Enumeration of dicirculant digraphs

Let T4p=⟨a,b∣a2p=1,ap=b2,b-1ab=a-1⟩ be the dicyclic group of order 4p. A Cayley digraph over T4p is called a dicirculant digraph. In this paper, we calculate the number of (connected) dicirculant digraphs of order 4p (p prime) up to isomorphism by using the Pólya Enumeration Theorem. Moreover, we get the number of (connected) dicirculant digraphs of order 4p (p prime) and out-degree k for every k.