The semidirect product of a group N with a group G acting on N via a homomorphism a from G into the automorphism group of N is the cartesian product G ×N with the multiplication (g,n)·(h,m) = (gh,n(ha)m).
SemidirectProduct( G, alpha, N ) O
constructs the semidirect product of N with G acting via alpha. alpha must be a homomorphism from G into a group of automorphisms of N.
gap> n:=AbelianGroup(IsPcGroup,[2,2]); <pc group of size 4 with 2 generators> gap> au:=AutomorphismGroup(n); <group with 4 generators> gap> apc:=IsomorphismPcGroup(au); CompositionMapping( Pcgs([ (2,3), (1,3,2) ]) -> [ f1, f2 ], <action homomorphism> ) gap> g:=Image(apc);; Group( [ f1*f2^2, f1*f2^2, f1*f2, <identity> of ... ] ) gap> apci:=InverseGeneralMapping(apc); InverseGeneralMapping( CompositionMapping( Pcgs([ (2,3), (1,3,2) ]) -> [ f1, f2 ], <action homomorphism> ) ) gap> IsGroupHomomorphism(apci); true gap> p:=SemidirectProduct(g,apci,n); <pc group of size 24 with 4 generators> gap> IsomorphismGroups(p,Group((1,2,3,4),(1,2))); [f1,f2,f3,f4]->[(3,4),(2,4,3),(1,2)(3,4),(1,3)(2,4)]
For the semidirect product P of G with N, Embedding(P,1) embeds
G, Embedding(P,2) embeds N. The operation Projection(P) returns
the projection of P onto G
(see Embeddings and Projections for Group Products).
gap> Size(Image(Embedding(p,1))); 6 gap> Embedding(p,2); [f1,f2]->[f3,f4] gap> Projection(p); [f1,f2,f3,f4]->[f1,f2,<identity>of...,<identity>of...]
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GAP 4 manual