Hurwitz group

Definition
A Hurwitz group is a finite group that occurs (up to isomorphism) as the automorphism group of a Riemann surface of genus $$g > 1$$, and has the maximum possible order $$84(g - 1)$$ for such a group. Note that it is in general true that for any Riemann surface of genus $$g > 1$$, the conformal automorphism group is finite of order at most $$84(g - 1)$$, and Hurwitz groups are groups where equality is attained. The corresponding Riemann surfaces are termed Hurwitz surfaces.

Note that, in particular, the strong symmetric genus of a Hurwitz group is this genus $$g$$. Working backward, if the group is $$G$$, then its strong symmetric genus $$\sigma^\circ(G) = g = 1 + (|G|/84)$$.

By definition, the order of any Hurwitz group must be a multiple of 84.

Hurwitz groups can be obtained as suitably constructed quotients of the (7,3,2)-von Dyck group.

Families of examples

 * Alternating groups of sufficiently large degree are Hurwitz groups: All except 64 alternating groups are Hurwitz groups. The smallest Hurwitz alternating group is alternating group:A15 and the largest non-Hurwitz alternating group is the alternating group of degree 167.
 * There are congruence conditions that describe when the projective special linear group of degree two over a finite field is Hurwitz.
 * Twelve of the sporadic simple groups are Hurwitz: Janko group:J1, Janko group:J2, Janko group:J4, Fischer group:Fi22, and others.