If S is a Beauville surface (C 1 × C 2 )/G, then the Hurwitz bound implies that |G| ≤ 1764 χ(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. 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 ≤ 7), 2 G 2 (3 e ), 2 F 4 (2 e ), 2 F 4 (2) ′ , G 2 (q) or 3 D 4 (q) of small Lie rank.

We prove that the smallest non-trivial quotient of the mapping class group of a connected orientable surface of genus g \geq 3 without punctures is Sp _{2g}(2) , thus confirming a conjecture of Zimmermann. In the process, we generalise Korkmaz’s results on \mathbb C -linear representations of mapping class groups to projective representations over any field.

We show that the smallest non‐abelian quotient of Aut(Fn) is prefixPSLnfalse(double-struckZ/2double-struckZfalse)=prefixLnfalse(2false), thus confirming a conjecture of Mecchia–Zimmermann. In the course of the proof we give an exponential (in n) lower bound for the cardinality of a set on which SAut(Fn), the unique index 2 subgroup of Aut(Fn), can act non‐trivially. We also offer new results on the representation theory of SAut(Fn) in small dimensions over small, positive characteristics and on rigidity of maps from SAut(Fn) to finite groups of Lie type and algebraic groups in characteristic 2.

We generalise a construction of mixed Beauville groups first given by Bauer, Catanese and Grunewald. We go on to give several examples of infinite families of characteristically simple groups that satisfy the hypotheses of our theorem and thus provide a wealth of new examples of mixed Beauville groups.

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