ATLAS: Linear group L₂(229)

The following information is available for L₂(229):

• Standard generators.
• Automorphisms.
• Black box algorithms (not yet available).
• Representations.
• Maximal subgroups.

Standard generators

Standard generators of L₂(229) are a of order 2, b of order 3 such that ab has order 229.
Standard generators of the double cover 2.L₂(229) = SL₂(229) are preimages A and B such that B has order 3 and AB has order 229.

Standard generators of L₂(229):2 = PGL₂(229) are not yet defined.
Standard generators of the double cover 2.L₂(229):2 = 2.PGL₂(229) are not yet defined.

Black box algorithms

Representations

• Some primitive permutation representations
  • Permutations on 230 points: a and b (GAP).
• Some matrix representations in characteristic 0:
  • Dimension 229 over Z: a and b (GAP).
  • Dimension 230 over Z(z114) (monomial): a and b (GAP).

• Some matrix representations in characteristic 0:
  • Dimension 230 over Z(z228) (monomial): A and B (GAP).

• none

• none

The maximal subgroups of L₂(229) are as follows.

• 229:114, with generators a, ababab²abababab
• D₂₃₀, with generators a, a^ababab
• D₂₂₈, with generators a, a^{abab2abababab}
• A₅, with standard generators a, b^ab2abab2abab
• A₅, with standard generators a, b^{abababab2ab2ab2ab}

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