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18.114 (2014)

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Does there exist an irreducible $5'$-subgroup $G$ of $\text{GL}(V)$ for some finite $\mathbb{F}_5$-space $V$ such that the number of conjugacy classes of the semidirect product $V \rtimes G$ is equal to $|V|$ but $G$ is not cyclic?

Such a subgroup exists if 5 is replaced by $p = 2, 3$ but does not exist for primes $p > 5$ (J. Group Theory, 14 (2011), 175–199).

Contributor: P. Schmid

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