Does anybody know the answer to this or a good way to go about working this out? I have a list for $GL\_2(Z/pZ)$ and I am trying to lift it to this; I have mostly been using fairly elementary algebraic methods. Thank you.



List of Conjugacy Classes of the General Linear group over $\mathbb{Z}/p^2\mathbb{Z}$
I don't have time to give a full answer now, but here's how I would try and answer this question:


You first need to establish what the normal subgroups of $GL\_2(\mathbb{Z}/p^2\mathbb{Z})$ are. This is done in a fair amount of detail at [this MO question](https://mathoverflow.net/questions/104262/normal-subgroup-of-classical-groups/109886) (at least for $SL\_2$, and the same method will generalize to $GL\_2$).


Now, as explained in the linked answer above, $G=GL\_2(\mathbb{Z}/p^2\mathbb{Z})$ has a normal subgroup $N$ of order $p^3$ which can be thought of as a 3-dimensional module for the group $G/N\cong GL\_2(\mathbb{Z}/p\mathbb{Z})$. To get the full-list of conjugacy classes, one needs to examine each coset $gN$ in turn, considering it as a module for the group $C\_{G/N}(gN)$.


