  A classification is given for factorizations of almost simple groups with at
least one factor solvable, and it is then applied to characterize
$s$-arc-transitive Cayley graphs of solvable groups, leading to a striking
corollary: Except the cycles, every non-bipartite connected 3-arc-transitive
Cayley graph of a solvable group is a cover of the Petersen graph or the
Hoffman-Singleton graph.
