Discrete gauge theories of charge conjugation

We define gauge theories whose gauge group includes charge conjugation as well as standard $\mathrm{SU}(N)$ transformations. When combined, these transformations form a novel type of group with a semidirect product structure. For $N$ even, we show that there are exactly two possible such groups which we dub $\widetilde{\mathrm{SU}}(N)_{\mathrm{I,II}}$. We construct the transformation rules for the fundamental and adjoint representations, allowing us to explicitly build four-dimensional $\mathcal{N}=2$ supersymmetric gauge theories based on $\widetilde{\mathrm{SU}}(N)_{\mathrm{I,II}}$ and understand from first principles their global symmetry. We compute the Haar measure on the groups, which allows us to quantitatively study the operator content in protected sectors by means of the superconformal index. In particular, we find that both types of $\widetilde{\mathrm{SU}}(N)_{\mathrm{I,II}}$ groups lead to non-freely generated Coulomb branches.