Suppose that $V$ is a finite-dimensional complex vector space, that $m\ge 2$ is an integer and that $G\subset \operatorname{GL}(V)$ is a finite subgroup such that $V$ is an irreducible ${\mathbb{C}}[G]$-module and $G$ is generated by a single $G$-conjugacy class of complex reflections of order $m$.
Of course, the finite complex reflection groups were classified 70 years ago by Shephard and Todd. However, the hypothesis that there is a single conjugacy class of generating reflections is slightly stronger than usual. Is there a reference for the classification of the pairs $(G,m)$?
