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 $\Gamma$ 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!

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!

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 degree one irreps (i.e., $Z(G)$ is a subgroup of the commutator group $[G, G]$);
2. There exists a degree $n>1$ 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 $\Gamma$ 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!

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 $\Gamma$ 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!