All crossed products of two cyclic groups are explicitly described using generators and relations. A necessary and sufficient condition for an extension of a group by a group to be a cyclic group is given.

Let $G(\circ )$ and $G(*)$ be two groups of finite order $n$, and suppose that they share a normal subgroup $S$ such that $u\circ v=u*v$ if $u\in S$ or $v\in S$. Cases when $G/S$ is cyclic or dihedral and when $u\circ v\ne u*v$ for exactly $n^2/4$ pairs $(u,v)\in G\times G$ have been shown to be of crucial importance when studying pairs of 2-groups with the latter property. In such cases one can describe two general constructions how to get all possible $G(*)$ from a given $G=G(\circ )$. The constructions, denoted by $G[\alpha ,h]$ and $G[\beta ,\gamma ,h]$, respectively, depend on a coset $\alpha$ (or two cosets $\beta$ and $\gamma$) modulo $S$, and on an...