We investigate the cubic Cayley graph CP3(G,M) of cyclic group G of order p3q3, where p and q are two distinct prime numbers and M consists of all elements in G of order p3 or q3. We present some characteristics of CP3(G,M) including its connectivity, diameter, regularity, Eulerianity, girth, and Hamiltonicity. We show that the graph is Eulerian and has girth 3, and prove that the Cayley graph is a Hamiltonian or a semi-Hamiltonian graph. Furthermore we discuss the independence, chromatic, clique and domination numbers of the graph.

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