><td><pre>
gap> gens1max2:= AtlasGenerators( "A5", 1, 2 );
rec( generators := [ (1,2)(3,4), (2,3)(4,5) ], standardization := 1,
repnr := 1, identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5,
2 ], p := 5, id := "", size := 10 )
gap> id:= gens1max2.identifier;;
gap> gens1max2 = AtlasGenerators( id );
true
gap> max2:= Group( gens1max2.generators );;
gap> Size( max2 );
10
gap> IdGroup( max2 ) = IdGroup( DihedralGroup( 10 ) );
true
</pre></td></tr></table>
<p>The elements stored in <code class="code">gens1max2.generators</code> describe the restriction of the first representation of A_5 to a group in the second class of maximal subgroups of A_5 according to the list in the <strong class="pkg">ATLAS</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCNPW85]</a>; this subgroup is isomorphic to the dihedral group D_10.</p>
<p><a id="X801F2E657C8A79ED" name="X801F2E657C8A79ED"></a></p>
<h5>2.5-3 AtlasProgram</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AtlasProgram</code>( <var class="Arg">gapname[, std], ...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AtlasProgram</code>( <var class="Arg">identifier</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b>a record containing a program, or <code class="keyw">fail</code>.</p>
<p>In the first form, <var class="Arg">gapname</var> must be a string denoting a <strong class="pkg">GAP</strong> name (see Section <a href="chap2.html#X81BF52FC7B8C08D4"><b>2.2</b></a>) of a group G, say. If the <strong class="pkg">ATLAS</strong> of Group Representations contains a straight line program (see Section <a href="../../../doc/htm/ref/CHAP035.htm#SECT008"><b>Reference: Straight Line Programs</b></a>) or straight line decision (see Section <a href="chap4.html#X8121E9567A7137C9"><b>4.1</b></a>) or black box program (see Section <a href="chap4.html#X7BE856BC785A9E8F"><b>4.2</b></a>) as described by the remaining arguments (see below) then <code class="func">AtlasProgram</code> returns an immutable record containing this program. Otherwise <code class="keyw">fail</code> is returned.</p>
<p>If the optional argument <var class="Arg">std</var> is given, only those straight line programs/decisions are considered that take generators from the <var class="Arg">std</var>-th set of standard generators of G as input, see Section <a href="chap2.html#X795DB7E486E0817D"><b>2.3</b></a>.</p>
<p>The result record has the following components.</p>
<dl>
<dt><strong class="Mark"><code class="code">program</code></strong></dt>
<dd><p>the required straight line program/decision, or black box program,</p>
</dd>
<dt><strong class="Mark"><code class="code">standardization</code></strong></dt>
<dd><p>the positive integer denoting the underlying standard generators of G,</p>
</dd>
<dt><strong class="Mark"><code class="code">identifier</code></strong></dt>
<dd><p>a <strong class="pkg">GAP</strong> object (a list of filenames plus additional information) that uniquely determines the program; the value can be used as <var class="Arg">identifier</var> argument of <code class="func">AtlasProgram</code> (see below).</p>
</dd>
</dl>
<p>In the first form, the last arguments must be as follows.</p>
<dl>
<dt><strong class="Mark">(the string <code class="code">"maxes"</code> and) a positive integer <var class="Arg">maxnr</var>
</strong></dt>
<dd><p>the required program computes generators of the <var class="Arg">maxnr</var>-th maximal subgroup of the group with <strong class="pkg">GAP</strong> name <var class="Arg">gapname</var>.</p>
<p>In this case, the result record of <code class="func">AtlasProgram</code> also may contain a component <code class="code">size</code>, whose value is the order of the maximal subgroup in question.</p>
</dd>
<dt><strong class="Mark">one of the strings <code class="code">"classes"</code> or <code class="code">"cyclic"</code></strong></dt>
<dd><p>the required program computes representatives of conjugacy classes of elements or representatives of generators of maximally cyclic subgroups of G, respectively.</p>
<p>See <a href="chapBib.html#biBBSW01">[BSWW01]</a> and <a href="chapBib.html#biBSWW00">[SWW00]</a> for the background concerning these straight line programs. In these cases, the result record of <code class="func">AtlasProgram</code> also contains a component <code class="code">outputs</code>, whose value is a list of class names of the outputs, as described in Section <a href="chap2.html#X861CD545803B97E8"><b>2.4</b></a>.</p>
</dd>
<dt><strong class="Mark">the strings <code class="code">"automorphism"</code> and <var class="Arg">autname</var></strong></dt>
<dd><p>the required program computes images of standard generators under the outer automorphism of G that is given by this string.</p>
</dd>
<dt><strong class="Mark">the string <code class="code">"check"</code></strong></dt>
