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

4d $\mathcal{N}=3$ indices via discrete gauging