Examples
The Hurwitz surface of least genus is the Klein quartic of genus 3, with automorphism group the projective special linear group PSL(2,7), of order 84(3−1) = 168 = 22·3·7, which is a simple group; (or order 336 if one allows orientation-reversing isometries). The next possible genus is 7, possessed by the Macbeath surface, with automorphism group PSL(2,8), which is the simple group of order 84(7−1) = 504 = 22·32·7; if one includes orientation-reversing isometries, the group is of order 1,008.
An interesting phenomenon occurs in the next possible genus, namely 14. Here there is a triple of distinct Riemann surfaces with the identical automorphism group (of order 84(14−1) = 1092 = 2·3·7·13). The explanation for this phenomenon is arithmetic. Namely, in the ring of integers of the appropriate number field, the rational prime 13 splits as a product of three distinct prime ideals. The principal congruence subgroups defined by the triplet of primes produce Fuchsian groups corresponding to the first Hurwitz triplet.
Read more about this topic: Hurwitz Surface
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