<dd><p>the required result is a straight line decision that takes a list of generators for G and returns <code class="keyw">true</code> if these generators are standard generators w.r.t. the standardization <var class="Arg">std</var>, and <code class="keyw">false</code> otherwise.</p>
</dd>
<dt><strong class="Mark">the string <code class="code">"presentation"</code></strong></dt>
<dd><p>the required result is a straight line decision that takes a list of group elements and returns <code class="keyw">true</code> if these elements are standard generators of G w.r.t. the standardization <var class="Arg">std</var>, and <code class="keyw">false</code> otherwise.</p>
</dd>
<dt><strong class="Mark">the string <code class="code">"find"</code></strong></dt>
<dd><p>the required result is a black box program that takes G and returns a list of standard generators of G, w.r.t. the standardization <var class="Arg">std</var>.</p>
</dd>
<dt><strong class="Mark">the string <code class="code">"restandardize"</code> and an integer <var class="Arg">std2</var></strong></dt>
<dd><p>the required result is a straight line program that computes standard generators of G w.r.t. the <var class="Arg">std2</var>-th set of standard generators of G; in this case, the argument <var class="Arg">std</var> must be given.</p>
</dd>
<dt><strong class="Mark">the strings <code class="code">"other"</code> and <var class="Arg">descr</var></strong></dt>
<dd><p>the required program is described by <var class="Arg">descr</var>.</p>
</dd>
</dl>
<p>The second form of <code class="func">AtlasProgram</code>, with only argument the list <var class="Arg">identifier</var>, can be used to fetch the result record with <code class="code">identifier</code> value equal to <var class="Arg">identifier</var>.</p>
<table class="example">
<tr><td><pre>
gap> prog:= AtlasProgram( "A5", 2 );
rec( program := <straight line program>, standardization := 1,
identifier := [ "A5", "A5G1-max2W1", 1 ], size := 10, groupname := "A5" )
gap> StringOfResultOfStraightLineProgram( prog.program, [ "a", "b" ] );
"[ a, bbab ]"
gap> gens1:= AtlasGenerators( "A5", 1 );
rec( generators := [ (1,2)(3,4), (1,3,5) ], groupname := "A5",
standardization := 1, repnr := 1,
identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ], p := 5,
id := "", size := 60 )
gap> maxgens:= ResultOfStraightLineProgram( prog.program, gens1.generators );
[ (1,2)(3,4), (2,3)(4,5) ]
gap> maxgens = gens1max2.generators;
true
</pre></td></tr></table>
<p>The above example shows that for restricting representations given by standard generators to a maximal subgroup of A_5, we can also fetch and apply the appropriate straight line program. Such a program (see <a href="../../../doc/htm/ref/CHAP035.htm#SECT008"><b>Reference: Straight Line Programs</b></a>) takes standard generators of a group --in this example A_5-- as its input, and returns a list of elements in this group --in this example generators of the D_10 subgroup we had met above-- which are computed essentially by evaluating structured words in terms of the standard generators.</p>
<table class="example">
<tr><td><pre>
gap> prog:= AtlasProgram( "J1", "cyclic" );
rec( program := <straight line program>, standardization := 1,
identifier := [ "J1", "J1G1-cycW1", 1 ],
outputs := [ "6A", "7A", "10B", "11A", "15B", "19A" ], groupname := "J1" )
gap> gens:= GeneratorsOfGroup( FreeGroup( "x", "y" ) );;
gap> ResultOfStraightLineProgram( prog.program, gens );
[ x*y*x*y^2*x*y*x*y^2*x*y*x*y*x*y^2*x*y^2, x*y, x*y*x*y^2*x*y*x*y*x*y^2*x*y^2,
x*y*x*y*x*y^2*x*y^2*x*y*x*y^2*x*y*x*y*x*y^2*x*y^2*x*y*x*y^2*x*y*x*y*x*y^
2*x*y^2, x*y*x*y*x*y^2*x*y^2, x*y*x*y^2 ]
</pre></td></tr></table>
<p>The above example shows how to fetch and use straight line programs for computing generators of representatives of maximally cyclic subgroups of a given group.</p>
<p><a id="X841478AB7CD06D44" name="X841478AB7CD06D44"></a></p>
<h5>2.5-4 OneAtlasGeneratingSetInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> OneAtlasGeneratingSetInfo</code>( <var class="Arg">[gapname, ][std, ][...]</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b>a record describing a representation that satisfies the conditions, or <code class="keyw">fail</code>.</p>
