Conway group Co3

In the area of modern algebra known as group theory, the Conway group [math]\displaystyle{ \mathrm{Co}_3 }[/math] is a sporadic simple group of order

   2¹⁰ · 3⁷ · 5³ · 7 · 11 · 23

= 495766656000

≈ 5×10¹¹.

History and properties

[math]\displaystyle{ \mathrm{Co}_3 }[/math] is one of the 26 sporadic groups and was discovered by John Horton Conway (1968, 1969) as the group of automorphisms of the Leech lattice [math]\displaystyle{ \Lambda }[/math] fixing a lattice vector of type 3, thus length √6. It is thus a subgroup of [math]\displaystyle{ \mathrm{Co} 0 }[/math]. It is isomorphic to a subgroup of [math]\displaystyle{ \mathrm{Co}_1 }[/math]. The direct product [math]\displaystyle{ 2\times \mathrm{Co}_3 }[/math] is maximal in [math]\displaystyle{ \mathrm{Co}_0 }[/math].

The Schur multiplier and the outer automorphism group are both trivial.

Representations

Co₃ acts on the unique 23-dimensional even lattice of determinant 4 with no roots, given by the orthogonal complement of a norm 4 vector of the Leech lattice. This gives 23-dimensional representations over any field; over fields of characteristic 2 or 3 this can be reduced to a 22-dimensional faithful representation.

Co₃ has a doubly transitive permutation representation on 276 points.

Walter Feit (1974) showed that if a finite group has an absolutely irreducible faithful rational representation of dimension 23 and has no subgroups of index 23 or 24 then it is contained in either [math]\displaystyle{ \Z/2\Z \times \mathrm{Co}_2 }[/math] or [math]\displaystyle{ \Z/2\Z \times \mathrm{Co}_3 }[/math].

Maximal subgroups

Some maximal subgroups fix or reflect 2-dimensional sublattices of the Leech lattice. It is usual to define these planes by h-k-l triangles: triangles including the origin as a vertex, with edges (differences of vertices) being vectors of types h, k, and l.

Larry Finkelstein (1973) found the 14 conjugacy classes of maximal subgroups of [math]\displaystyle{ \mathrm{Co}_3 }[/math] as follows:

• McL:2 – McL fixes a 2-2-3 triangle. The maximal subgroup also includes reflections of the triangle. [math]\displaystyle{ \mathrm{Co}_3 }[/math] has a doubly transitive permutation representation on 276 type 2-2-3 triangles having as an edge a type 3 vector fixed by [math]\displaystyle{ \mathrm{Co}_3 }[/math].
• HS – fixes a 2-3-3 triangle.
• U₄(3).2²
• M₂₃ – fixes a 2-3-4 triangle.
• 3⁵:(2 × M₁₁) - fixes or reflects a 3-3-3 triangle.
• 2.Sp₆(2) – centralizer of involution class 2A (trace 8), which moves 240 of the 276 type 2-2-3 triangles
• U₃(5):S₃
• 3¹⁺⁴:4S₆
• 2⁴.A₈
• PSL(3,4):(2 × S₃)
• 2 × M₁₂ – centralizer of involution class 2B (trace 0), which moves 264 of the 276 type 2-2-3 triangles
• [2¹⁰.3³]
• S₃ × PSL(2,8):3 - normalizer of 3-subgroup generated by class 3C (trace 0) element
• A₄ × S₅

Conjugacy classes

Traces of matrices in a standard 24-dimensional representation of Co₃ are shown. The names of conjugacy classes are taken from the Atlas of Finite Group Representations. The cycle structures listed act on the 276 2-2-3 triangles that share the fixed type 3 side.

