Conway group Co3

For general background and history of the Conway sporadic groups, see Conway group.

In the area of modern algebra known as group theory, the Conway group${\displaystyle \mathrm {Co} _{3}}$ is a sporadic simple group of order

   495,766,656,000

= 2¹⁰ ·3⁷ ·5³ ·7 ·11 ·23

≈ 5×10¹¹.

History and properties

${\displaystyle \mathrm {Co} _{3}}$ is one of the 26 sporadic groups and was discovered by John HortonConway  ( 1968 , 1969 ) as the group of automorphisms of the Leech lattice ${\displaystyle \Lambda }$ fixing a lattice vector of type 3, thus length √6. It is thus a subgroup of ${\displaystyle \mathrm {Co} _{0}}$. It is isomorphic to a subgroup of ${\displaystyle \mathrm {Co} _{1}}$. The direct product ${\displaystyle 2\times \mathrm {Co} _{3}}$ is maximal in ${\displaystyle \mathrm {Co} _{0}}$.

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.

WalterFeit  ( 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 ${\displaystyle \mathbb {Z} /2\mathbb {Z} \times \mathrm {Co} _{2}}$ or ${\displaystyle \mathbb {Z} /2\mathbb {Z} \times \mathrm {Co} _{3}}$.

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.

LarryFinkelstein ( 1973 ) found the 14 conjugacy classes of maximal subgroups of ${\displaystyle \mathrm {Co} _{3}}$ as follows:

Maximal subgroups of Co₃

No.	Structure	Order	Index	Comments
1	McL:2	1,796,256,000 = 28·36·53·7·11	276 = 22·3·23	McL fixes a 2-2-3 triangle. The maximal subgroup also includes reflections of the triangle. ${\displaystyle \mathrm {Co} _{3}}$ has a doubly transitive permutation representation on 276 type 2-2-3 triangles having as an edge a type 3 vector fixed by ${\displaystyle \mathrm {Co} _{3}}$.
2	HS	44,352,000 = 29·32·53·7·11	11,178 = 2·35·23	fixes a 2-3-3 triangle
3	U4(3).22	13,063,680 = 29·36·5·7	37,950 = 2·3·52·11·23
4	M23	10,200,960 = 27·32·5·7·11·23	48,600 = 23·35·52	fixes a 2-3-4 triangle
5	35:(2 × M11)	3,849,120 = 25·37·5·11	128,800 = 25·52·7·23	fixes or reflects a 3-3-3 triangle
6	2·Sp6(2)	2,903,040 = 210·34·5·7	170,775 = 33·52·11·23	centralizer of an involution of class 2A (trace 8), which moves 240 of the 276 type 2-2-3 triangles
7	U3(5):S3	756,000 = 25·33·53·7	655,776 = 25·34·11·23
8	31+4 +:4S6	699,840 = 26·37·5	708,400 = 24·52·7·11·23	normalizer of a subgroup of order 3 (class 3A)
9	24·A8	322,560 = 210·32·5·7	1,536,975 = 35·52·11·23
10	PSL(3,4):(2 × S3)	241,920 = 28·33·5·7	2,049,300 = 22·34·52·11·23
11	2 × M12	190,080 = 27·33·5·11	2,608,200 = 23·34·52·7·23	centralizer of an involution of class 2B (trace 0), which moves 264 of the 276 type 2-2-3 triangles
12	[210.33]	27,648 = 210·33	17,931,375 = 34·53·7·11·23
13	S3 × PSL(2,8):3	9,072 = 24·34·7	54,648,000 = 26·33·53·11·23	normalizer of a subgroup of order 3 (class 3C, trace 0)
14	A4 × S5	1,440 = 25·32·5	344,282,400 = 25·35·52·7·11·23

Conjugacy classes

Traces of matrices in a standard 24-dimensional representation of Co₃ are shown. ^[1] The names of conjugacy classes are taken from the Atlas of Finite Group Representations. ^{[2] [3]} The cycle structures listed act on the 276 2-2-3 triangles that share the fixed type 3 side. ^[4]

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 ${\displaystyle T_{4A}(\tau )}$ where one can set the constant term a(0) = 24 ( OEIS:  A097340 ),

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

and η(τ) is the Dedekind eta function.

References

1. Conway et al. (1985)
2. "ATLAS: Conway group Co3".
3. "ATLAS: Conway group Co1".
4. "ATLAS: Co3 — Permutation representation on 276 points".

• 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, Bibcode:1968PNAS...61..398C, doi: 10.1073/pnas.61.2.398 , MR   0237634, PMC   225171 , PMID   16591697
• 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, MR   0248216
• Conway, John Horton (1971), "Three lectures on exceptional groups", in Powell, M. B.; Higman, Graham (eds.), 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, vol. 290 (3rd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4757-2016-7, ISBN   978-0-387-98585-5, MR   0920369
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• 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, MR   0346046
• Thompson, Thomas M. (1983), From error-correcting codes through sphere packings to simple groups, Carus Mathematical Monographs, vol. 21, Mathematical Association of America, ISBN   978-0-88385-023-7, MR   0749038
• 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   0827219
• 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, MR   1707296
• Wilson, Robert A. (2009), The finite simple groups., Graduate Texts in Mathematics 251, vol. 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN   978-1-84800-987-5, Zbl   1203.20012

External links

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