<p>Let <var class="Arg">gapname</var> be a string denoting a <strong class="pkg">GAP</strong> name (see Section <a href="chap2.html#X81BF52FC7B8C08D4"><b>2.2</b></a>) of a group G, say. If the <strong class="pkg">ATLAS</strong> of Group Representations contains at least one representation for G with the required properties then <code class="func">OneAtlasGeneratingSetInfo</code> returns a record <var class="Arg">r</var> whose components are the same as those of the records returned by <code class="func">AtlasGenerators</code> (<a href="chap2.html#X7D1CCCF8852DFF39"><b>2.5-2</b></a>), except that the component <code class="code">generators</code> is not contained; the component <code class="code">identifier</code> of <var class="Arg">r</var> can be used as input for <code class="func">AtlasGenerators</code> (<a href="chap2.html#X7D1CCCF8852DFF39"><b>2.5-2</b></a>) in order to fetch the generators. If no representation satisfying the given conditions ia available then <code class="keyw">fail</code> is returned.</p>
<p>If the argument <var class="Arg">std</var> is given then it must be a positive integer or a list of positive integers, denoting the sets of standard generators w.r.t. which the representation shall be given (see Section <a href="chap2.html#X795DB7E486E0817D"><b>2.3</b></a>).</p>
<p>The argument <var class="Arg">gapname</var> can be missing (then all available groups are considered), or a list of group names can be given instead.</p>
<p>Further restrictions can be entered as arguments, with the same meaning as described for <code class="func">DisplayAtlasInfo</code> (<a href="chap2.html#X79DACFFA7E2D1A99"><b>2.5-1</b></a>). The result of <code class="func">OneAtlasGeneratingSetInfo</code> describes the first generating set for G that matches the restrictions, in the ordering shown by <code class="func">DisplayAtlasInfo</code> (<a href="chap2.html#X79DACFFA7E2D1A99"><b>2.5-1</b></a>).</p>
<p>Note that even in the case that the user parameter "remote" has the value <code class="keyw">true</code> (see Section <a href="chap1.html#X7E1934D2780D108F"><b>1.7-1</b></a>), <code class="func">OneAtlasGeneratingSetInfo</code> does <em>not</em> attempt to <em>transfer</em> remote data files, just the table of contents is evaluated. So this function (as well as <code class="func">AllAtlasGeneratingSetInfos</code> (<a href="chap2.html#X84C2D76482E60E42"><b>2.5-5</b></a>)) can be used to check for the availability of certain representations, and afterwards one can call <code class="func">AtlasGenerators</code> (<a href="chap2.html#X7D1CCCF8852DFF39"><b>2.5-2</b></a>) for those representations one wants to work with.</p>
<p>In the following example, we try to access information about permutation representations for the alternating group A_5.</p>
<table class="example">
<tr><td><pre>
gap> info:= OneAtlasGeneratingSetInfo( "A5" );
rec( groupname := "A5", standardization := 1, repnr := 1,
identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ], p := 5,
id := "", size := 60 )
gap> gens:= AtlasGenerators( info.identifier );
rec( generators := [ (1,2)(3,4), (1,3,5) ], groupname := "A5",
standardization := 1, repnr := 1,
identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ], p := 5,
id := "", size := 60 )
gap> info = OneAtlasGeneratingSetInfo( "A5", IsPermGroup, true );
true
gap> info = OneAtlasGeneratingSetInfo( "A5", NrMovedPoints, "minimal" );
true
gap> info = OneAtlasGeneratingSetInfo( "A5", NrMovedPoints, [ 1 .. 10 ] );
true
gap> OneAtlasGeneratingSetInfo( "A5", NrMovedPoints, 20 );
fail
</pre></td></tr></table>
<p>Note that a permutation representation of degree 20 could be obtained by taking twice the primitive representation on 10 points; however, the <strong class="pkg">ATLAS</strong> of Group Representations does not store this imprimitive representation (cf. Section <a href="chap2.html#X87D26B13819A8209"><b>2.1</b></a>).</p>
<p>We continue this example a little. Next we access matrix representations of A_5.</p>
<table class="example">
<tr><td><pre>
gap> info:= OneAtlasGeneratingSetInfo( "A5", IsMatrixGroup, true );
rec( groupname := "A5", standardization := 1, repnr := 4,
identifier := [ "A5", [ "A5G1-f2r4aB0.m1", "A5G1-f2r4aB0.m2" ], 1, 2 ],
dim := 4, id := "a", ring := GF(2), size := 60 )
gap> gens:= AtlasGenerators( info.identifier );
rec(
generators := [ <an immutable 4x4 matrix over GF2>, <an immutable 4x4 matrix\
over GF2> ], groupname := "A5", standardization := 1, repnr := 4,
identifier := [ "A5", [ "A5G1-f2r4aB0.m1", "A5G1-f2r4aB0.m2" ], 1, 2 ],
dim := 4, id := "a", ring := GF(2), size := 60 )
gap> info = OneAtlasGeneratingSetInfo( "A5", Dimension, 4 );
true
gap> info = OneAtlasGeneratingSetInfo( "A5", Characteristic, 2 );
true
gap> info = OneAtlasGeneratingSetInfo( "A5", Ring, GF(2) );
true
gap> OneAtlasGeneratingSetInfo( "A5", Characteristic, [2,5], Dimension, 2 );
rec( groupname := "A5", standardization := 1, repnr := 8,
identifier := [ "A5", [ "A5G1-f4r2aB0.m1", "A5G1-f4r2aB0.m2" ], 1, 4 ],
dim := 2, id := "a", ring := GF(2^2), size := 60 )
gap> OneAtlasGeneratingSetInfo( "A5", Characteristic, [2,5], Dimension, 1 );
fail
gap> info:= OneAtlasGeneratingSetInfo( "A5", Characteristic, 0, Dimension, 4 );
rec( groupname := "A5", standardization := 1, repnr := 14,
