Conway group Co1

is obtained as the quotient of Co₀ (group of automorphisms of the Leech lattice Λ that fix the origin) by its center, which consists of the scalar matrices ±1. It also appears at the top of the automorphism group of the even 26-dimensional unimodular lattice II_25,1.

Representations

The smallest non-trivial representation of Co₀ over any field is the 24-dimensional one coming from the Leech lattice, and this is faithful over fields of characteristic other than 2.

The smallest faithful permutation representation of Co₁ is on the 98280 pairs {v,–v} of norm 4 vectors.

The centralizer of an involution of type 2B in the monster group is of the form 2¹⁺²⁴Co₁.

The Dynkin diagram of the even Lorentzian unimodular lattice II_1,25 is isometric to the (affine) Leech lattice Λ, so the group of diagram automorphisms is split extension Λ,Co₀ of affine isometries of the Leech lattice.

Maximal subgroups of Co₁

Wilson (1983) classified the maximal subgroup, though there were some errors in this list corrected by Wilson (1988). Co₁ has 22 conjugacy classes of maximal subgroups as follows.

Co₂ The lift to Aut(Λ) fixes a norm 4 vector.
3.Suz:2 The lift to Aut(Λ) fixes a complex structure or changes it to the complex conjugate structure. Suzuki chain; see below
2¹¹:M₂₄ The lift to Aut(Λ) fixes a frame; see below.
Co₃ The lift to Aut(Λ) fixes a norm 6 vector.
2¹⁺⁸.O₈⁺(2)
U₆(2):S₃
(A₄ × G₂(4)):2 Suzuki chain.
2²⁺¹²:(A₈ × S₃)
2⁴⁺¹².(S₃ × 3.S₆)
3².U₄(3).D₈
3⁶:2.M₁₂ (holomorph of ternary Golay code)
(A₅ × J₂):2 Suzuki chain
3¹⁺⁴:2.Sp₄(3).2
(A₆ × U₃(3)).2 Suzuki chain
3³⁺⁴:2.(S₄ × S₄)
A₉ × S₃ Suzuki chain
(A₇ × L₂(7)):2 Suzuki chain
(D₁₀ × (A₅ × A₅).2).2
5¹⁺²:GL₂(5)
5³:(4 × A₅).2
7²:(3 × 2.S₄)
5²:2A₅

Maximal subgroup N = 2¹²M₂₄ of Co₀

Conway started his investigation with a subgroup called N. The Leech lattice is defined by use of the binary Golay code, whose automorphism group is the Mathieu group M₂₄. For any norm 8 vector of the Leech lattice, there are exactly 48 norm 8 vectors congruent to it modulo 2, falling into 24 orthogonal pairs {v,–v}. A set of 48 such vectors is called a frame. The subgroup fixing a frame is a group N which is a split extension of the form 2¹².M₂₄, where the 2¹² is isomorphic to the Golay code and acts as sign changes on vectors of the frame, while the M₂₄ permutes the 24 pairs of the frame. Conway found that N is a maximal subgroup of Co₀ and contains 2-Sylow subgroups of Co₀. He used N to deduce the order of Co₀, as the order 2¹²|M₂₄| of N times the number of frames, where the number of frames is the number of norm 8 vectors divided by 48.

Suzuki chain of product groups

Co₀ (as well as its quotient Co₁) has 4 conjugacy classes of elements of order 3. One of these commutes with a double cover of the alternating group A₉. In fact the normalizer of that 3-element has the form 2.A₉ x S₃.

John Thompson pointed out that was fruitful to investigate the normalizers of smaller subgroups of the form 2.A_n (Conway 1971, p.242). Several other maximal subgroups of Co₀ are found in this way. Moreover, two sporadic groups appear in the resulting chain.

There is a subgroup 2.A₈ x S₄, but it is not maximal in Co₀. Next there is the subgroup (2.A₇ x PSL₂(7)):2. This group and the rest of the chain are maximal in Co₀. Next comes (2.A₆ x SU₃(3)):2. The unitary group SU₃(3) (order 6048) possesses a graph of 36 vertices, in anticipation of the next subgroup. That subgroup is (2.A₅ o 2.HJ):2. The aforementioned graph expands to the Hall-Janko graph, with 100 vertices. The Hall-Janko group HJ makes its appearance here. Next comes (2.A₄ o 2.G₂(4)):2. G₂(4) is an exceptional group of Lie type.

The chain ends with 6.Suz:2 (Suz=Suzuki group), which, as mentioned above, respects a complex representaion of the Leech Lattice.

References

Conway, John Horton (1968), "A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups", Proceedings of the National Academy of Sciences of the United States of America 61 (2): 398–400, doi:10.1073/pnas.61.2.398, MR 0237634
Brauer, R.; Sah, Chih-han, eds. (1969), Theory of finite groups: A symposium, W. A. Benjamin, Inc., New York-Amsterdam, MR 0240186
Conway, John Horton (1969), "A group of order 8,315,553,613,086,720,000", The Bulletin of the London Mathematical Society 1: 79–88, ISSN 0024-6093, MR 0248216
Conway, John Horton (1971), "Three lectures on exceptional groups", in Powell, M. B.; Higman, Graham, Finite simple groups, Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969., Boston, MA: Academic Press, pp. 215–247, ISBN 978-0-12-563850-0, MR 0338152 Reprinted in Conway & Sloane (1999, 267-298)
Conway, John Horton; Sloane, Neil J. A. (1999), Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften 290 (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98585-5, MR 0920369
Thompson, Thomas M. (1983), From error-correcting codes through sphere packings to simple groups, Carus Mathematical Monographs 21, Mathematical Association of America, ISBN 978-0-88385-023-7, MR 749038
Conway, John Horton; Parker, Richard A.; Norton, Simon P.; Curtis, R. T.; Wilson, Robert A. (1985), Atlas of finite groups, Oxford University Press, ISBN 978-0-19-853199-9, MR 827219
Griess, Robert L. Jr. (1998), Twelve sporadic groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62778-4, MR 1707296
Atlas of Finite Group Representations: Co₁ version 2
Atlas of Finite Group Representations: Co₁ version 3
Wilson, Robert A. (1983), "The maximal subgroups of Conway's group Co₁", Journal of Algebra 85 (1): 144–165, doi:10.1016/0021-8693(83)90122-9, ISSN 0021-8693, MR 723071
Wilson, Robert A. (1988), "On the 3-local subgroups of Conway's group Co₁", Journal of Algebra 113 (1): 261–262, doi:10.1016/0021-8693(88)90192-5, ISSN 0021-8693, MR 928064
Wilson, Robert A. (2009), The finite simple groups., Graduate Texts in Mathematics 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 05622792

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