Hyperbolic triangular prisms with one ideal vertex

In this paper, we classify all of the five-sided three-dimensional hyperbolic polyhedra with one ideal vertex, which have the shape of a triangular prism. 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 ${\mathrm{P}\mathrm{S}\mathrm{L}}_2(\mathbb{C})$ for the orientation-preserving subgroup of each corresponding reflection group.