Hurwitz bound (awaiting review)

For any integer $g > 1$, the automorphism group of any curve of genus $g$ over $\C$ has order at most $84(g-1)$. This is not best possible for every $g$, but there do exist infinitely many $g$ for which the bound is achieved; examples include Klein's quartic curve ($g=3$) and the Fricke-Macbeath curve ($g=7$). See also Wikipedia.