Doubly Hurwitz Beauville groups

If $\mathcal S$ is a Beauville surface $({\mathcal C}_1\times{\mathcal C}_2)/G$, then the Hurwitz bound implies that $|G|\le 1764\,\chi({\mathcal S})$, with equality if and only if the Beauville group $G$ acts as a Hurwitz group on both curves ${\mathcal C}_i$. 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. We show that this property is satisfied by alternating groups $A_n$, their double covers $2.A_n$, and special linear groups $SL_n(q)$ if $n$ is sufficiently large, but by no sporadic simple groups or simple groups $L_n(q)$ ($n\le 7$), ${}^2G_2(3^e)$, ${}^2F_4(2^e)$, ${}^2F_4(2)'$, $G_2(q)$ or ${}^3D_4(q)$ of small Lie rank.