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| McLaughlin group | |
|---|---|
| Name | McLaughlin group |
| Order | 898128000 |
| Notation | McL |
| Discovered | 1968–1969 |
| Discoverer | Jack McLaughlin |
McLaughlin group
The McLaughlin group is a sporadic simple group of order 898,128,000 and is one of the 26 sporadic groups associated with the classification of finite simple groups. It sits in the constellation of sporadic examples alongside Monster group, Conway group Co1, Conway group Co2, Conway group Co3 and Fischer group Fi22, and it has connections to permutation actions on 275 points related to the Steiner system S(3,5,22), the Higman-Sims graph, and structures linked to the Leech lattice. The group played a role in the development of the Atlas of Finite Groups and has been studied through methods developed by John G. Thompson, Bertram Huppert, Daniel Gorenstein, and others.
The McLaughlin group was constructed by Jack McLaughlin via considerations of permutation groups and 3-transpositions arising from investigations into rank-3 actions and designs related to the Higman–Sims group and the Mathieu group M24. One construction realizes it as a subgroup of the automorphism group of the Leech lattice by examining stabilizers of certain lattice configurations and using techniques similar to those used for Conway group Co0 and Conway group Co1. Alternative constructions use generators and relations coming from amalgams related to Suzuki sporadic group, Janko group J2, and centralizer fusion patterns observed in the work of Bernd Fischer and Ronald Solomon.
The McLaughlin group is simple, non-abelian, and has Schur multiplier of order 1 and outer automorphism group of order 2 in some extensions; it admits a double cover and fits into the framework of covering groups studied by Richard Brauer and E. C. Titchmarsh. Its element orders include 2, 3, 5, 7, 11 and 23, matching prime divisors appearing in the orders of other sporadic groups such as Held group He and O'Nan group O'N. The group's Sylow subgroups and local subgroups exhibit patterns analogous to those in Mathieu group M22 and Higman–Sims group HS, with centralizers of involutions investigated through techniques from the work of Walter Feit and John Conway.
Maximal subgroups of the McLaughlin group include copies of U4(3) (unitary group), M22:2 (an extension of Mathieu group M22), and subgroups isomorphic to L3(4) and 2^4:A8 appearing as point stabilizers in the rank-3 action on 275 points. The subgroup lattice displays connections to classical groups such as PSL2(11), PSU4(3), and symmetric groups like S8 arising in embeddings comparable to those studied for Janko group J3 and Conway group Co3. The interplay of these subgroups under conjugacy and fusion is illuminated by publications of Robert Curtis and computations in the Atlas of Finite Groups project.
The complex character table of the McLaughlin group contains irreducible characters of degrees including 1, 22, 252, 275, 770, and larger degrees that mirror patterning found in Fischer group Fi23 and Conway group Co2. Modular representations over fields of characteristic 2, 3, 5, and 11 have been studied in the context of block theory developed by Richard Brauer and Graham Higman, with decomposition matrices computed using algorithms stemming from work by John Conway and computational algebra systems such as GAP and Magma. The permutation representation on 275 points yields a rank-3 character that decomposes into principal and two nontrivial constituents closely examined by Daniel Wales and contributors to the Atlas of Finite Groups.
Geometric realizations link the McLaughlin group to combinatorial designs and graphs: it acts transitively on a set related to a Steiner system with parameters reminiscent of Steiner system S(3,6,22) and preserves structures in the Higman–Sims graph, the McLaughlin graph, and configurations in the Leech lattice and associated sphere packings studied by John Leech and Conway group Co0. Its action produces strongly regular graphs with parameters comparable to those arising from Petersen graph constructs and ties to block designs investigated by E. S. Barnes and R. T. Curtis.
The McLaughlin group was discovered in 1968–1969 by Jack McLaughlin while analyzing rank-3 permutation groups and was announced following communications with researchers such as John Conway, Bernd Fischer, and Daniel Gorenstein. Further developments and classification of its properties were recorded in the Atlas of Finite Groups and expanded upon in papers by McLaughlin (1969), Gorenstein (1983), and subsequent computational studies by teams using GAP and Magma. Its discovery contributed to the broader project of classifying sporadic simple groups alongside work by Jean-Pierre Serre, John Thompson, and Bertrand Russell-era collaborators in finite group theory.
Category:Sporadic groups