{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39407",
  "author_total": 3,
  "authors": [
    {
      "name": "Wenwen Fan"
    },
    {
      "name": "Yuan Shi"
    },
    {
      "name": "Yun Tian Wu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e408h119n26kj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39407",
  "fulltext_url": null,
  "id": "e408h119n26kj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39407",
  "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.GR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39407",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e408h119n26kj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.39407",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39407"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Complete regular dessins and non-metacyclic bicyclic 2-groups with cyclic commutator subgroup",
  "title_text": "Complete regular dessins and non-metacyclic bicyclic 2-groups with cyclic commutator subgroup",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/e408h119n26kj",
  "venue": "arXiv",
  "versions": []
}