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 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 a 23-dimensional even lattice with no roots, given by the orthogonal complement of a norm 6 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 ${\displaystyle \mathbb {Z} /2\mathbb {Z} \times \mathrm {Co} _{2}}$ 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:

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.