Conway group Co3

Last updated May 16, 2024

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

History and properties

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

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

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 $\mathrm {Co} _{3}$ as follows:

McL:2 – McL fixes a 2-2-3 triangle. The maximal subgroup also includes reflections of the triangle. $\mathrm {Co} _{3}$ has a doubly transitive permutation representation on 276 type 2-2-3 triangles having as an edge a type 3 vector fixed by $\mathrm {Co} _{3}$ .
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	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	power equivalent
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	power equivalent
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	power equivalent
23A	23	2¹⁰·3⁷·5³·7·11	1	23¹²	power equivalent
23B	23	2¹⁰·3⁷·5³·7·11	1	23¹²	power equivalent
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 $T_{4A}(\tau )$ where one can set the constant term a(0) = 24 ( OEIS: A097340 ),

{\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.

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References

↑ Conway et al. (1985)
↑ "ATLAS: Conway group Co3".
↑ "ATLAS: Conway group Co1".
↑ "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
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, 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, 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

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[1] Conway et al. (1985)

[2] "ATLAS: Conway group Co3".

[3] "ATLAS: Conway group Co1".

[4] "ATLAS: Co3 — Permutation representation on 276 points".

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