{
  "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 $e\\ge f\\ge4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22620",
  "author_total": 1,
  "authors": [
    {
      "name": "Kan Hu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dtjp6xe6vwhhc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22620",
  "fulltext_url": null,
  "id": "dtjp6xe6vwhhc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22620",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7222941999",
  "pdf_url": "https://arxiv.org/pdf/2610.22620",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dtjp6xe6vwhhc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.22620",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22620"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Complete regular dessins: the nonmetacyclic case",
  "title_text": "Complete regular dessins: the nonmetacyclic case",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/dtjp6xe6vwhhc",
  "venue": "arXiv",
  "versions": []
}