Two infinite families of regular $3$-polytopes of type $\{3, 8m\}$

Abstract

We construct two infinite families of regular $3$-polytopes of type $\{3, 8m\}$ with $192m^2$ and $384m^2$ automorphisms for every positive integer $m$, respectively. The automorphism groups of these polytopes are solvable groups, and when $m$ is a power of $2$, they provide examples with automorphism groups of order $3\cdot2^n$ where $n$ can be any integer greater than $5$. In particular, our two families give a partial answer to a problem proposed by Schulte and Weiss.