Let $G= C_m \wr C_n$ be a wreath product of cyclic groups $C_m$ and $C_n$. I am interested to find all irreducible representation of $G$.
Update: My thoughts: If I take $H:=\underbrace{C_m \times \cdots C_m}_{n \ times}$, then we know that $G=C_m \wr C_n=H \rtimes C_n$. Also since we know every irreducible characters of cyclic group $C_m$, the irreducible characters of $H$ is known. Now for every $\psi \in {\rm Irr}(H)$, first we need to find $I_G(\psi)$, (the inertia group of $\psi$ in $G$) and then we must find $\varphi \in {\rm Irr}(I)$ that lies over $\psi$, and finally we must apply Clifford's correspondence. I have difficulty in finding $I_G(\psi)$ and $\varphi$.