Which group of order 96 is this group?

Question

I have a group of order 96, and I am wondering which combination of familiar groups it might be. I have tried and failed to identify it with a semidirect product of cyclic groups.

$$G_1 = \langle a, b \mid a^8 = b^3 = (ab) ^2 = (a^2b^2)^3 = (a^4b^2)^3 = 1 \rangle$$

Thank you for your help.

Answer 1

Here are the GAP commands I used:

f:=FreeGroup("a","b");
a:=f.1;b:=f.2;
g:=f/[a^8,b^3,(a*b)^2,(a*a*b*b)^3,(a^4*b^2)^3];
IdGroup(g);
StructureDescription(g);

GAP reports this is small group [96,64] with structure description $((C_4\times C_4):C_3):C_2)$