  We construct two infinite families of locally toroidal chiral polytopes of
type $\{4,4,4\}$, with $1024m^2$ and $2048m^2$ 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 $2^n$ where $n$ can be any integer greater than
$9$. (On the other hand, no chiral polytopes of type $[4,4,4]$ exist for $n
\leq 9$.) In particular, our two families give a partial answer to a problem
proposed by Schulte and Weiss in [Problems on polytopes, their groups, and
realizations, {\em Periodica Math.\ Hungarica\} 53 (2006), 231-255].
