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We classify those 2-groups G which factorise as a product of two disjoint cyclic subgroups A and B , transposed by an automorphism of order 2 . The case where G is metacyclic having been dealt with elsewhere, we show that for each e ≥ 3 there are exactly three such non-metacyclic groups G with ∣ A ∣ = ∣ B ∣ = 2 e , and for e = 2 there is one. These groups appear in a classification by Berkovich and Janko of 2 -groups with one non-metacyclic maximal subgroup; we enumerate these groups, give simpler presentations for them, and determine their automorphism groups.

2-Groups that factorise as products of cyclic groups, and regular embeddings of complete bipartite graphs