The 3D hexagonal torus is presented as a natural extension of the hexagonal torus. Hexagonal tori are proved to be a type of Cayley graph. This paper develops a new type of 3D hexagonal torus based on Cayley graph. An addressing scheme for this type of network is developed. Based on this addressing scheme, the distance formula between any two nodes is derived. Then, a distributed optimal routing algorithm is developed. This distributed routing algorithm is optimal, which means it can be executed on any node in the network to construct a shortest path between any pair of nodes. A broadcasting algorithm is also presented according to the coset graph theory. The upper bound and lower bound of the diameter are also proposed in this paper.