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Dear GAP-Forum,

(I had answered this already in private, but just noted the question

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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