GAP WreathProduct with fixed points

Question

Gap always takes the domain of a permutation group to be the set of points moved by its elements. The documentation of WreathProduct thus includes this comment explaining how form the wreath product between permutation groups whose domains include fixed points:

"If fixed points are desired the wreath product $G \wr T$ has to be formed with a transitive overgroup $T$ of $P$ and then the pre-image of $P$ under the projection $G \wr T \rightarrow T$ has to be taken."

Can someone more well versed in GAP than me demonstrate how to do this in practice? Say e.g. I want to compute WreathProduct(Group([(1,2)(3)]), Group([(1,2)])), i.e. I want to compute the wreath product as if the first group had domain $\{1,2,3\}$ instead $\{1,2\}$. How do I do that?

Answer 1

You could take the wreath product of a trasitive overgroup (in your example $S_3$) by your second factor, and then take the group generated by the image of your embedded group $G$, together with (the image of) $T$:

gap> G:=Group((1,2));;
gap> S:=SymmetricGroup(3);
Sym( [ 1 .. 3 ] )
gap> T:=Group((1,2));;
gap> WS:=WreathProduct(S,T);
Group([ (1,2,3), (1,2), (4,5,6), (4,5), (1,4)(2,5)(3,6) ])
gap> Gemb:=Image(Embedding(WS,1),G);
Group([ (1,2) ])
gap> Temb:=Image(Embedding(WS,3),T);  # 3=1+LargestMovedPoint(T);
Group([ (1,4)(2,5)(3,6) ])
gap> W:=ClosureGroup(Gemb,Temb);
Group([ (1,2), (1,4)(2,5)(3,6) ])
gap> Size(W);
8

You could do analogously if $T$ is not transitive (though then you would take embeddings of $G$ for every orbit of $T$.