Complete regular dessins and non-metacyclic bicyclic 2-groups with cyclic commutator subgroup

We study complete regular dessins with underlying graph ${\mathcal K}_{2^e,2^f}$, where $e\geq f\geq 2$, whose automorphism groups are non-metacyclic. Together with the cases of coprime orders and odd-prime-power orders, this is one of the basic classes needed for Jones's open classification problem for regular dessins of complete bipartite graphs. Within the 2-power case, the present paper treats the subcase in which the commutator subgroup is cyclic; it is the first of three papers devoted to the non-metacyclic case. Our approach is group-theoretic. The automorphism group $G$ of a regular dessin of ${\mathcal K}_{2^e,2^f}$ admits an exact bicyclic factorization $G=\langle b\rangle\langle w\rangle$ of type $(2^e,2^f)$ with $\langle b\rangle\cap\langle w\rangle=1$. For a fixed group $G$, two exact generating pairs give isomorphic dessins exactly when they lie in the same orbit under ${\rm A}ut(G)$. Thus the classification reduces to three group-theoretic tasks: determining the isomorphism classes of the possible groups $G$, characterizing their exact bicyclic generating pairs, and computing $|{\rm A}ut(G)|$. Berkovich and Janko divided the relevant non-metacyclic groups, through parameterized presentations, according to whether the commutator subgroup is cyclic, contains a cyclic subgroup of index $2$, or contains a cyclic subgroup of index $2^r$ with $r>1$. However, their classification does not identify which parameter values in these presentations define isomorphic groups; this problem was raised by Janko in {J}. We carry out these steps for the cyclic-commutator case. This settles Janko's isomorphism-classification problem in this case and yields a complete classification and enumeration of the corresponding regular dessins.