Doubly Hurwitz Beauville groups

A Beauville surface is a complex algebraic surface of general type, of the form $S = (C_{i} \times C_{2})/G$, where $C_{1}$ and $C_{2}$ are the Belyi curves underlying two regular dessins of genera $g_{1}, g_{2} > 1$, which have the same automorphism $G$ which acts freely on their product. (These surfaces were introduced by Beauville in 1978, and subsequently studied by algebraic geometers such as Catanese, along with Bauer and Grunewald.) The Hurwitz bound implies that $|G| \leq 1764 \chi(S)$, with equality if and only if the Beauville group $G$ acts as a Hurwitz group on both curves $C_{i}$. Equivalently, $G$ has two generating triples of type (2, 3, 7), such that no generator in one triple is conjugate to a power of a generator in the other. In joint work with Emilio Pierro, and in answer to a question of Sasha Zvonkin, we show that this property is satisfied by all alternating groups of sufficiently large degree, together with their double covers, but by no sporadic simple groups or simple groups in various families of small Lie rank.