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We construct two infinite families of regular 3-polytopes of type \3, 8m\ with 192m² and 384m² 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 32ⁿ 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.
Kong et al. (2024) studied this question.