Regular embeddings of Kn, n where n is an odd prime power

We show that if n=p^e where p is an odd prime and e>=1, then the complete bipartite graph K"n","n has p^e^-^1 regular embeddings in orientable surfaces. These maps, which are Cayley maps for cyclic and dihedral groups, have type {2n,n} and genus (n-1)(n-2)/2; one is reflexible, and the rest are chiral. The method involves groups which factorise as a product of two cyclic groups of order n. We deduce that if n is odd then K"n","n has at least n/@?"p"|"np orientable regular embeddings, and that this lower bound is attained if and only if no two primes p and q dividing n satisfy p=1mod(q).