ATLAS: Linear group L₂(227)

The following information is available for L₂(227):

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

Standard generators

Standard generators of L₂(227) are a of order 2, b of order 3 such that ab has order 227.
Standard generators of the double cover 2.L₂(227) = SL₂(227) are preimages A and B such that B has order 3 and AB has order 227.

Standard generators of L₂(227):2 = PGL₂(227) are not yet defined.
Standard generators of the double cover 2.L₂(227):2 = 2.PGL₂(227) are not yet defined.

Black box algorithms

Representations

• Some primitive permutation representations
  • Permutations on 228 points: a and b (GAP).
• Some matrix representations in characteristic 0:
  • Dimension 227 over Z: a and b (GAP).
  • Dimension 228 over Z(z113) (monomial): a and b (GAP).

• Some matrix representations in characteristic 0:
  • Dimension 228 over Z(z226) (monomial): A and B (GAP).

• none

• none

The maximal subgroups of L₂(227) are as follows.

• 227:113, with generators ab², abab²abab²ababab²abab²abababab
• D₂₂₈, with generators a, a^ab
• D₂₂₆, with generators a, a^ab2abab
• A₄, with generators a, b^{ab2ababab2ababab2ab}

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