Class	Order of centralizer	Size of class	Trace	Cycle type
1A	all Co3	1	24
2A	2,903,040	33·52·11·23	8	136,2120
2B	190,080	23·34·52·7·23	0	112,2132
3A	349,920	25·52·7·11·23	-3	16,390
3B	29,160	27·3·52·7·11·23	6	115,387
3C	4,536	27·33·53·11·23	0	392
4A	23,040	2·35·52·7·11·23	-4	116,210,460
4B	1,536	2·36·53·7·11·23	4	18,214,460
5A	1500	28·36·7·11·23	-1	1,555
5B	300	28·36·5·7·11·23	4	16,554
6A	4,320	25·34·52·7·11·23	5	16,310,640
6B	1,296	26·33·53·7·11·23	-1	23,312,639
6C	216	27·34·53·7·11·23	2	13,26,311,638
6D	108	28·34·53·7·11·23	0	13,26,33,642
6E	72	27·35·53·7·11·23	0	34,644
7A	42	29·36·53·11·23	3	13,739
8A	192	24·36·53·7·11·23	2	12,23,47,830
8B	192	24·36·53·7·11·23	-2	16,2,47,830
8C	32	25·37·53·7·11·23	2	12,23,47,830
9A	162	29·33·53·7·11·23	0	32,930
9B	81	210·33·53·7·11·23	3	13,3,930
10A	60	28·36·52·7·11·23	3	1,57,1024
10B	20	28·37·52·7·11·23	0	12,22,52,1026
11A	22	29·37·53·7·23	2	1,1125	power equivalent
11B	22	29·37·53·7·23	2	1,1125
12A	144	26·35·53·7·11·23	-1	14,2,34,63,1220
12B	48	26·36·53·7·11·23	1	12,22,32,64,1220
12C	36	28·35·53·7·11·23	2	1,2,35,43,63,1219
14A	14	29·37·53·11·23	1	1,2,751417
15A	15	210·36·52·7·11·23	2	1,5,1518
15B	30	29·36·52·7·11·23	1	32,53,1517
18A	18	29·35·53·7·11·23	2	6,94,1813
20A	20	28·37·52·7·11·23	1	1,53,102,2012	power equivalent
20B	20	28·37·52·7·11·23	1	1,53,102,2012
21A	21	210·36·53·11·23	0	3,2113
22A	22	29·37·53·7·23	0	1,11,2212	power equivalent
22B	22	29·37·53·7·23	0	1,11,2212
23A	23	210·37·53·7·11	1	2312	power equivalent
23B	23	210·37·53·7·11	1	2312
24A	24	27·36·53·7·11·23	-1	124,6,1222410
24B	24	27·36·53·7·11·23	1	2,32,4,122,2410
30A	30	29·36·52·7·11·23	0	1,5,152,308

Generalized Monstrous Moonshine

In analogy to monstrous moonshine for the monster M, for Co₃, the relevant McKay-Thompson series is [math]\displaystyle{ T_{4A}(\tau) }[/math] where one can set the constant term a(0) = 24 (OEIS: A097340),

[math]\displaystyle{ \begin{align}j_{4A}(\tau) &=T_{4A}(\tau)+24\\ &=\Big(\tfrac{\eta^2(2\tau)}{\eta(\tau)\,\eta(4\tau)} \Big)^{24} \\ &=\Big(\big(\tfrac{\eta(\tau)}{\eta(4\tau)}\big)^{4}+4^2 \big(\tfrac{\eta(4\tau)}{\eta(\tau)}\big)^{4}\Big)^2\\ &=\frac{1}{q} + 24+ 276q + 2048q^2 +11202q^3+49152q^4+\dots \end{align} }[/math]

and η(τ) is the Dedekind eta function.

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, PMID 16591697, Bibcode: 1968PNAS...61..398C 
• 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, doi:10.1112/blms/1.1.79, ISSN 0024-6093 
• 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, https://books.google.com/books?id=TPPkAAAAIAAJ  Reprinted in (Conway Sloane)
• 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, doi:10.1007/978-1-4757-2016-7, ISBN 978-0-387-98585-5, https://books.google.com/books?id=upYwZ6cQumoC 
• Feit, Walter (1974), "On integral representations of finite groups", Proceedings of the London Mathematical Society, Third Series 29 (4): 633–683, doi:10.1112/plms/s3-29.4.633, ISSN 0024-6115 
• Finkelstein, Larry (1973), "The maximal subgroups of Conway's group C₃ and McLaughlin's group", Journal of Algebra 25: 58–89, doi:10.1016/0021-8693(73)90075-6, ISSN 0021-8693 
• 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, https://books.google.com/books?id=ggqxuG31B3cC 
• 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, https://books.google.com/books?id=38fEMl2-Fp8C 
• Griess, Robert L. Jr. (1998), Twelve sporadic groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-662-03516-0, ISBN 978-3-540-62778-4 
• Wilson, Robert A. (2009), The finite simple groups., Graduate Texts in Mathematics 251, 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5 

External links

• MathWorld: Conway Groups
• Atlas of Finite Group Representations: Co₃ version 2
• Atlas of Finite Group Representations: Co₃ version 3