Conway group Co2

Co₂ is one of the 26 sporadic groups and was discovered by John Horton Conway as the group of automorphisms of the Leech lattice Λ fixing a lattice vector of type 2.

Representations

Co₂ acts as a rank 3 permutation group on 2300 points.

Co₂ acts on the 23-dimensional even integral lattice with no roots of determinant 4, given as a sublattice of the Leech lattice orthogonal to a norm 4 vector. Over the field with 2 elements it has a 22-dimensional faithful representation; this is the smallest faithful representation over any field.

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 Z/2Z × Co₂ or Z/2Z × Co₃.

Maximal subgroups

Wilson (1983) showed that there are 11 conjugacy classes of maximal subgroups as follows.

U₆(2):2 Fixes a point of the rank 3 permutation representation on 2300 points.
2¹⁰:M₂₂:2
McL (fixing 2-2-3 triangle)
2¹⁺⁸:Sp₆(2)
HS:2 (can transpose type 3 vertices of conserved 2-3-3 triangle)
(2⁴ × 2¹⁺⁶).A₈
U₄(3):D₈
2⁴⁺¹⁰.(S₅ × S₃)
M₂₃
3¹⁺⁴.2¹⁺⁴.S₅
5¹⁺²:4S₄

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, MR 0237634
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, 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, 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
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 749038
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 827219
Griess, Robert L. Jr. (1998), Twelve sporadic groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62778-4, MR 1707296
Wilson, Robert A. (1983), "The maximal subgroups of Conway's group ·2", Journal of Algebra 84 (1): 107–114, doi:10.1016/0021-8693(83)90069-8, ISSN 0021-8693, MR 716772
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 05622792

External links

MathWorld: Conway Groups
Atlas of Finite Group Representations: Co₂ version 2
Atlas of Finite Group Representations: Co₂ version 3