identifier := [ "A5", "A5G1-Zr4B0.g", 1, 4 ], dim := 4, id := "",
ring := Integers, size := 60 )
gap> gens:= AtlasGenerators( info.identifier );
rec(
generators := [ [ [ 1, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [ -1, -1,
-1, -1 ] ],
[ [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 1, 0, 0, 0 ] ] ],
groupname := "A5", standardization := 1, repnr := 14,
identifier := [ "A5", "A5G1-Zr4B0.g", 1, 4 ], dim := 4, id := "",
ring := Integers, size := 60 )
gap> info = OneAtlasGeneratingSetInfo( "A5", Ring, Integers );
true
gap> info = OneAtlasGeneratingSetInfo( "A5", Ring, CF(37) );
true
gap> OneAtlasGeneratingSetInfo( "A5", Ring, Integers mod 77 );
fail
gap> info:= OneAtlasGeneratingSetInfo( "A5", Ring, CF(5), Dimension, 3 );
rec( groupname := "A5", standardization := 1, repnr := 17,
identifier := [ "A5", "A5G1-Ar3aB0.g", 1, 3 ], dim := 3, id := "a",
ring := NF(5,[ 1, 4 ]), size := 60 )
gap> gens:= AtlasGenerators( info.identifier );
rec(
generators := [ [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ -E(5)-E(5)^4, -E(5)-E(5)^4,
1 ] ], [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ],
groupname := "A5", standardization := 1, repnr := 17,
identifier := [ "A5", "A5G1-Ar3aB0.g", 1, 3 ], dim := 3, id := "a",
ring := NF(5,[ 1, 4 ]), size := 60 )
gap> OneAtlasGeneratingSetInfo( "A5", Ring, GF(17) );
fail
</pre></td></tr></table>
<p><a id="X84C2D76482E60E42" name="X84C2D76482E60E42"></a></p>
<h5>2.5-5 AllAtlasGeneratingSetInfos</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AllAtlasGeneratingSetInfos</code>( <var class="Arg">[gapname, ][std, ][...]</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b>the list of all records describing representations that satisfy the conditions.</p>
<p><code class="func">AllAtlasGeneratingSetInfos</code> is similar to <code class="func">OneAtlasGeneratingSetInfo</code> (<a href="chap2.html#X841478AB7CD06D44"><b>2.5-4</b></a>). The difference is that the list of <em>all</em> records describing the available representations with the given properties is returned instead of just one such component. In particular an empty list is returned if no such representation is available.</p>
<table class="example">
<tr><td><pre>
gap> AllAtlasGeneratingSetInfos( "A5", IsPermGroup, true );
[ rec( groupname := "A5", standardization := 1, repnr := 1,
identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
p := 5, id := "", size := 60 ),
rec( groupname := "A5", standardization := 1, repnr := 2,
identifier := [ "A5", [ "A5G1-p6B0.m1", "A5G1-p6B0.m2" ], 1, 6 ],
p := 6, id := "", size := 60 ),
rec( groupname := "A5", standardization := 1, repnr := 3,
identifier := [ "A5", [ "A5G1-p10B0.m1", "A5G1-p10B0.m2" ], 1, 10 ],
p := 10, id := "", size := 60 ) ]
</pre></td></tr></table>
<p>Note that a matrix representation in any characteristic can be obtained by reducing a permutation representation or an integral matrix representation; however, the <strong class="pkg">ATLAS</strong> of Group Representations does not <em>store</em> such a representation (cf. Section <a href="chap2.html#X87D26B13819A8209"><b>2.1</b></a>).</p>
<p><a id="X80AABEE783363B70" name="X80AABEE783363B70"></a></p>
<h5>2.5-6 AtlasGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AtlasGroup</code>( <var class="Arg">[gapname, ][std, ][...]</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b>a group that satisfies the conditions, or <code class="keyw">fail</code>.</p>
<p><code class="func">AtlasGroup</code> takes the same arguments as <code class="func">OneAtlasGeneratingSetInfo</code> (<a href="chap2.html#X841478AB7CD06D44"><b>2.5-4</b></a>), and returns the group generated by the <code class="code">generators</code> component of the record that is returned by <code class="func">OneAtlasGeneratingSetInfo</code> (<a href="chap2.html#X841478AB7CD06D44"><b>2.5-4</b></a>) with these arguments; if <code class="func">OneAtlasGeneratingSetInfo</code> (<a href="chap2.html#X841478AB7CD06D44"><b>2.5-4</b></a>) returns <code class="keyw">fail</code> then also <code class="func">AtlasGroup</code> returns <code class="keyw">fail</code>.</p>
<p>Alternatively, a record as returned by <code class="func">OneAtlasGeneratingSetInfo</code> (<a href="chap2.html#X841478AB7CD06D44"><b>2.5-4</b></a>) or <code class="func">AllAtlasGeneratingSetInfos</code> (<a href="chap2.html#X84C2D76482E60E42"><b>2.5-5</b></a>) can be given as the only argument.</p>
<table class="example">
<tr><td><pre>
gap> g:= AtlasGroup( "A5" );
Group([ (1,2)(3,4), (1,3,5) ])
</pre></td></tr></table>
<p><a id="X7A3E460C82B3D9A3" name="X7A3E460C82B3D9A3"></a></p>
<h5>2.5-7 AtlasSubgroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AtlasSubgroup</code>( <var class="Arg">gapname[, std][, ...], maxnr</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b>a group that satisfies the conditions, or <code class="keyw">fail</code>.</p>
