Engineer here.
I'm confused about representation theory of finite groups. For some program I am writing, I have to work with arithmetic modulo $p^k,$ where $p$ is a prime, and $k \ge 2.$ That is, numbers from $\mathbb{Z}/p^k\mathbb{Z} = \{ 0, 1, \ldots, p^k - 1\}.$ If I understand it correctly, linear representation theory allows me to map my computations from scalars into the domain of matrices through a homomorphism: $h : \mathbb{Z}/p^k\mathbb{Z} \to {\rm GL}(n, \mathbb{C})$
Quote from the wikipedia article:
Thus to specify a representation, we just assign a square matrix to each element of the group, in such a way that the matrices behave in the same way as the group elements when multiplied together.
which is great since modular reduction computations are slow/hard in my target platform, but linear algebra is easy/fast. So I really want to replace my scalaraarithmetic operations $\{+, -, \times\} \bmod {p^k}$ with matrix operations $\{+, -, \times\}.$
My questions:
Is linear representation what I need?
Given $p, k,$ how can I choose $n$?
How can I construct the mapping $h$ and its inverse $h^{-1}$?
References are welcome as answers. But I'm really interested in the computational solutions that work (even if it fails few times.)