$2$-groups behaving as automorphism groups of regular $3$-polytopes

Dong-Dong Hou, Yan-Quan Feng, Dimitri Leemans

Published 2018-02-25Version 1

In this paper, we classify regular polytopes with automorphism groups of order $2^n$ and Schl\"afli types $\{4, 2^{n-3}\}, \{4, 2^{n-4}\}$ and $\{4, 2^{n-5}\}$ for $n \geq 10$, therefore giving a partial answer to a problem proposed by Schulte and Weiss in [Problems on polytopes, their groups, and realizations, Periodica Math. Hungarica 53(2006) 231-255].