The final chapter concerns two families of regular 5-polytopes, the second consisting of the double covers of the first. The starting point is a simple group, which is the automorphism group of a regular quotient of a 5-dimensional hyperbolic honeycomb. Armed with only modest information, it is first shown that the realization domain of this polytope is very simple. Two defining relations for the quotient were initially provided; subsequently, it is seen geometrically that one of them is redundant. As in the previous chapter, there is an extended family, among which there are polytopes whose facets and vertex-figures beong to the pentagonal 4-polytopes of Chapter 7 and the family of 4-polytopes described in Chapter 16. The initial quotients are non-orientable; the family of their double covers contains members that are universal as amalgamations. These families of polytopes correspond to actions of the automorphism groups on two of their maximal subgroups. The groups have another maximal subgroup, and though there are no nice related polytopes, nevertheless this gives rise to interesting symmetric sets. There is another quotient and a close relative which one might initially think belong to one of the families; that they do not, with a completely unrelated group, is perhaps surprising.

The bridging concept between the abstract and geometric is the theory of realizations. This chapter concentrates on symmetric sets, namely, finite sets on which a group of permutations acts transitively. After a discussion of their basic properties, the concept of their realizations is introduced, with operations on them (such as blending) showing that the family of their congruences classes has the structure of a convex cone. A key idea is that of the inner product and cosine vectors of realizations, which define them up to congruence. The theory up to this point is then illustrated by some examples. It is next shown that, corresponding to the tensor product of representations, there is a product of realizations. Another fundamental notion is that of orthogonality relations for cosine vectors. The different realizations derived from an irreducible representation of the abstract group may form a subcone of the realization cone that is more than 1-dimensional. These are looked at more closely, leading to a definition of cosine matrices for the general realization domain. There follows a discussion of cuts and their relationship with duality. Cosine vectors may have entries in some subfield of the real numbers, with implications for the corresponding realizations. The chapter ends with a brief account of how representations of groups are related to realizations.