An integer n\geq2 is said to be a genus of a finite group G if there is a compact Riemann surface of genus n on which G acts as a group of automorphisms. In this paper, formulae are given for the minimum genus, minimum stable genus and the gap sequence, i.e., the (finite) set of non-genera, for a split metacyclic group of order pq, where p and q are primes. This information completely determines the genus spectrum for such groups.