ATLAS: Conway group Co₂

The following information is available for Co₂:

• Standard generators.
• Black box algorithms.
• Presentation.
• Representations.
• Maximal subgroups.
• Conjugacy classes.

Standard generators

Black box algorithms

Finding generators

To find standard generators for Co₂:

1. Find any element of order 16, 18 or 28. This powers up to a 2A­element x.
   [The probability of success at each attempt is 155 in 1008 (about 1 in 7).]
2. Find any element of order 15 or 30, This powers up to y of order 5.
   [The probability of success at each attempt is 1 in 5, and the probability that y ends up being in class 5A is 2 in 3.]
3. Find a conjugate a of x and a conjugate b of y, whose product has order 28.
   [If y is in class 5A, then the probability of success at each attempt is 40 in 759 (about 1 in 19).
   If y is in class 5B, then the probability of success at each attempt is 8 in 759 (about 1 in 95).]
   If you have still not succeeded after (say) 35 attempts at this step, you begin to suspect that y is in the wrong conjugacy class, so go back to Step 2.
4. If abb has order 15, then y is in the wrong conjugacy class, so go back to Step 2.
5. Otherwise abb has order 9 and standard generators for Co₂ have been obtained.

Checking generators

• Check o(x) = 2
• Check o(y) = 5
• Check o(xy) = 28
• Check o(x(xy)¹⁴) = 3

Presentation

< a, b | a² = b⁵ = (ab²)⁹ = [a, b]⁴ = [a, b²]⁴ = [a, bab]³ = [a, bab²ab]² = [a, bab^-2]³ = [a, b^-2abab^-2]² = (abab²ab^-1ab^-2)⁷ = 1 >.

This presentation is available in Magma format as follows: Co2 on a and b.

Representations

• Permutations on 2300 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 4600 points - imprimitive: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 22 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 24 over GF(2) - reducible: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 230 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 748 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 748 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 23 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 253 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 275 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 23 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 253 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 275 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 23 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 253 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 275 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 23 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 253 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 275 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 23 over GF(23): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 253 over GF(23): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 274 over GF(23): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).

Maximal subgroups

• U₆(2):2, with generators (ab)^7, abababbababb)^-5(abababbab)^-1(ababb)^4abababbab(abababbababb)^5.
  Er . . . try this one - it's shorter (a, abab).
• 2¹⁰:M₂₂:2, with generators here.
• McL, with generators here.
• 2¹⁺⁸.S₆(2), with generators here.
• HS:2, with generators here.
• (2⁴ × 2¹⁺⁶).A₈, with generators here.
  The quotient of this group by its centre is a split extension 2¹⁰:A₈, and the 2¹⁰ regared as a module for the A₈ is a uniserial module 4.6.
  There is just one class of complementary A₈ in this image, and they become 2.A₈ in (2⁴ × 2¹⁺⁶).A₈.
• U₄(3):D₈, with generators here
• 2⁴⁺¹⁰.(S₅ × S₃), with generators [a, abbbababbb], abababbabbab.
• M₂₃, with standard generators here.
• 3¹⁺⁴.2¹⁺⁴.S₅, with generators here.
• 5¹⁺²:4S₄, with generators (abb)^3, (ab)^-6(abbbb)^8(abababbbabbb)^5(abbbb)^-8(ab)^6.

Conjugacy classes

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