Four infinite families of chiral 3-polytopes of type {4,8} with solvable automorphism groups

Abstract

We construct four infinite families of chiral 3-polytopes of type {4,8}, with 1024m4, 2048m4, 4096m4 and 8192m4 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 2n where n≥10. (On the other hand, no chiral polytopes of type {4,8} exist for n≤9.) In particular, our families give a partial answer to a problem proposed by Schulte and Weiss (2006) [15] and a problem proposed by Pellicer (2012) [13].