Hyperbolic triangular prisms with one ideal vertex

We classify all of the five-sided three-dimensional hyperbolic polyhedra with one ideal vertex, which have the shape of a triangular prism, and which give rise to a discrete reflection group. We show how to find each such polyhedron in the upper half-space model by considering lines and circles in the plane. Finally, we give matrix generators in PSL2(C) for the orientation-preserving subgroup of each corresponding reflection group.