TITLE: Classifying the irreducible representations of $\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/n \mathbb{Z}$
QUESTION [11 upvotes]: I need help with the following problem:

Suppose $n$ is some positive integer, and $n|p-1$.  Classify all irreducible representations of  $$\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/n \mathbb{Z}.$$

I am not sure where to start,
Thank you in advance!!

REPLY [7 votes]: Have a look at Weintraub - "Representation theory of finite group" in the section Mackey Machine. They give the right tool for exactly studying the irreducible representation of a semidirect product $H \rtimes G$, where $H$ is abelian.
In fact, the Mackey Machine holds in greater generality for an exact sequence of locally compact groups
$ 1 \rightarrow H \rightarrow K \rightarrow G \rightarrow 1,$
with some mild conditions on $H$ (being type 1).