I am currently working on trying to get a solvable doubly transitive permutation group using GAP. So, I am trying to create the semidirect product of a subgroup of a general linear group and a vector space. Currently I am trying to work with the group GL(2,3) and vector space V of dimension 2 over the field of 3 elements. I don't know if this is possible, since it seems like both parts of the semidirect product needs to be a group and I need a homomorphism from GL to V. I am new to GAP so I don't know what I can do to achieve this.
For the special case of a semidirect product of a matrix group with its natural vector space you can use
SemidirectProduct without homomorphism:
gap> matgrp:=GL(2,3); GL(2,3) gap> sdp:=SemidirectProduct(matgrp,GF(3)^2); <matrix group of size 432 with 3 generators>
The result is the semidirect product as an affine matrix group, that is the upper left corner is the matrix part and the last row (except from the last entry $1$) is the vector space part.
To get the permutation action on 9 points, you need to get the conjugation action (
OnPoints) of the matrix part, together with the translation action (multiplication,
OnRight) of the vector space part:
gap> normal:=Image(Embedding(sdp,2));; gap> Size(normal); 9 gap> normalelms:=Elements(normal);; gap> matrixpart:=Image(Embedding(sdp,1));; gap> act1:=Action(matrixpart,normalelms,OnPoints); Group([ (4,7)(5,8)(6,9), (2,7,6)(3,4,8) ]) gap> act2:=Action(normal,normalelms,OnRight); Group([ (1,4,7)(2,5,8)(3,6,9), (1,2,3)(4,5,6)(7,8,9) ])
Together this gets the 2-transitive permutation action
gap> permrep:=ClosureGroup(act1,act2); Group([ (4,7)(5,8)(6,9), (2,7,6)(3,4,8), (1,4,7)(2,5,8)(3,6,9) ]) gap> Size(permrep); 432 gap> Transitivity(permrep); 2