<p>The arguments of <code class="func">AtlasSubgroup</code>, except the last argument <var class="Arg">maxn</var>, are the same as for <code class="func">AtlasGroup</code> (<a href="chap2.html#X80AABEE783363B70"><b>2.5-6</b></a>). If the <strong class="pkg">ATLAS</strong> of Group Representations provides a straight line program for restricting representations of the group with name <var class="Arg">gapname</var> (given w.r.t. the <var class="Arg">std</var>-th standard generators) to the <var class="Arg">maxnr</var>-th maximal subgroup and if a representation with the required properties is available, in the sense that calling <code class="func">AtlasGroup</code> (<a href="chap2.html#X80AABEE783363B70"><b>2.5-6</b></a>) with the same arguments except <var class="Arg">maxnr</var> yields a group, then <code class="func">AtlasSubgroup</code> returns the restriction of this representation to the <var class="Arg">maxnr</var>-th maximal subgroup. In all other cases, <code class="keyw">fail</code> is returned.</p>
<p>Note that the conditions refer to the group and not to the subgroup. It may happen that in the restriction of a permutation representation to a subgroup, fewer points are moved, or that the restriction of a matrix representation turns out to be defined over a smaller ring. Here is an example.</p>
<table class="example">
<tr><td><pre>
gap> g:= AtlasSubgroup( "A5", NrMovedPoints, 5, 1 );
Group([ (1,5)(2,3), (1,3,5) ])
gap> NrMovedPoints( g );
4
</pre></td></tr></table>
<p><a id="X87ACE06E82B68589" name="X87ACE06E82B68589"></a></p>
<h4>2.6 <span class="Heading">Examples of Using the <strong class="pkg">AtlasRep</strong> Package</span></h4>
<p><a id="X8563D96878AC685C" name="X8563D96878AC685C"></a></p>
<h5>2.6-1 <span class="Heading">Example: Class Representatives</span></h5>
<p>First we show the computation of class representatives of the Mathieu group M_11, in a 2-modular matrix representation. We start with the ordinary and Brauer character tables of this group.</p>
<table class="example">
<tr><td><pre>
gap> tbl:= CharacterTable( "M11" );;
gap> modtbl:= tbl mod 2;;
gap> CharacterDegrees( modtbl );
[ [ 1, 1 ], [ 10, 1 ], [ 16, 2 ], [ 44, 1 ] ]
</pre></td></tr></table>
<p>The output of <code class="func">CharacterDegrees</code> (<a href="../../../doc/htm/ref/CHAP069.htm#SECT008"><b>Reference: CharacterDegrees</b></a>) means that the 2-modular irreducibles of M_11 have degrees 1, 10, 16, 16, and 44.</p>
<p>Using <code class="func">DisplayAtlasInfo</code> (<a href="chap2.html#X79DACFFA7E2D1A99"><b>2.5-1</b></a>), we find out that matrix generators for the irreducible 10-dimensional representation are available in the database.</p>
<table class="example">
<tr><td><pre>
gap> DisplayAtlasInfo( "M11", Characteristic, 2 );
Representations for G = M11: (all refer to std. generators 1)
----------------------------
6: G <= GL(10,2)
7: G <= GL(32,2)
8: G <= GL(44,2)
16: G <= GL(16a,4)
17: G <= GL(16b,4)
</pre></td></tr></table>
<p>So we decide to work with this representation. We fetch the generators and compute the list of class representatives of M_11 in the representation. The ordering of class representatives is the same as that in the character table of the <strong class="pkg">ATLAS</strong> of Finite Groups (<a href="chapBib.html#biBCCN85">[CCNPW85]</a>), which coincides with the ordering of columns in the <strong class="pkg">GAP</strong> table we have fetched above.</p>
<table class="example">
<tr><td><pre>
gap> info:= OneAtlasGeneratingSetInfo( "M11", Characteristic, 2,
> Dimension, 10 );;
gap> gens:= AtlasGenerators( info.identifier );;
gap> ccls:= AtlasProgram( "M11", gens.standardization, "classes" );
rec( program := <straight line program>, standardization := 1,
identifier := [ "M11", "M11G1-cclsW1", 1 ],
outputs := [ "1A", "2A", "3A", "4A", "5A", "6A", "8A", "8B", "11A", "11B" ],
groupname := "M11" )
gap> reps:= ResultOfStraightLineProgram( ccls.program, gens.generators );;
</pre></td></tr></table>
<p>If we would need only a few class representatives, we could use the <strong class="pkg">GAP</strong> library function <code class="func">RestrictOutputsOfSLP</code> (<a href="../../../doc/htm/ref/CHAP035.htm#SECT008"><b>Reference: RestrictOutputsOfSLP</b></a>) to create a straight line program that computes only specified outputs. Here is an example where only the class representatives of order eight are computed.</p>
<table class="example">
<tr><td><pre>
gap> ord8prg:= RestrictOutputsOfSLP( ccls.program,
> Filtered( [ 1 .. 10 ], i -> ccls.outputs[i][1] = '8' ) );
<straight line program>
gap> ord8reps:= ResultOfStraightLineProgram( ord8prg, gens.generators );;
