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