  We study a rich set of four-dimensional superconformal field theories (SCFTs)
with both central charges identical: $a = c$. These are constructed via the
diagonal $\mathcal{N}=2$ or $\mathcal{N}=1$ gauging of the flavor symmetry $G$
of a collection of $\mathcal{N}=2$ Argyres-Douglas theories of type
$\mathcal{D}_p(G)$, with or without adjoint chiral multiplets, in
arXiv:2106.12579 and arXiv:2111.12092. We compute superconformal indices of
some theories where the rank of $G$ is low, performing a refined test for
unitarity, and further determine the relevant and marginal operator content in
detail. We find that most of these theories flow to interacting SCFTs with
$a=c$ in the infrared.
