Let us consider two singularities $\mathbb C^n/G$ and $\mathbb C^n/G'$, where $G$ and $G'$ are finite subgroups of $\mathrm{GL}(n,\mathbb{C})$ acting linearly.

It is easy too see, that a different groups may give the same singularities. For example, $G'=G/\langle g\rangle$, where $\langle g\rangle$ is a subgroup of $G$ generated by a reflection $g$.

Which groups $G$ and $G'$ give the same singularity?