Suppose $\varphi: Z(G) \to GL(V)$ is an irreducible representation of the center of a non-abelian group $G$. I want to show that $Ind^G_{Z(G)} \varphi$ is not irreducible. Any hints?
So far, I have consider that $Z(G)$ is a normal subgroup. Moreover, the character is given by $$ \chi_{Ind^G_{Z(G)}} \varphi(g) = Ind^G_{Z(G)}(g) = \frac{1}{|Z(G)|} \sum_{x \in G} \dot{\chi}(x^{-1}gx). $$