Alternating groups, Hurwitz groups and H*-groups

Riemann surfaces of genus g admit at most 84(g−1) automorphisms. The group attaining this bound is called a Hurwitz group. A group G is a Hurwitz group if and only if it has a pair of generators of order 2 and 3 whose product has order 7. Each alternating group An for n>168 is a Hurwitz group and most cases with n<168 are too, but the suitable generators are not explicitly known. In this paper we obtain all pairs of such generators of Hurwitz groups An for n<35, namely A15, A21, A22, A28, A29. These results are used to deal with the corresponding problem on non-orientable surfaces. In this case the question is stated in terms of finding a third element of order 2 whose products with the previous elements have also order two. In particular, we obtain that A15 and A28 match the bound for non-orientable surfaces (that is to say they are H*-groups) whilst A21, A22, and A29 do not. As a byproduct we obtain other Hurwitz groups which are proper subgroups of An and give some examples of such generators for them.