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Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam asa class of Hadamard matrices with interesting algebraic properties. \'OCath\'ain and R\"oder described a classification algorithm for CHMs of order$4n$ based on relative difference sets in groups of order $8n$; this led to theclassification of all CHMs of order at most 36. Based on work of de Launey andFlannery, we describe a classification algorithm for CHMs of order $4p$ with$p$ a prime; we prove refined structure results and provide a classificationfor $p \leqslant 13$. Our analysis shows that every CHM of order $4p$ with$p\equiv 1\bmod 4$ is equivalent to a Hadamard matrix with one of five distinctblock structures, including Williamson type and (transposed) Ito matrices. If$p\equiv 3 \bmod 4$, then every CHM of order $4p$ is equivalent to a Williamsontype or (transposed) Ito matrix.

Constructing cocyclic Hadamard matrices of order 4p