gap> List( ord8reps, m -> Position( reps, m ) );
[ 7, 8 ]
</pre></td></tr></table>
<p>Let us check that the class representatives have the right orders.</p>
<table class="example">
<tr><td><pre>
gap> List( reps, Order ) = OrdersClassRepresentatives( tbl );
true
</pre></td></tr></table>
<p>From the class representatives, we can compute the Brauer character we had started with. This Brauer character is defined on all classes of the 2-modular table. So we first pick only those representatives, using the <strong class="pkg">GAP</strong> function <code class="func">GetFusionMap</code> (<a href="../../../doc/htm/ref/CHAP071.htm#SECT002"><b>Reference: GetFusionMap</b></a>); in this situation, it returns the class fusion from the Brauer table into the ordinary table.</p>
<table class="example">
<tr><td><pre>
gap> fus:= GetFusionMap( modtbl, tbl );
[ 1, 3, 5, 9, 10 ]
gap> modreps:= reps{ fus };;
</pre></td></tr></table>
<p>Then we call the <strong class="pkg">GAP</strong> function <code class="func">BrauerCharacterValue</code> (<a href="../../../doc/htm/ref/CHAP070.htm#SECT015"><b>Reference: BrauerCharacterValue</b></a>), which computes the Brauer character value from the matrix given.</p>
<table class="example">
<tr><td><pre>
gap> char:= List( modreps, BrauerCharacterValue );
[ 10, 1, 0, -1, -1 ]
gap> Position( Irr( modtbl ), char );
2
</pre></td></tr></table>
<p><a id="X81C9233778A3A817" name="X81C9233778A3A817"></a></p>
<h5>2.6-2 <span class="Heading">Example: Permutation and Matrix Representations</span></h5>
<p>The second example shows the computation of a permutation representation from a matrix representation. We work with the 10-dimensional representation used above, and consider the action on the 2^10 vectors of the underlying row space.</p>
<table class="example">
<tr><td><pre>
gap> grp:= Group( gens.generators );;
gap> v:= GF(2)^10;;
gap> orbs:= Orbits( grp, AsList( v ) );;
gap> List( orbs, Length );
[ 1, 396, 55, 330, 66, 165, 11 ]
</pre></td></tr></table>
<p>We see that there are six nontrivial orbits, and we can compute the permutation actions on these orbits directly using <code class="func">Action</code> (<a href="../../../doc/htm/ref/CHAP039.htm#SECT006"><b>Reference: Action</b></a>). However, for larger examples, one cannot write down all orbits on the row space, so one has to use another strategy if one is interested in a particular orbit.</p>
<p>Let us assume that we are interested in the orbit of length 11. The point stabilizer is the first maximal subgroup of M_11, thus the restriction of the representation to this subgroup has a nontrivial fixed point space. This restriction can be computed using the <strong class="pkg">AtlasRep</strong> package.</p>
<table class="example">
<tr><td><pre>
gap> gens:= AtlasGenerators( "M11", 6, 1 );;
</pre></td></tr></table>
<p>Now computing the fixed point space is standard linear algebra.</p>
<table class="example">
<tr><td><pre>
gap> id:= IdentityMat( 10, GF(2) );;
gap> sub1:= Subspace( v, NullspaceMat( gens.generators[1] - id ) );;
gap> sub2:= Subspace( v, NullspaceMat( gens.generators[2] - id ) );;
gap> fix:= Intersection( sub1, sub2 );
<vector space of dimension 1 over GF(2)>
</pre></td></tr></table>
<p>The final step is of course the computation of the permutation action on the orbit.</p>
<table class="example">
<tr><td><pre>
gap> orb:= Orbit( grp, Basis( fix )[1] );;
gap> act:= Action( grp, orb );; Print( act, "\n" );
Group( [ ( 1, 2)( 4, 6)( 5, 8)( 7,10), ( 1, 3, 5, 9)( 2, 4, 7,11) ] )
</pre></td></tr></table>
<p>Note that this group is <em>not</em> equal to the group obtained by fetching the permutation representation from the database. This is due to a different numbering of the points, so the groups are permutation isomorphic.</p>
<table class="example">
<tr><td><pre>
gap> permgrp:= Group( AtlasGenerators( "M11", 1 ).generators );;
gap> Print( permgrp, "\n" );
Group( [ ( 2,10)( 4,11)( 5, 7)( 8, 9), ( 1, 4, 3, 8)( 2, 5, 6, 9) ] )
gap> permgrp = act;
false
gap> IsConjugate( SymmetricGroup(11), permgrp, act );
true
</pre></td></tr></table>
<p><a id="X8284D7E87D38889C" name="X8284D7E87D38889C"></a></p>
<h5>2.6-3 <span class="Heading">Example: Outer Automorphisms</span></h5>
<p>The straight line programs for applying outer automorphisms to standard generators can of course be used to define the automorphisms themselves as <strong class="pkg">GAP</strong> mappings.</p>
<table class="example">
<tr><td><pre>
gap> DisplayAtlasInfo( "G2(3)", IsStraightLineProgram );
Programs for G = G2(3): (all refer to std. generators 1)
-----------------------
available maxes of G: [ 1 .. 10 ] (all)
class repres. of G available
repres. of cyclic subgroups of G available
available automorphisms: [ "2" ]
standard generators checker available
presentation available
