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I have been searching for a finite non-abelian group $G$ with the following properties:

  1. Its center $Z(G)$ acts as the identity in all dimension one irreps (i.e., $Z(G)$ is a subgroup of the commutator group $[G, G]$);
  2. There exists a dimension $n>1$ irrep ${\Gamma}$ for which at least one element $z\in Z(G)$ satisfies $\mathrm{order}(\chi_\Gamma(z)/n) \neq \gcd (\,\mathrm{order}(\chi_\Gamma(z)/n),n\,)$. Here, $\chi_\Gamma(z)$ is the character of $z$ in the irrep $\Gamma$ (i.e., the trace of the $n\times n$ matrix $\Gamma(z)$) and $\mathrm{order}(e^{i\theta})$ is the smallest positive integer $m$ such that $e^{i\theta m} = 1$

I have tried using GAP to brute search for such a group but have had no luck. I am starting to wonder if there is a basic reason why such a group cannot exist, and I would appreciate any help!

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    $\begingroup$ By the first assumption every element of the center has trivial determinant in every representation, so in an $n$-dimensional irreducible representation it has to act by a scalar which is an $n$-th root of unity, which is to say that the multiplicative order of this scalar $\chi_{\Gamma}(z)/n$ divides $n$. So you are right that such a group does not exist. $\endgroup$
    – SashaP
    Commented Sep 7 at 3:39
  • $\begingroup$ @SashaP Ah, it is now totally obvious. Thank you for answering my very basic question! $\endgroup$
    – Sal Pace
    Commented Sep 7 at 13:14

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