  Within the Schottky problem, the study of special subvarieties of the Torelli
locus has long been of great interest. We describe a representation-theoretic
criterion for a Jacobian variety arising from a $G$-Galois cover of
$\mathbb{P}^1$ branched at $3$ points to have complex multiplication (CM). For
$G$ faithful metacyclic or dicyclic, we classify all such covers with Galois
group $G$, identifying those that have CM. We compute the CM-field and type of
Jacobian varieties arising from these covers, applying the representation
theory of $G$ over $\mathbb{Q}$ and $\mathbb{Q}(\zeta_4)$. In particular,
symplectic irreducible representations of $G$ are afforded by the Jacobian
variety in the dicyclic case, giving rise to new examples of CM abelian
varieties.
