Hurwitz's automorphisms theorem

The Hurwitz's automorphisms theorem is a theorem about the automorphisms of compact Riemann surfaces of genus &geq; 2, with implications for regular maps.

Regular maps
An orientable regular map, of genus $$g \geq 2$$, can have at most $$168(g-1)$$ flags. A non-orientable regular map, of $$g \geq 2$$, can have at most $$84(g-1)$$ flags

Automorphism groups
The automorphism group, $$\mathrm{Aut}(X)$$, of a compact Riemann manifold, $X$, is the group of orientation-preserving conformal mappings from $X$ to $X$. For a smooth connected Riemann manifold of genus $g$ &geq; 2, its automorphism group, $$\mathrm{Aut}(X)$$, has at most order $$84(g-1)$$.

Examples
For regular maps of genus 0 and 1 the Hurwitz bound does not apply. Regular maps of these genera can can have arbitrarily high flag counts. For example in $n$-gonal duocombs are genus 1 maps with $$8n^2$$ flags.