Let $\mathcal O$ be an order in an imaginary quadratic field $K$. Let $n$ be a positive integer. The multiplicative group $(\mathcal O/n\mathcal O)^\times$ acts on the module $\mathcal O/n\mathcal O\cong \mathbb Z / n\mathbb Z\times \mathbb Z / n\mathbb Z$. We get thus an embedding of $(\mathcal O/n\mathcal O)^\times$ into $\operatorname{GL}_2(\mathbb Z / n\mathbb Z)$. Let $C_n$ be the image.
Is the centralizer of $C_n$ in $\operatorname{GL}_2(\mathbb Z / n\mathbb Z)$ equal to $C_n$ itself?
I am interested especially in the case when $n$ is a power of two.
