It is well-known that every non-linear character $\chi$ of a finite group $G$ vanishes on some elements of $G\setminus Z(\chi)$. The question is
What can be said about a finite group $G$ for which $\chi(x)=0$ for all irreducible character $\chi$ and elements $x\in G\setminus Z(\chi)$?
Here, $Z(\chi)=\{g\in G:|\chi(g)|=\chi(1)\}$. So, $Z(\chi)/\ker\chi=Z(G/\ker\chi)$.