Wreathing, discrete gauging, and non-invertible symmetries

’t Hooft anomalies of discrete global symmetries and gaugings thereof have rich mathematical structures and far-reaching physical consequences. We examine each subgroup G, up to automorphisms, of the permutation group S4 that acts on the four legs of the affine D4 quiver diagram, which is mirror dual to the 3d N = 4 SU(2) gauge theory with four flavours. These actions are studied in terms of how each permutation cycle acts on the superconformal index of the theory in question. We present a prescription for refining the index with respect to the fugacities associated with the Abelian discrete symmetries that are subgroups of G. This allows us to study sequential gauging of various subgroups of G and construct symmetry webs. We study the effects of ’t Hooft anomalies and non-invertible symmetries that arise from discrete gauging on the index. When the whole symmetry G is gauged, our results are in perfect agreement with a type of discrete operations on the quiver, known as wreathing, discussed in the literature. We provide a general prescription for computing the index for any wreathed quivers that contain unitary or special unitary gauge groups. We demonstrate this in an example of the 3d N = 4 U(N) gauge theory with n flavours and compare the results with gauging the charge conjugation symmetry associated with the flavour symmetry of such a theory.