Two infinite families of chiral polytopes of type {4,4,4} with solvable automorphism groups

We construct two infinite families of locally toroidal chiral polytopes type { 4 , } with 1024 m 2 and 2048 automorphisms for every positive integer respectively. The automorphism groups these are solvable groups, when is a power 2, they provide examples order n where can be any greater than 9. (On the other hand, no [ ] exist ≤ 9 .) In particular, our give partial answer to problem proposed by Schulte Weiss in [21] .