A large class of Polyhedra which includes the Dodecadodecahedron and Great Icosidodecahedron. No complete enumeration (even with restrictive uniqueness conditions) has been worked out.
References
Coxeter, H. S. M.; Longuet-Higgins, M. S.; and Miller, J. C. P. ``Uniform Polyhedra.''
Phil. Trans. Roy. Soc. London Ser. A 246, 401-450, 1954.
Wenninger, M. J. Polyhedron Models. New York: Cambridge University Press, pp. 66-72, 1989.