Eulerian and Hamiltonian Graphs · Superchallenge problem

The hexagonal prism graph has two regular hexagons (top vertices $0,\ldots,5$ and bottom vertices $6,\ldots,11$) connected by six rungs where vertex $i$ (on top) is joined to vertex $i+6$ (on bottom). Every vertex has degree $3$. Counting two Hamiltonian cycles as the same when they use the same set of edges, how many distinct Hamiltonian cycles does the hexagonal prism have?