Conway group Co1
Algebraic structure → Group theory Group theory |
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In the area of modern algebra known as
- 221 · 39 · 54 · 72 · 11 · 13 · 23
- = 4157776806543360000
- ≈ 4×1018.
History and properties
Co1 is one of the 26 sporadic groups and was discovered by
The outer automorphism group is trivial and the Schur multiplier has order 2.
Involutions
Co0 has 4 conjugacy classes of involutions; these collapse to 2 in Co1, but there are 4-elements in Co0 that correspond to a third class of involutions in Co1.
An image of a dodecad has a centralizer of type 211:M12:2, which is contained in a maximal subgroup of type 211:M24.
An image of an octad or 16-set has a centralizer of the form 21+8.O8+(2), a maximal subgroup.
Representations
The smallest faithful permutation representation of Co1 is on the 98280 pairs {v,–v} of norm 4 vectors.
There is a matrix representation of dimension 24 over the field .
The centralizer of an involution of type 2B in the monster group is of the form 21+24Co1.
The Dynkin diagram of the even Lorentzian
Maximal subgroups
Wilson (1983) found the 22 conjugacy classes of maximal subgroups of Co1, though there were some errors in this list, corrected by Wilson (1988).
- Co2
- 3.Suz:2 The lift to Aut(Λ) = Co0 fixes a complex structure or changes it to the complex conjugate structure. Also, top of Suzuki chain.
- 211:M24 Image of monomial subgroup from Aut(Λ), that subgroup stabilizing the standard frame of 48 vectors of form (±8,023) .
- Co3
- 21+8.O8+(2) centralizer of involution class 2A (image of octad from Aut(Λ))
- Fi21:S3 ≈ U6(2):S3 The lift to Aut(Λ) is the symmetry group of a coplanar hexagon of 6 type 2 points.
- (A4 × G2(4)):2 in Suzuki chain.
- 22+12:(A8 × S3)
- 24+12.(S3 × 3.S6)
- 32.U4(3).D8
- 36:2.M12 (holomorph of ternary Golay code)
- (A5 × J2):2 in Suzuki chain
- 31+4:2.PSp4(3).2
- (A6 × U3(3)).2 in Suzuki chain
- 33+4:2.(S4 × S4)
- A9 × S3 in Suzuki chain
- (A7 × L2(7)):2 in Suzuki chain
- (D10 × (A5 × A5).2).2
- 51+2:GL2(5)
- 53:(4 × A5).2
- 72:(3 × 2.S4)
- 52:2A5
References
- PMID 16591697
- MR 0240186
- MR 0248216
- MR 0338152 Reprinted in Conway & Sloane (1999, 267-298)
- MR 0920369
- Thompson, Thomas M. (1983), From error-correcting codes through sphere packings to simple groups, Carus Mathematical Monographs, vol. 21, MR 0749038
- MR 0827219
- MR 1707296
- Wilson, Robert A. (1983), "The maximal subgroups of Conway's group Co₁", MR 0723071
- Wilson, Robert A. (1988), "On the 3-local subgroups of Conway's group Co₁", MR 0928064
- Wilson, Robert A. (2009), The finite simple groups., Graduate Texts in Mathematics 251, vol. 251, Berlin, New York: Zbl 1203.20012