# Which group of order 96 is this group?

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.

• Is there a particular reason you expect this group to be a nice combination of commonly considered groups? After all, there are 231 groups of order 96. Jun 30, 2018 at 3:12
• Not a precise reason. It arises as an automorphism group of a Riemann surface and I would like to know which of the 231 groups it is in order to compare it to some automorphism groups of related Riemann surfaces. I was unable to figure out how to specify enough information for GAP to identify the group. Jun 30, 2018 at 3:14
• GAP id's it as small group [96,64], and the structure description calls it ((C4 x C4) : C3) : C2 Jun 30, 2018 at 3:24
• people.maths.bris.ac.uk/~matyd/GroupNames/73/Dic6sC4.html Jun 30, 2018 at 3:37
• Which Riemann surface? How do you know it is the automorphism group of that surface? Possibly a knowledge of the properties of the surface could help answer the question. Jun 30, 2018 at 3:51

f:=FreeGroup("a","b");

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