I think both the genus g and the Euler characteristic bear on what symmetries a surface can have. The (orientable, compact) surface M_g of genus g always has a metric with an isometry of order g: Just arrange the holes in a circle. The surface of genus g = K + 1 always has a metric with an isometry of order K: Arrange one hole in the middle and the other K around it. The surface M_3 of genus 3 (with X(M_g) = -4) has a metric with an isometry of order 7. M_3 cannot be exhibited in 3-space with this 7-fold symmetry. It has other metrics, like one having 192 symmetries, including a subgroup of order 64. It has no metric with both a 7-fold symmetry and a subgroup of order 64, since a metric surface of genus g can have at most 168 * (g-1) symmetries and 7*64 = 448 > 336 would violate that theorem. —Dan Allan Wechsler wrote: ----- My thought is that if the genus is 73, the Euler characteristic is -144. 73 may seem like a number that is not very conducive to fancy symmetry, but -144 seems a lot more promising. I'm guessing that the Euler characteristic is more important to allowable symmetries than the genus is. Other examples can be harvested from the thickened skeleta of the platonic solids treated as surfaces. I expect the genus plus one to be a nicely divisible number. The tetrahedron's skeleton has genus 3; 4 has better divisibility. The cube has genus 5 but 6 is better. The octahedron: 7 versus 8. The dodecahedron: 11 versus 12. The icosahedron: 19 versus 20. In other words, I don't think the symmetries are juggling a set whose size is the genus; I think the relevant size is the genus plus 1. -----