Metacyclic groups as automorphism groups of compact Riemann surfaces

Let X be a compact Riemann surface of genus (gge 2), and let G be a subgroup of (mathrm{Aut}(X)). We show that if the Sylow 2-subgroups of G are cyclic, then (|G|le 30(g-1)). If all Sylow subgroups of G are cyclic, then, with two exceptions, (|G|le 10(g-1)). More generally, if G is metacyclic, then, with one exception, (|G|le 12(g-1)). Each of these bounds is attained for infinitely many values of g.