the surfaces (triangulation, tiling by heptagons, etc.). From the arguments above it can be inferred that a Hurwitz group G is characterized by the property that it is a finite quotient of the group with two generators a and b and three relations thus G is a finite group generated by two elements of orders two and three, whose product is of order seven. More precisely, any Hurwitz surface, that is, a hyperbolic surface that realizes the maximum order of the automorphism group for the surfaces of a given genus, can be obtained by the construction given. This is the last part of the theorem of Hurwitz. Examples of Hurwitz groups and surfaces The smallest Hurwitz group is the projective special linear group PSL(2,7), of order 168, and the corresponding curve is the Klein quartic curve. This group is also isomorphic to PSL(3,2). Next is the Macbeath curve, with automorphism group PSL(2,8) of order 504. Many more finite simple groups are Hurwitz groups; for instance all but 64 of the alternating groups are Hurwitz groups, the largest non-Hurwitz example being of degree 167. The smallest alternating group that is a Hurwitz group is A15. Most projective special linear groups of large rank are Hurwitz groups, . For lower ranks, fewer such groups are Hurwitz. For np the order of p modulo 7, one has that PSL(2,q) is Hurwitz if and only if either q=7 or q = pnp. Indeed, PSL(3,q) is Hurwitz if and only if q = 2, PSL(4,q) is never Hurwitz, and PSL(5,q) is Hurwitz if and only if q = 74 or q = pnp, . Similarly, many groups of Lie type are Hurwitz. The finite classical groups of large rank are Hurwitz, . The exceptional Lie groups of type G2 and the Ree groups of type 2G2 are nearly always Hurwitz, . Other families of exceptional and twisted Lie groups of low rank are shown to be Hurwitz in . There are 12 sporadic groups that can be generated as Hurwitz groups: the Janko groups J1, J2 and J4, the Fischer groups Fi22 and Fi'24, the Rudvalis group, the Held group, the Thompson group, the Harada–Norton group, the third Conway group Co3, the Lyons group, and the Monster, . Automorphism groups in low genus The largest |Aut(X)| can get for a Riemann surface X of genus g is shown below, for 2≤g≤10, along with a surface