Perfect one-factorizations of K24 with automorphism group of order greater than three: a complete catalogue.

A complete catalogue of the perfect one-factorizations of the complete graph K24 whose automorphism group has order greater than three: 561 isomorphism classes, each given once in canonical form as its 23 factors. A one-factorization of K24 splits the 276 edges into 23 perfect matchings, called factors; it is perfect when the union of any two of them is a single Hamiltonian cycle on all 24 vertices. Two factorizations belong to the same class when some relabeling of the 24 vertices carries one to the other, and |Aut| is the order of the group of relabelings carrying a factorization to itself. The catalogue covers |Aut| > 3 and is complete for that range: every isomorphism class of perfect one-factorization of K24 whose automorphism group has order greater than three appears in the file exactly once. Classes with |Aut| equal to 1, 2 or 3 are not included. The distribution by group order is 11:531, 22:21, 23:4, 253:4, 506:1. #7 has automorphism group AGL(1,23) of order 506. The nine one-rotational classes are #7, #180, #445, #522, #554, #558, #559, #560 and #561, and the 22 classes with an automorphism of order 22 are #1 to #22; both counts agree with Pike and Shalaby (2003). All 561 classes were found by our own exhaustive search. Every class built from a starter has |Aut| of at least 22, so all classes obtained by Pike and Shalaby from starters are in this catalogue: these 30. The other 531, all with |Aut| = 11, cannot be built from a starter. Among the 561 classes, exactly four fold to an atomic Latin square of order 23 (in the sense of Bryant, Maenhaut and Wanless, J. Combin. Theory Ser. A 113 (2006) 608-624): #7 and #8 at vertex 0, #522 at vertex 14 and #561 at vertex 19. No other class gives an atomic square at any vertex. The records are sorted in ascending order of the canonical form and numbered contiguously from #1 to #561, so a record number is a position in this version of the catalogue rather than a permanent name; cite the version you used. Every record was re-verified from its matrix alone: 561 valid perfect one-factorizations, no recomputed group order disagreeing with the one recorded, and no class recorded twice. The accompanying README documents the file format, the record numbering and the checks performed. The engine, the run scripts and the argument that the searches cover every group of order greater than three are public at https://github.com/LeAndr-Graph/P1F-Census. The engine extensions for K22 and above, the run scripts and much of the documentation were developed with AI coding assistants (Google Antigravity, Anthropic Claude Code); the data were checked independently as described above.