Superelliptic curves with many automorphisms and CM Jacobians

Let $\mathcal{C}$ be a smooth, projective, genus $g\geq 2$ curve, defined over $\mathbb{C}$. Then has \emph{many automorphisms} if its corresponding moduli point $p \in \mathcal{M}_g$ neighborhood $U$ in the complex topology, such that all curves to points $U \setminus \{p \}$ have strictly fewer automorphisms than $\mathcal{C}$. We compute completely list of superelliptic having many automorphisms. For each these curves, we determine whether Jacobian multiplication. As consequence, prove converse Streit's multiplication criterion for curves.