  Motivated by computational efficiency in algebraic automata theory here we
define the cascade product of permutation groups as an external product, as a
generic extension. It is the most general hierarchical product that uses
arbitrary total functions to combine a linearly ordered set of permutation
groups. Algebraically speaking, cascade products are explicitly constructed
substructures of the iterated wreath product. We show how direct, semidirect
and wreath products can be described as cascade products and we also discuss
examples of composite groups that can only be constructed exactly as generic
extensions by cascade products. The cascade construction naturally generalizes
to the transformation semigroup case by leaving out the details of defining
inverse operations.
