$\mathcal{N}=5$ SCFTs and quaternionic reflection groups

It was previously noted that for 3d SCFTs with $\mathcal{N}\ge 6$ the moduli space has the form of ${\mathbb{C}}^{4r}\mathrm{/}\mathrm{\Gamma }$, where $\mathrm{\Gamma }$ is a complex reflection group, at least following suitable gauging of finite symmetries. Here we argue that this observation can be extended also to 3d SCFTs with $\mathcal{N}\ge 5$ SUSY, where $\mathrm{\Gamma }$ is now a quaternionic reflection group. To do this, we study the moduli space of the known 3d $\mathcal{N}=5$ SCFTs. For Lagrangian cases, the results for the moduli space are further checked using the superconformal index.