[GAP Forum] Semidirect Product of C_2^4 and A_5

[Stefan Kohl]

Dear Forum,

Zeinab Foruzanfar asked:

>  I'm trying to write a Gap-program for  (Z2×Z2×Z2×Z2)⋊φA5. Please Help me.

The easiest is of course to construct the direct product, but I guess
you rather want one of the perfect groups of the structure you give.

To construct them, you can do the following:

1. Define the normal subgroup:

gap> N := Group((1,2),(3,4),(5,6),(7,8));;

2. Compute the automorphism group:

gap> A := AutomorphismGroup(N);
<group with 4 generators>
gap> Size(A);
20160

3. Embed A5:

gap> phi := IsomorphismPermGroup(A);;
gap> Ap := Image(phi);
Group([ (4,8)(5,9)(6,10)(7,11), (1,8,4,2)(3,9,12,6)(5,10)(7,11,13,14),
        (8,12)(9,13)(10,14)(11,15), () ])
gap> A5 := IsomorphicSubgroups(Ap,AlternatingGroup(5));
[ [ (1,4,5,3,2), (1,5)(2,4) ] -> [ (1,2,5,9,15)(3,7,12,6,14)(4,11,10,8,13),
    (1,5)(2,15)(3,10)(6,11)(7,14)(8,12) ],
  [ (1,4,5,3,2), (1,5)(2,4) ] -> [ (1,2,11,12,4)(3,9,7,8,5)(6,10,14,15,13),
    (1,5)(2,15)(3,10)(6,11)(7,14)(8,12) ] ]

--> Up to conjugacy there are 2 embeddings of A5 into Aut(N).

4. Construct the corresponding semidirect products:

gap> G := SemidirectProduct(PreImage(phi,Image(A5[1])),N);
<permutation group with 6 generators>
gap> Size(G);
960
gap> IsPerfect(G);
true
gap> IdGroup(G);
[ 960, 11358 ]

gap> H := SemidirectProduct(PreImage(phi,Image(A5[2])),N);
<permutation group with 6 generators>
gap> Size(H);
960
gap> IsPerfect(H);
true
gap> IdGroup(H);
[ 960, 11357 ]

5. These are all such perfect groups:

gap> NrPerfectGroups(960);
2

Hope this helps,

    Stefan