Direct and semidirect products in GAP

Question

Can someone please tell me the necessary codes/commands to enter the below group in GAP.

Thanks a lot in advance.

Answer 1

For things like this, I like to work with generators and relators, because then it's easy for me to be very clear about what the automorphisms are doing to the normal subgroup:

> f := FreeGroup("x", "y", "z");
> AssignGeneratorVariables(f);
> relations := "x^7=y^7=1, [x,y]=1, z^3=1, x^z=foo, y^z=bar";
> r := ParseRelators([x,y,z], relations);
> g := f/r;

Here foo and bar are the expressions in x and y that determine your action.

Of course, this gets more tedious when your base groups aren't abelian, but for this kind of group, it's not too bad.