gap> prog:= AtlasProgram( "G2(3)", "automorphism", "2" ).program;;
gap> info:= OneAtlasGeneratingSetInfo( "G2(3)", Dimension, 7 );;
gap> gens:= AtlasGenerators( info ).generators;;
gap> imgs:= ResultOfStraightLineProgram( prog, gens );;
</pre></td></tr></table>
<p>If we are not suspicious whether the script really describes an automorphism then we should tell this to <strong class="pkg">GAP</strong>, in order to avoid the expensive checks of the properties of being a homomorphism and bijective (see Section <a href="../../../doc/htm/ref/CHAP038.htm#SECT001"><b>Reference: Creating Group Homomorphisms</b></a>). This looks as follows.</p>
<table class="example">
<tr><td><pre>
gap> g:= Group( gens );;
gap> aut:= GroupHomomorphismByImagesNC( g, g, gens, imgs );;
gap> SetIsBijective( aut, true );
</pre></td></tr></table>
<p>If we are suspicious whether the script describes an automorphism then we might have the idea to check it with <strong class="pkg">GAP</strong>, as follows.</p>
<table class="example">
<tr><td><pre>
gap> aut:= GroupHomomorphismByImages( g, g, gens, imgs );;
gap> IsBijective( aut );
true
</pre></td></tr></table>
<p>(Note that even for a comparatively small group such as G_2(3), this was a difficult task for <strong class="pkg">GAP</strong> before version 4.3.)</p>
<p>Often one can form images under an automorphism alpha, say, without creating the homomorphism object. This is obvious for the standard generators of the group G themselves, but also for generators of a maximal subgroup M computed from standard generators of G, provided that the straight line programs in question refer to the same standard generators. Note that the generators of M are given by evaluating words in terms of standard generators of G, and their images under alpha can be obtained by evaluating the same words at the images under alpha of the standard generators of G.</p>
<table class="example">
<tr><td><pre>
gap> max1:= AtlasProgram( "G2(3)", 1 ).program;;
gap> mgens:= ResultOfStraightLineProgram( max1, gens );;
gap> comp:= CompositionOfStraightLinePrograms( max1, prog );;
gap> mimgs:= ResultOfStraightLineProgram( comp, gens );;
</pre></td></tr></table>
<p>The list <code class="code">mgens</code> is the list of generators of the first maximal subgroup of G_2(3), <code class="code">mimgs</code> is the list of images under the automorphism given by the straight line program <code class="code">prog</code>. Note that applying the program returned by <code class="func">CompositionOfStraightLinePrograms</code> (<a href="../../../doc/htm/ref/CHAP035.htm#SECT008"><b>Reference: CompositionOfStraightLinePrograms</b></a>) means to apply first <code class="code">prog</code> and then <code class="code">max1</code>, Since we have already constructed the <strong class="pkg">GAP</strong> object representing the automorphism, we can check whether the results are equal.</p>
<table class="example">
<tr><td><pre>
gap> mimgs = List( mgens, x -> x^aut );
true
</pre></td></tr></table>
<p>However, it should be emphasized that using <code class="code">aut</code> requires a huge machinery of computations behind the scenes, whereas applying the straight line programs <code class="code">prog</code> and <code class="code">max1</code> involves only elementary operations with the generators. The latter is feasible also for larger groups, for which constructing the <strong class="pkg">GAP</strong> automorphism might be impossible.</p>
<p><a id="X794D669E7A507310" name="X794D669E7A507310"></a></p>
<h5>2.6-4 <span class="Heading">Example: Using Semi-presentations and Black Box Programs</span></h5>
<p>Let us suppose that we want to restrict a representation of the Mathieu group M_12 to a non-maximal subgroup of the type L_2(11). The idea is that this subgroup can be found as a maximal subgroup of a maximal subgroup of the type M_11, which is itself maximal in M_12. For that, we fetch a representation of M_12 and use a straight line program for restricting it to the first maximal subgroup, which has the type M_11.</p>
<table class="example">
<tr><td><pre>
gap> info:= OneAtlasGeneratingSetInfo( "M12", NrMovedPoints, 12 );
rec( groupname := "M12", standardization := 1, repnr := 1,
identifier := [ "M12", [ "M12G1-p12aB0.m1", "M12G1-p12aB0.m2" ], 1, 12 ],
p := 12, id := "a", size := 95040 )
gap> gensM12:= AtlasGenerators( info.identifier );;
gap> restM11:= AtlasProgram( "M12", "maxes", 1 );;
gap> gensM11:= ResultOfStraightLineProgram( restM11.program,
> gensM12.generators );
[ (3,9)(4,12)(5,10)(6,8), (1,4,11,5)(2,10,8,3) ]
</pre></td></tr></table>
<p>Now we cannot simply apply a straight line program for M_11 to these generators of M_11, since they are not necessarily <em>standard</em> generators of M_11. We check this using a semi-presentation for M_11.</p>
