> < ^ Date: Tue, 24 Aug 1999 10:22:43 +0100 (BST)
> < ^ From: Alexander Hulpke <hulpke@math.colostate.edu >
< ^ Subject: Re: ManualGAP4

Dear GAP-Forum,

actually went to the forum and not to gap-trouble. Anyhow, in case somebody
else is interested, for completeness here's the answer:)

Kurt Ewald wrote:
> I have read the Chapter 44.2 SemidirectProducts.
> But I cannot see how to calculate a SemidirectProduct of two explicitly
> given groups.

You have to give a homomorphism into a group of automorphisms:

```gap> a5:=AlternatingGroup(5);
Alt( [ 1 .. 5 ] )
gap> aut:=AutomorphismGroup(a5); # we could also take a smaller subgroup
<group of size 120 with 3 generators>
gap> z:=Group((1,2));
Group([ (1,2) ])
gap> Order(aut.3); # it turns out that aut.3 has order 2 and thus is a
potential image. The choice of this image determines the isomorphism type of
the prduct.
2
gap> hom:=GroupHomomorphismByImagesNC(z,aut,[(1,2)],[aut.3]);
[ (1,2) ] ->
[ GroupHomomorphismByImages( AlternatingGroup( [ 1 .. 5 ] ),
AlternatingGroup([ 1 .. 5 ] ), [ (1,4)(2,5), (1,2,3) ], [ (1,5)(2,4), (1,2,3) ] ) ]
gap> p:=SemidirectProduct(z,hom,a5);
<permutation group with 3 generators>
gap> Size(p); #test
120
gap> IsomorphismGroups(p,SymmetricGroup(5)); # so the result is isomorphic to a5
[ ( 1,11)( 2,12)( 3,10)( 4, 6)( 7, 9)(13,50)(14,51)(15,49)(16,55)(17,57)
(18,56)(19,58)(20,60)(21,59)(22,52)(23,53)(24,54)(25,27)(28,34)(29,36)
(30,35)(32,33)(37,39)(40,46)(41,48)(42,47)(44,45),
( 1,14, 3,13, 2,15)( 4,19,10,16, 7,22)( 5,21,11,17, 9,23)( 6,20,12,18,
8,24)
(25,40,49,28,37,52)(26,42,50,29,39,53)(27,41,51,30,38,54)(31,47,55,32,46,
56)(33,48,57)(34,44,58,35,43,59)(36,45,60) ] -> [ (3,4), (1,3,2)(4,5) ]
```

How can I calculate f.i. the SemidirectProct of A4 and Z2, expecting S4?

You cannot expect to get S_4 in the natural action, you will only get an
isomorphic group.

Best wishes,

Alexander Hulpke

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