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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, $Z(\chi)/\ker\chi=Z(G/\ker\chi)$.

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  • $\begingroup$ As an example, any split metacyclic $p$-group of odd order has the mentioned property. Also, the same holds for split metacyclic $2$-groups with few exceptions. $\endgroup$ Jun 12, 2015 at 9:17

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These appear to be the groups that I was asking about in this older MO question -- see in particular the answers by Isaacs and Ladisch.

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