Exceptional moduli spaces for exceptional $$ \mathcal{N} $$ = 3 theories

It is expected on general grounds that the moduli space of 4d $\mathcal{N}=3$ theories form $\mathbb{C}^{3r}/\Gamma$, with $r$ rank and $\Gamma$ a crystallographic complex reflection group (CCRG). As in case Lie algebras, CCRGs consists several infinite families, together some exceptionals. To date, no theory labelled by an exceptional CCRG (excluding Weyl groups) has been identified. In this work we show proposed \cite{Garcia-Etxebarria:2016erx}, constructed via non-geometric quotients type-$\mathfrak{e}$ 6d (2,0) theories, realize nearly all such spaces. addition, introduce extension construction to allow for twists outer automorphism symmetries. This gives new examples going beyond simple S-folds.