Dihedral symmetry in SU (N) Yang-Mills theory

We point out that charge conjugation and coordinate reflection symmetries do not commute with the center symmetry of SU(N) Yang-Mills (YM) theory when N>2. As a result, for generic values of the θ angle, the group of discrete 0-form symmetries of YM theory on, e.g., the spacetime manifold R3×S1 includes the dihedral group D2N, which is non-Abelian for N>2. At θ=π, the non-Abelian factor in the symmetry group is enhanced to D4N due to discrete 't Hooft anomaly considerations. We illustrate these results in YM theory as well as in a simple quantum mechanical model, where we study representation theory as a function of the θ angle.