# Dimension of faithful irreducible representations of $\mathbb{Z}_q\rtimes \mathbb{Z}_{p^2}$ in characteristic p,q
Let $p,q$ be primes s.t. $q=np+1$. Denote $m=p^2$. Then $\mathbb{Z}_p$ acts non-trivially on $\mathbb{Z}_q$, so we have a non-abelian semi-direct product $\mathbb{Z}_q\rtimes \mathbb{Z}_m$, with the action of $\mathbb{Z}_p$. This group has a faithful irreducible representation.
Over $\mathbb{C}$, every faithful irreducible representation must be of dimension $p$, because the dimension must divide $p^2q$, it cannot be 1 since the group is non-abelian, and it cannot be greater than $p$ because there is an abelian subgroup of order $pq$.
Is it possible to bound the dimensions of the faithful irreducible representations in characteristic $p,q$ as well?
Also, is there a simple way to check if the faithful irreducible representation I have is also irreducible in characteristic $p,q$?