When the standard representation of a crystallographic Coxeter group Γ
is reduced modulo an odd prime p, a finite representation in some orthogonal
space over Zp​ is obtained. If Γ has a string diagram, the
latter group will often be the automorphism group of a finite regular polytope.
In Part I we described the basics of this construction and enumerated the
polytopes associated with the groups of rank 3 and the groups of spherical or
Euclidean type. In this paper, we investigate such families of polytopes for
more general choices of Γ, including all groups of rank 4. In
particular, we study in depth the interplay between their geometric properties
and the algebraic structure of the corresponding finite orthogonal group.Comment: 30 pages (Advances in Applied Mathematics, to appear