Orientable vertex transitive embeddings of $ {{\sf K}}_p $

In [J. Combin. Theory Ser. B, 99 (2009), 447-454)], Li characterized the classification of vertex-transitive embeddings of complete graphs, and proposed how to enumerate such maps. In this paper, we study the counting problem of orientable vertex-transitive embeddings of $ {{\sf K}}_p $, where $ p\geq 5 $ is a prime. Moreover, we obtain the number of non-isomorphic orientable vertex-transitive complete maps with $ p $ vertices.