Semidirect product command in GAP

Question

I need to type the group

in GAP. Is it correct if I type as, "SemidirectProduct(ZmodnZ(3),AbelianGroup([7,7]));"? Will it recognize that AbelianGroup([7,7]) is the normal subgroup?

Answer 1

You will need to also specify the action of $Z_3$ on $Z_7^2$. Generically, you would do this by specifying as second argument (i.e. arguments are subgroup, map, normal subgroup) a homomorphism from $Z_3$ into the automorphism group os $Z_7^2$.

As your normal subgroup is a vector space, this whole construction however is easier done by describing the action through matrices. for example (not sure whether this is the action you want):

gap> m:=[[4,0],[0,2]]*One(GF(7));
[ [ Z(7)^4, 0*Z(7) ], [ 0*Z(7), Z(7)^2 ] ]
gap> s:=SemidirectProduct(Group(m),GF(7)^2);
<matrix group of size 147 with 3 generators>