This chapter is devoted to a family of abstract regular 4-polytopes, which display remarkable parallels with the 4-dimensional pentagonal polytopes of Chapter 7. Two basic members of the family are quotients of 4-dimensional regular hyperbolic honeycombs. Their common automorphism group, of order 8160, is an extension by an involutory outer automorphism of a simple group. Part of the discussion centres on a certain regular polyhedron, which is closely related to the facet of the sole regular polytope of rank 4 dealt with in Chapter 13 whose symmetry group consists entirely of rotations. The treatment makes substantial use of a permutation representation of the automorphism group.