String C-groups of 2-power order project onto a common string C-group

TL;DR

This paper explores the structure of string C-groups of 2-power order, revealing a unique minimal regular polytope with extremal properties and universal covering characteristics among such groups.

Contribution

It demonstrates that all finite regular d-polytopes with 2-power automorphism groups cover a specific minimal polytope, establishing a universal property of these groups.

Findings

The automorphism group of C_d is a quotient of every finite string C-group of the same rank and 2-power order.

C_d has the smallest number of flags among all regular d-polytopes of high ranks.

Every finite regular d-polytope with 2-power automorphism group covers C_d.

Abstract

String C-groups are precisely the automorphism groups of abstract regular polytopes. A certain regular d-polytope C_d with an automorphism group of order 2^{2d-1}, discovered by Conder and shown to have the smallest number of flags among all regular d-polytopes of high ranks, also has the important extremal property to be the unique minimal d-polytope, with respect to combinatorial covering, among all finite regular d-polytopes with 2-power automorphism groups. In other words, the automorphism group of C_d is a quotient group of every finite string C-group of rank d and 2-power order; and every finite regular d-polytope with an automorphism groups of 2-power order covers C_d. The existence of a unique minimal element among string C-groups of 2-power order and given rank is remarkable in itself.