Editing Hurwitz's automorphisms theorem (section)

== Examples of Hurwitz groups and surfaces ==
[[File:Small cubicuboctahedron.png|thumb|The [[small cubicuboctahedron]] is a polyhedral immersion of the tiling of the [[Klein quartic]] by 56 triangles, meeting at 24 vertices.<ref>{{Harv|Richter}} Note each face in the polyhedron consist of multiple faces in the tiling – two triangular faces constitute a square face and so forth, as per [http://homepages.wmich.edu/~drichter/images/mathieu/hypercolors.jpg this explanatory image].</ref>]]

The smallest Hurwitz group is the projective special linear group [[PSL(2,7)]], of order 168, and the corresponding curve is the [[Klein quartic|Klein quartic curve]].  This group is also isomorphic to [[PSL(2,7)|PSL(3,2)]].

Next is the [[Macbeath surface|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 group]]s are Hurwitz groups, the largest non-Hurwitz example being of degree 167. The smallest alternating group that is a Hurwitz group is A<sub>15</sub>.

Most [[projective special linear group]]s of large rank are Hurwitz groups, {{harv|Lucchini|Tamburini|Wilson|2000}}. For lower ranks, fewer such groups are Hurwitz.  For ''n''<sub>''p''</sub> the order of ''p'' modulo 7, one has that PSL(2,''q'') is Hurwitz if and only if either ''q''=7 or ''q'' = ''p''<sup>''n''<sub>''p''</sub></sup>.  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'' = 7<sup>4</sup> or ''q'' = ''p''<sup>''n''<sub>''p''</sub></sup>, {{harv|Tamburini|Vsemirnov|2006}}.

Similarly, many [[group of Lie type|groups of Lie type]] are Hurwitz.  The finite [[classical group]]s of large rank are Hurwitz, {{harv|Lucchini|Tamburini|1999}}.  The [[exceptional Lie group]]s of type G2 and the [[Ree group]]s of type 2G2 are nearly always Hurwitz, {{harv|Malle|1990}}.  Other families of exceptional and twisted Lie groups of low rank are shown to be Hurwitz in {{harv|Malle|1995}}.

There are 12 [[sporadic groups]] that can be generated as Hurwitz groups: the [[Janko group]]s J<sub>1</sub>, J<sub>2</sub> and J<sub>4</sub>, the [[Fischer group]]s Fi<sub>22</sub> and Fi'<sub>24</sub>, the [[Rudvalis group]], the [[Held group]], the [[Thompson group (finite)|Thompson group]], the [[Harada–Norton group]], the third [[Conway group]] Co<sub>3</sub>, the [[Lyons group]], and  the [[Monster group|Monster]], {{harv|Wilson|2001}}.