  A class of 4d $\mathcal{N}=3$ SCFTs can be obtained from gauging a discrete
subgroup of the global symmetry group of $\mathcal{N}=4$ Super Yang-Mills
theory. This discrete subgroup contains elements of both the $SU(4)$ R-symmetry
group and the $SL(2,\mathbb{Z})$ S-duality group of $\mathcal{N}=4$ SYM. We
give a prescription for how to perform the discrete gauging at the level of the
superconformal index and Higgs branch Hilbert series. We interpret and match
the information encoded in these indices to known results for rank one
$\mathcal{N}=3$ theories. Our prescription is easily generalised for the
Coloumb branch and the Higgs branch indices of higher rank theories, allowing
us to make new predictions for these theories. Most strikingly we find that the
Coulomb branches of higher rank theories are generically not-freely generated.

High Energy Physics - Theory