  We introduce an external version of the internal r-fold semidirect product of
groups (SDP) of Carrasco and Cegarra. Just as for the classical external SDP,
certain algebraic data are required to guarantee associativity of the
construction. We give an algorithmic procedure for computing axioms
characterizing these data. Additionally, we give criteria for determining when
a family of homomorphisms from the factors of an SDP into a monoid or group
assemble into a homomorphism on the entire SDP. These tools will be used
elsewhere to give explicit algebraic axioms for hypercrossed complexes, which
are algebraic models for classical homotopy types introduced by Carrasco and
Cegarra.
