Let $G = Af$ $f(\mathbb{F}_p)$ be the group of affine transformation, i.e., $G = \{f: \mathbb{F}_p \to \mathbb{F}_p, x \mapsto ax+b| a, b \in \mathbb{F}_p, a\ne 0\}$ Let $H$ be the cyclic subgroup of order $p$. Find the decomposition of $Ind_H^G V$ into sum of irreducibles for an arbitrary irreducible representation $V$ of $H$.
I have found out that $G$ has $p-1$ one dimensional representations and one $p-1$-dimensional representation. But after that, I can't progress. Any help will be appreciated.
The representation $V$ is a character $\chi\colon H=\mathbb F_p\to\mathbb C^\times$.
If $\chi=1$, then $Ind_H^G1_H$ then each character of $G$ is trivial when restricted to $H$, so each of the $p-1$ characters are a sub-representation of $Ind_H^G1_H$. Since $Ind_H^G1_H$ is $(p-1)$-dimensional, $Ind_H^G1=\bigoplus_{\chi\colon G/H\to\mathbb C^\times}\chi$.
Otherwise if $\chi\ne1$ then $Ind_H^G\chi$ is irreducible, by a Mackey theory computation (and does not depend on the choice of $\chi$, since $G$ acts transitively on non-trivial characters of $H$).