I am trying to use GAP to identify subgroups of $S_n$ with two generators. As an example, in one case the generators are $a=(1~2)(3~4)$ and $b=(1~3)$, in which case $\langle a,b \rangle \cong D_4$. However, many of the groups I'd like to identify are much larger.
A group that I have not yet identified has generators $a=(1~2~3)(4~5~6)(7~8~9)(10~11~12)$ and $b=(1~7)(2~10)(4~8)(5~11)$. Using the IdGroup function in GAP, I have determined that it is the $72$nd group of order $96$ in the SmallGroup library. If I ask GAP for the $72$nd group of order $96$ in the SmallGroup library, it says it is a polycyclic group with $6$ generators, but that does not tell me the structure of the group.
Are there some other techniques I can use in GAP to understand the group $\langle a,b\rangle$ in terms of familiar groups (that are smaller than $S_{12}$)? I would like a technique that is as general as possible, since I will be trying to identify many more groups with different generators.