<table class="example">
<tr><td><pre>
gap> checkM11:= AtlasProgram( "M11", "check" );
rec( program := <straight line decision>, standardization := 1,
identifier := [ "M11", "M11G1-check1", 1, 1 ], groupname := "M11" )
gap> ResultOfStraightLineDecision( checkM11.program, gensM11 );
true
</pre></td></tr></table>
<p>So we are lucky that applying the appropriate program for M_11 will give us the required generators for L_2(11).</p>
<table class="example">
<tr><td><pre>
gap> restL211:= AtlasProgram( "M11", "maxes", 2 );;
gap> gensL211:= ResultOfStraightLineProgram( restL211.program, gensM11 );
[ (3,9)(4,12)(5,10)(6,8), (1,11,9)(2,12,8)(3,6,10) ]
gap> G:= Group( gensL211 );; Size( G ); IsSimple( G );
660
true
</pre></td></tr></table>
<p>Usually representations are not given in terms of standard generators. For example, let us take the M_11 type group returned by the <strong class="pkg">GAP</strong> function <code class="func">MathieuGroup</code> (<a href="../../../doc/htm/ref/CHAP048.htm#SECT001"><b>Reference: MathieuGroup</b></a>).</p>
<table class="example">
<tr><td><pre>
gap> G:= MathieuGroup( 11 );;
gap> gens:= GeneratorsOfGroup( G );
[ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) ]
gap> ResultOfStraightLineDecision( checkM11.program, gens );
false
</pre></td></tr></table>
<p>If we want to compute an L_2(11) type subgroup of this group, we can use a black box program for computing standard generators, and then apply the straight line program for computing the restriction.</p>
<table class="example">
<tr><td><pre>
gap> find:= AtlasProgram( "M11", "find" );
rec( program := <black box program>, standardization := 1,
identifier := [ "M11", "M11G1-find1", 1, 1 ], groupname := "M11" )
gap> stdgens:= ResultOfBBoxProgram( find.program, Group( gens ) );;
gap> List( stdgens, Order );
[ 2, 4 ]
gap> ResultOfStraightLineDecision( checkM11.program, stdgens );
true
gap> gensL211:= ResultOfStraightLineProgram( restL211.program, stdgens );;
gap> List( gensL211, Order );
[ 2, 3 ]
gap> G:= Group( gensL211 );; Size( G ); IsSimple( G );
660
true
</pre></td></tr></table>
<p><a id="X7CE7C2068017525C" name="X7CE7C2068017525C"></a></p>
<h5>2.6-5 <span class="Heading">Example: Using the <strong class="pkg">GAP</strong> Library of Tables of Marks</span></h5>
<p>The <strong class="pkg">GAP</strong> library of tables of marks provides, for many almost simple groups, information for constructing representatives of all conjugacy classes of subgroups. If this information is compatible with the standard generators of the <strong class="pkg">ATLAS</strong> of Group Representations then we can use it to restrict any representation from the <strong class="pkg">ATLAS</strong> to prescribed subgroups. This is useful in particular for those subgroups for which the <strong class="pkg">ATLAS</strong> of Group Representations itself does not contain a straight line program.</p>
<table class="example">
<tr><td><pre>
gap> tom:= TableOfMarks( "A5" );
TableOfMarks( "A5" )
gap> info:= StandardGeneratorsInfo( tom );
[ rec( generators := "a, b", description := "|a|=2, |b|=3, |ab|=5",
script := [ [ 1, 2 ], [ 2, 3 ], [ 1, 1, 2, 1, 5 ] ], ATLAS := true ) ]
</pre></td></tr></table>
<p>The <code class="keyw">true</code> value of the component <code class="code">ATLAS</code> indicates that the information stored on <code class="code">tom</code> refers to the standard generators of type 1 in the <strong class="pkg">ATLAS</strong> of Group Representations.</p>
<p>We want to restrict a 4-dimensional integral representation of A_5 to a Sylow 2 subgroup of A_5, and use <code class="func">RepresentativeTomByGeneratorsNC</code> (<a href="../../../doc/htm/ref/CHAP068.htm#SECT011"><b>Reference: RepresentativeTomByGeneratorsNC</b></a>) for that.</p>
<table class="example">
<tr><td><pre>
gap> info:= OneAtlasGeneratingSetInfo( "A5", Ring, Integers, Dimension, 4 );;
gap> stdgens:= AtlasGenerators( info.identifier );
rec(
generators := [ [ [ 1, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [ -1, -1,
-1, -1 ] ],
[ [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ], [ 1, 0, 0, 0 ] ] ],
groupname := "A5", standardization := 1, repnr := 14,
identifier := [ "A5", "A5G1-Zr4B0.g", 1, 4 ], dim := 4, id := "",
ring := Integers, size := 60 )
gap> orders:= OrdersTom( tom );
[ 1, 2, 3, 4, 5, 6, 10, 12, 60 ]
gap> pos:= Position( orders, 4 );
4
gap> sub:= RepresentativeTomByGeneratorsNC( tom, pos, stdgens.generators );
<matrix group of size 4 with 2 generators>
gap> GeneratorsOfGroup( sub );
[ [ [ 1, 0, 0, 0 ], [ -1, -1, -1, -1 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ] ],
[ [ 1, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [ -1, -1, -1, -1 ] ] ]
</pre></td></tr></table>
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