Conway group Co2

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In mathematics, the Conway group Co2 is a sporadic group of order 42,305,421,312,000 discovered by (Conway 1968, 1969) as the group of automorphisms of the Leech lattice Λ fixing a lattice vector of type 2.

Representations[edit]

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 of Co2[edit]

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[edit]