Enumeration of polyhedra with triangular and hexagonal faces and three faces around each vertex

We give an exact count of the number of trivalent graphs whose faces all have 3 or 6 sides, or equivalently, the number of polyhedra with triangular and hexagonal faces and three faces around each vertex. The count is given in terms of the prime factorization of the number of vertices. We also enumerate graphs of this type with mirror symmetry, with 3-fold rotational symmetry, and with both types of symmetry.