Let $G=H\times K$ be the direct product of finite groups. Let $\varphi\in Irr(H)$ and $\eta\in Irr(K)$ be faithful. Show that $\varphi\times\eta$ is faithful if and only if $(|Z(H)|,|Z(K)|)=1$.
Here, $\varphi\times \eta$ is a character of $G$ defined by $(\varphi\times\eta)(h,k)=\varphi(h)\eta(k)$.
Thanks in advance.