Complete regular dessins: the nonmetacyclic case

Source abstract

We classify regular dessins on $K_{2^e,2^f}$ with nonmetacyclic colour-preserving automorphism group, retaining the two specified vertex rotations. We give nonredundant presentations and exact counts in both the abelian and nonabelian Frattini cases. For orientation- and colour-preserving isomorphisms, the number is $2^{2e-2}+16$ when $e=f\ge4$ and $3\cdot2^{2f-3}+16$ when $e>f\ge4$, with explicit counts for all smaller exponents. The construction uses a parity criterion in the abelian case and a uniquely determined odd quadratic root with two lifts in the nonabelian case. A dessin in the classification is reflexible exactly when its Frattini subgroup is abelian. All symmetric classes are circular. We determine every circular subfamily, including its face orders and genera; the circular count is $2^e+8$ on the diagonal and $2^{f-2}(2^{f-1}+1)+8$ off the diagonal for $e\ge f\ge4$.