Constructing cocyclic Hadamard matrices of order 4p

Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices (HMs) with interesting algebraic properties. Ó Catháin and Röder described a classification algorithm for CHMs of order 4n based on relative difference sets in groups of order 8n; this led to the classification of all CHMs of order at most 36. On the basis of work of de Launey and Flannery, we describe a classification algorithm for CHMs of order 4p with pa prime; we prove refined structure results and provide a classification for p ≤ 13. Our analysis shows that every CHM of order 4p with p ≡ 1 is equivalent to an HM with one of five distinct block structures, including Williamson-type and (transposed) Ito matrices. If p ≡ 3, then every CHM of order 4p is equivalent to a Williamson-type or (transposed) Ito matrix.