In this paper, we show that no Hadamard matrix constructed using the twin prime power method is cocyclic, with the exception of the matrix of order 16. We achieve this by showing that the action of the automorphism group of a cocyclic twin prime power matrix induces a 2-transitive action on the rows of the matrix. We then use Ito's classification of Hadamard matrices with 2-transitive automorphism groups to show that no twin prime power Hadamard matrix is cocyclic. This work answers a research problem posed by K.J. Horadam, and exhibits the first known infinite family of Hadamard matrices which are not cocyclic.

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