3 Strong Shoda pairs
  
  
  3.1 Computing strong Shoda pairs
  
  3.1-1 StrongShodaPairs
  
  > StrongShodaPairs( G ) ___________________________________________attribute
  Returns:  A list of pairs of subgroups of the input group.
  
  The input should be a finite group G.
  
  Computes  a  list  of  representatives  of the equivalence classes of strong
  Shoda pairs (7.15) of a finite group G.
  
  ---------------------------  Example  ----------------------------
    
    gap> StrongShodaPairs( SymmetricGroup(4) );
    [ [ Sym( [ 1 .. 4 ] ), Group([ (1,3)(2,4), (1,4)(2,3), (2,4,3), (1,2) ]) ],
      [ Sym( [ 1 .. 4 ] ), Group([ (1,3)(2,4), (1,4)(2,3), (2,4,3) ]) ],
      [ Group([ (1,2)(3,4), (1,3,2,4), (3,4) ]), Group([ (1,2)(3,4), (1,3,2,4) ])
         ],
      [ Group([ (1,2)(3,4), (3,4), (1,3,2,4) ]), Group([ (1,2)(3,4), (3,4) ]) ],
      [ Group([ (1,4)(2,3), (1,3)(2,4), (2,4,3) ]),
          Group([ (1,4)(2,3), (1,3)(2,4) ]) ] ]
    gap> StrongShodaPairs( DihedralGroup(64) );
    [ [ <pc group of size 64 with 6 generators>,
          Group([ f6, f5, f4, f3, f1, f2 ]) ],
      [ <pc group of size 64 with 6 generators>, Group([ f6, f5, f4, f3, f1*f2 ])
         ],
      [ <pc group of size 64 with 6 generators>, Group([ f6, f5, f4, f3, f2 ]) ],
      [ <pc group of size 64 with 6 generators>, Group([ f6, f5, f4, f3, f1 ]) ],
      [ Group([ f1*f2, f4*f5*f6, f5*f6, f6, f3, f3 ]),
          Group([ f6, f5, f4, f1*f2 ]) ],
      [ Group([ f6, f5, f2, f3, f4 ]), Group([ f6, f5 ]) ],
      [ Group([ f6, f2, f3, f4, f5 ]), Group([ f6 ]) ],
      [ Group([ f2, f3, f4, f5, f6 ]), Group([  ]) ] ]
    
  ------------------------------------------------------------------
  
  
  3.2 Properties related with Shoda pairs
  
  3.2-1 IsStrongShodaPair
  
  > IsStrongShodaPair( G, K, H ) ____________________________________operation
  
  The first argument should be a finite group G, the second one a sugroup K of
  G and the third one a subgroup of K.
  
  Returns  true  if  (K,H)  is  a  strong  Shoda  pair  (7.15) of G, and false
  otherwise.
  
  ---------------------------  Example  ----------------------------
    
    gap> G:=SymmetricGroup(3);; K:=Group([(1,2,3)]);; H:=Group( () );;
    gap> IsStrongShodaPair( G, K, H );
    true
    gap> IsStrongShodaPair( G, G, H );
    false
    gap> IsStrongShodaPair( G, K, K );
    false
    gap> IsStrongShodaPair( G, G, K );
    true
    
  ------------------------------------------------------------------
  
  3.2-2 IsShodaPair
  
  > IsShodaPair( G, K, H ) __________________________________________operation
  
  The  first argument should be a finite group G, the second a subgroup K of G
  and the third one a subgroup of K.
  
  Returns true if (K,H) is a Shoda pair (7.14) of G.
  
  Note  that  every strong Shoda pair is a Shoda pair, but the converse is not
  true.
  
  ---------------------------  Example  ----------------------------
    
    gap> G:=AlternatingGroup(5);;
    gap> K:=AlternatingGroup(4);;
    gap> H := Group( (1,2)(3,4), (1,3)(2,4) );;
    gap> IsStrongShodaPair( G, K, H );
    false
    gap> IsShodaPair( G, K, H );
    true
    
  ------------------------------------------------------------------
  
  3.2-3 IsStronglyMonomial
  
  > IsStronglyMonomial( G ) _________________________________________operation
  
  The input G should be a finite group.
  
  Returns true if G is a strongly monomial (7.16) finite group.
  
  ---------------------------  Example  ----------------------------
    
    gap> S4:=SymmetricGroup(4);;
    gap> IsStronglyMonomial(S4);
    true
    gap> G:=SmallGroup(24,3);;
    gap> IsStronglyMonomial(G);
    false
    gap> IsMonomial(G);
    false
    gap> G:=SmallGroup(1000,86);;
    gap> IsMonomial(G);
    true
    gap> IsStronglyMonomial(G);
    false
    
  ------------------------------------------------------------------