  By developing a tangible way to decompose unipotent radicals into irreducible
submodules of Singer cycles, we achieve a classification of solvable factors of
finite classical groups of Lie type. This completes previous work on
factorizations of classical groups with a solvable factor. In particular, it
resolves the final uncertain case in the long-standing problem of determining
exact factorizations of almost simple groups. As a byproduct of the
classification, we also obtain a new characterization of one-dimensional
transitive groups, offering further insights into their group structure.
