ATLAS: Symplectic group S₁₀(2)

The following information is available for S₁₀(2):

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

Standard generators

Black box algorithms

Finding generators

• Find an element of order 34. Its 17th power, x say, is in class 2A.
  [The probability of success at each attempt is 1 in 17.]
• Find an element of order 11 or 33. This powers up to y of order 11.
  [The probability of success at each attempt is 1 in 11.]
• Find a conjugate a of x and a conjugate b of y such that ab has order 15.
  [The probability of success at each attempt is 1 in 31.]
• Now a and b are standard generators of S₁₀(2).

Checking generators

To check that elements x and y of S₁₀(2) are standard generators:

• Check o(x) = 2
• Check o(y) = 11
• Check o(xy) = 15
• Let z = xyxyy(xy)³(xyy)⁴
• Check o(z) = 34
• Check o(xz¹⁷) = 3

Presentation

< a, b | a² = b¹¹ = (ab)¹⁵ = (ab⁵)¹⁸ = [a, b]³ = [a, b²]² = [a, b³]² = [a, b⁴]² = [a, b⁵]³ = [a, (ab)⁵]² = 1 >.

This presentation is available in Magma format as follows: S10(2) on a and b.

Representations

• Some faithful permutation representations.
  • Permutations on 496 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Permutations on 528 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Permutations on 992 points - imprimitive: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Permutations on 1023 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Permutations on 1056 points - imprimitive: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some faithful irreducibles in characteristic 2.
  • Dimension 10 over GF(2) - the natural representation: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 32 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 44 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 100 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 164 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some faithful irreducibles in characteristic other than 2.
  • Dimension 155 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 155 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 155 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 155 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 155 over GF(17): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
  • Dimension 155 over GF(31): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some faithful irreducibles in characteristic 0
  • Dimension 155 over Z: a and b (GAP).
  • Dimension 187 over Z: a and b (GAP).
  • Dimension 341 over Z: a and b (GAP).

Maximal subgroups

• O10-(2):2.
  Order: 50030759116800.
  Index: 496.

• O10+(2):2.
  Order: 46998591897600.
  Index: 528.

• 2⁹:S₈(2).
  Order: 24257337753600.
  Index: 1023.

• 2¹⁵:L₅(2).
  Order: 327659028480.
  Index: 75735.

• [2¹⁵]:(S₆(2) × S₃).
  Order: 285380444160.
  Index: 86955.

• S₈(2) × S₃.
  Order: 284265676800.
  Index: 87296.

• [2¹⁸]:(S₆ × L₃(2)).
  Order: 31708938240.
  Index: 782595.

• [2¹⁸]:(S₃ × A₈).
  Order: 31708938240.
  Index: 782595.

• S₆(2) × S₆.
  Order: 1045094400.
  Index: 23744512.

• L₂(32):5.
  Order: 163680.
  Index: 151608360960.

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