A hyperelliptic variety is a quotient $X = T/G$ of a complex torus $T$ by a finite group $G$ acting freely and without translations. This site collects the classification of such varieties in dimensions $2$, $3$ and $4$, together with the invariants computed from the tangent representation $\rho\colon G \to \mathrm{GL}(V)$.
dimension 2
The seven Bagnera–De Franchis types of hyperelliptic (bielliptic) surfaces.
dimension 3
Hyperelliptic threefolds: $17$ holonomy groups, completely classified.
dimension 4
The $79$ hyperelliptic groups in dimension $4$ and their invariants.
The Hodge diamond, the order of the canonical bundle, the number of moduli, the irregularity, the polyvector fields and more depend only on $(G,\rho)$ and are computed by character theory; see the explained pages (starting with what we do and do not mean).