[GAP Forum] help

[Stephen Linton]

Dear GAP Forum, 

On 1 Jun 2011, at 06:59, mazaher rahimi wrote:

> How we can introduce the following group :
> $G=D\times H$ where $D$ is dihedral group of order 42 and $H$ is a
> semidirect product of vector space $V$ of dimension 3 over $GF(2)$ by
> a subgroup of order 21 from $GL(3,2)$ acting on $V$ naturally.Thanks
> 

There are many ways of obtaining groups isomorphic to your G. A very straightforward way, that depends on
no special knowledge about the groups concerned is shown below:

gap> d := DihedralGroup(42);          
<pc group of size 42 with 3 generators>
gap> v := ElementaryAbelianGroup(8);  
<pc group of size 8 with 3 generators>
gap> a := AutomorphismGroup(v);
<group of size 168 with 2 generators>
gap> ccs := ConjugacyClassesSubgroups(a);
[ Group( IdentityMapping( Group( [ f1, f2, f3 ] ) ) )^G, 
  Group( [ Pcgs([ f1, f2, f3 ]) -> [ f1*f2*f3, f2, f3 ] ] )^G, 
  Group( [ Pcgs([ f1, f2, f3 ]) -> [ f2, f1*f2*f3, f3 ] ] )^G, 
  Group( [ Pcgs([ f1, f2, f3 ]) -> [ f1*f2*f3, f2, f3 ], 
      Pcgs([ f1, f2, f3 ]) -> [ f1*f2*f3, f3, f2 ] ] )^G, 
  Group( [ Pcgs([ f1, f2, f3 ]) -> [ f1*f2*f3, f2, f3 ], 
      Pcgs([ f1, f2, f3 ]) -> [ f1*f3, f2, f3 ] ] )^G, 
  Group( [ Pcgs([ f1, f2, f3 ]) -> [ f1*f2*f3, f2, f3 ], 
      Pcgs([ f1, f2, f3 ]) -> [ f1*f3, f3, f2 ] ] )^G, 
  Group( [ Pcgs([ f1, f2, f3 ]) -> [ f2, f1*f2*f3, f3 ], 
      Pcgs([ f1, f2, f3 ]) -> [ f1*f2*f3, f2, f3 ] ] )^G, 
  Group( [ Pcgs([ f1, f2, f3 ]) -> [ f2, f2*f3, f1*f3 ] ] )^G, 
  Group( [ Pcgs([ f1, f2, f3 ]) -> [ f1*f2*f3, f2, f3 ], 
      Pcgs([ f1, f2, f3 ]) -> [ f1*f2*f3, f3, f2 ], 
      Pcgs([ f1, f2, f3 ]) -> [ f1*f2*f3, f1*f2, f1*f3 ] ] )^G, 
  Group( [ Pcgs([ f1, f2, f3 ]) -> [ f1*f2*f3, f2, f3 ], 
      Pcgs([ f1, f2, f3 ]) -> [ f1*f2*f3, f3, f2 ], 
      Pcgs([ f1, f2, f3 ]) -> [ f2, f1*f2, f1*f3 ] ] )^G, 
  Group( [ Pcgs([ f1, f2, f3 ]) -> [ f1*f2*f3, f2, f3 ], 
      Pcgs([ f1, f2, f3 ]) -> [ f1*f3, f2, f3 ], 
      Pcgs([ f1, f2, f3 ]) -> [ f1*f2, f3, f2*f3 ] ] )^G, 
  Group( [ Pcgs([ f1, f2, f3 ]) -> [ f2, f2*f3, f1*f3 ], 
      Pcgs([ f1, f2, f3 ]) -> [ f1*f2, f1*f3, f3 ] ] )^G, 
  Group( [ Pcgs([ f1, f2, f3 ]) -> [ f1*f2*f3, f2, f3 ], 
      Pcgs([ f1, f2, f3 ]) -> [ f1*f2*f3, f3, f2 ], 
      Pcgs([ f1, f2, f3 ]) -> [ f2, f1*f2, f1*f3 ], 
      Pcgs([ f1, f2, f3 ]) -> [ f1*f2*f3, f1*f2, f1*f3 ] ] )^G, 
  Group( [ Pcgs([ f1, f2, f3 ]) -> [ f1*f2*f3, f2, f3 ], 
      Pcgs([ f1, f2, f3 ]) -> [ f1*f3, f2, f3 ], 
      Pcgs([ f1, f2, f3 ]) -> [ f1*f2, f3, f2*f3 ], 
      Pcgs([ f1, f2, f3 ]) -> [ f1*f2, f2*f3, f3 ] ] )^G, 
  Group( [ Pcgs([ f1, f2, f3 ]) -> [ f1*f2, f2, f3 ], 
      Pcgs([ f1, f2, f3 ]) -> [ f3, f1, f2 ] ] )^G ]
gap> Filtered(ccs, c -> Size(Representative(c)) = 21);
[ Group( [ Pcgs([ f1, f2, f3 ]) -> [ f2, f2*f3, f1*f3 ], 
      Pcgs([ f1, f2, f3 ]) -> [ f1*f2, f1*f3, f3 ] ] )^G ]
gap> k := Representative(last[1]);
<group of size 21 with 2 generators>
gap> h := SemidirectProduct(k,v);
<pc group with 5 generators>
gap> g := DirectProduct(d,h);
<pc group of size 7056 with 8 generators>
gap> 

	Steve Linton