On Semi-isogenous mixed surfaces

Let $C$ be a smooth projective curve and $G$ a finite subgroup of $\mathrm{A}\mathrm{u}\mathrm{t}(C{)}^2⋊{\mathbb{Z}}_2$ whose action is $\mathit{\text{mixed}}$, i.e.~there are elements in $G$ exchanging the two isotrivial fibrations of $C\times C$. Let $G^0◃G$ be the index two subgroup $G\cap \mathrm{A}\mathrm{u}\mathrm{t}(C{)}^2$. If $G^0$ acts freely, then $X:=(C\times C)\mathrm{/}G$ is smooth and we call it $\mathit{\text{semi-isogenous}}\mathit{\text{mixed}}\mathit{\text{surface}}$. In this paper we give an algorithm to determine semi-isogenous mixed surfaces with given geometric genus, irregularity and self-intersection of the canonical class. As an application we classify irregular semi-isogenous mixed surfaces with $K^2>0$ and geometric genus equal to the irregularity; the regular case is subjected to some computational restrictions. In this way we construct new examples of surfaces of general type with $\chi =1$. We provide an example of a minimal surface of general type with $K^2=7$ and $p_g=q=2$.