Complete regular dessins: the nonmetacyclic case
Abstract
We classify regular dessins on K2e,2f 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 22e−2+16 when e=f≥4 and 3·22f−3+16 when e>f≥4, 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 2e+8 on the diagonal and 2f−2(2f−1+1)+8 off the diagonal for e≥ f≥4.
Related
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BibTeX
@misc{hu2026complete,
title = {{Complete regular dessins: the nonmetacyclic case}},
author = {Kan Hu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22620},
eprint = {2610.22620},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22620},
}
RIS
TY - UNPB
TI - Complete regular dessins: the nonmetacyclic case
AU - Kan Hu
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22620
UR - https://arxiv.org/abs/2610.22620
AB - 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≥4 and 3·2^2f−3+16 when e>f≥4, 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≥ f≥4.
ER -