More on quotient polytopes

Summary.

In an earlier paper, it was shown that every abstract polytope is a quotient ${\cal Q} = {\cal M}(W)/N$ of some regular polytope ${\cal M}(W)$ whose automorphism group is W, by a subgroup N of W. In this paper, attention is focussed on the quotient ${\cal Q}$, and various important structures relating to polytopes are described in terms of N′, the stabilizer of a flag of the quotient under an action of W (the ‘flag action’). It is pointed out how N′ may be assumed without loss of generality to equal N. The paper also shows what properties of N′ yield polytopes which are regular, section regular, chiral, locally regular, or locally universal. The aim is to make it more practical to study non-regular polytopes in terms of group theory.