Let $p \geq 5$ be a prime and let $n$ be positive integer. In his Ph.D thesis (See *The characters of binary modular congruence group*, Bulletin of the
American Mathematical society. 79 (1973), no. 4.) P. Kutzko wrote the character tables of the groups
`\[G_{n}:=\text{SL}_{2}(\mathbb{Z}/p^{n+1}\mathbb{Z})/\pm I.\]`

Let $C_{n} := \textbf{O}_{p}(G_n)$ - the $p$-core of $G_n$ -

**Question:** Does anyone know if the character tables of the groups $C_n$ have been written up, and if so where can I find them?

It might be possible that from knowing the character tables of the groups $G_{n}$ one can construct the character tables of the $C_n$'s (This doesn't sound right in general but the groups $(G_{n}, C_{n})$ have some nice properties). Here I list some properties of the groups $G_n$ and $C_n$ that I find interesting, and that I think could help in calculating the tables for $C_n$ having that one knows the ones for $G_n$.

The finite groups $G_n$ form a projective system (here $G_n \rightarrow G_m$ is the natural reduction map whenever $n \geq m$).

The $p$-core of $G_n$ is ``big'', meaning that for all $n$, and for all $P \in \text{Syl}_{p}(G_n)$ we have that $[P:C_n]=p$.

It is not hard to see that $C_n=Ker(G_n \rightarrow G_0)$. In fact, any normal subgroup of $G_n$ is the Kernel of some $G_n \rightarrow G_m$.

For all $n$ we have that $C_{n}/Z(C_{n}) \cong C_{n-1}$. Moreover, $Z(C_{n})$ is minimal normal subgroup of $G_n$ and $G_{n}/Z(C_{n}) \cong G_{n-1}$.