Homological models for semidirect products of finitely generated Abelian groups

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Let G be a semidirect product of finitely generated Abelian groups. We provide a method for constructing an explicit contraction (special homotopy equivalence) from the reduced bar construction of the group ring of G, $${\overline{B}(\mathsf{\textstyle Z\kern-0.4em Z}[G])}$$, to a much smaller DGA-module hG. Such a contraction is called a homological model for G and is used as the input datum in the methods described in Alvarez et al. (J Symb Comput 44:558–570, 2009; 2012) for calculating a generating set for representative 2-cocycles and n-cocycles over G, respectively. These computations have led to the finding of new cocyclic Hadamard matrices (Alvarez et al. in 2006).