Conway group Co₃

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

   2¹⁰ · 3⁷ · 5³ · 7 · 11 · 23

= 495766656000

≈ 5×10¹¹.

History and properties

${\displaystyle \mathrm {Co} _{3}}$ 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 ${\displaystyle \Lambda }$ 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.

(txt) harv error: no target: CITEREFtxt (help) 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.

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

• McL:2 – 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 ${\displaystyle \mathrm {Co} _{3}}$.
• 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.[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	3^3·5^2·11·23	8	1^36,2^120
2B	190,080	2^3·3^4·5^2·7·23	0	1^12,2^132
3A	349,920	2^5·5^2·7·11·23	-3	1^6,3^90
3B	29,160	2^7·3·5^2·7·11·23	6	1^15,3^87
3C	4,536	2^7·3^3·5^3·11·23	0	3^92
4A	23,040	2·3^5·5^2·7·11·23	-4	1^16,2^10,4^60
4B	1,536	2·3^6·5^3·7·11·23	4	1^8,2^14,4^60
5A	1500	2^8·3^6·7·11·23	-1	1,5^55
5B	300	2^8·3^6·5·7·11·23	4	1^6,5^54
6A	4,320	2^5·3^4·5^2·7·11·23	5	1^6,3^10,6^40
6B	1,296	2^6·3^3·5^3·7·11·23	-1	2^3,3^12,6^39
6C	216	2^7·3^4·5^3·7·11·23	2	1^3,2^6,3^11,6^38
6D	108	2^8·3^4·5^3·7·11·23	0	1^3,2^6,3^3,6^42
6E	72	2^7·3^5·5^3·7·11·23	0	3^4,6^44
7A	42	2^9·3^6·5^3·11·23	3	1^3,7^39
8A	192	2^4·3^6·5^3·7·11·23	2	1^2,2^3,4^7,8^30
8B	192	2^4·3^6·5^3·7·11·23	-2	1^6,2,4^7,8^30
8C	32	2^5·3^7·5^3·7·11·23	2	1^2,2^3,4^7,8^30
9A	162	2^9·3^3·5^3·7·11·23	0	3^2,9^30
9B	81	2^10·3^3·5^3·7·11·23	3	1^3,3,9^30
10A	60	2^8·3^6·5^2·7·11·23	3	1,5^7,10^24
10B	20	2^8·3^7·5^2·7·11·23	0	1^2,2^2,5^2,10^26
11A	22	2^9·3^7·5^3·7·23	2	1,11^25	power equivalent
11B	22	2^9·3^7·5^3·7·23	2	1,11^25
12A	144	2^6·3^5·5^3·7·11·23	-1	1^4,2,3^4,6^3,12^20
12B	48	2^6·3^6·5^3·7·11·23	1	1^2,2^2,3^2,6^4,12^20
12C	36	2^8·3^5·5^3·7·11·23	2	1,2,3^5,4^3,6^3,12^19
14A	14	2^9·3^7·5^3·11·23	1	1,2,7^514^17
15A	15	2^10·3^6·5^2·7·11·23	2	1,5,15^18
15B	30	2^9·3^6·5^2·7·11·23	1	3^2,5^3,15^17
18A	18	2^9·3^5·5^3·7·11·23	2	6,9^4,18^13
20A	20	2^8·3^7·5^2·7·11·23	1	1,5^3,10^2,20^12	power equivalent
20B	20	2^8·3^7·5^2·7·11·23	1	1,5^3,10^2,20^12
21A	21	2^10·3^6·5^3·11·23	0	3,21^13
22A	22	2^9·3^7·5^3·7·23	0	1,11,22^12	power equivalent
22B	22	2^9·3^7·5^3·7·23	0	1,11,22^12
23A	23	2^10·3^7·5^3·7·11	1	23^12	power equivalent
23B	23	2^10·3^7·5^3·7·11	1	23^12
24A	24	2^7·3^6·5^3·7·11·23	-1	1^24,6,12^224^10
24B	24	2^7·3^6·5^3·7·11·23	1	2,3^2,4,12^2,24^10
30A	30	2^9·3^6·5^2·7·11·23	0	1,5,15^2,30^8

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. http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Co3/#ccls
3. http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/Co1/#ccls
4. http://brauer.maths.qmul.ac.uk/Atlas/v3/permrep/Co3G1-p276B0

• 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, 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, 290 (3rd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4757-2016-7, ISBN 978-0-387-98585-5, MR 0920369
• Feit, Walter (1974), "On integral representations of finite groups", Proceedings of the London Mathematical Society, Third Series, 29: 633–683, doi:10.1112/plms/s3-29.4.633, ISSN 0024-6115, MR 0374248
• 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, 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, 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