Sporadic Mathieu group M₁₂

M₁₂: Length 87, 2-generator, 5-relator.

Remark: x and y are R.A.Wilson's standard generators for M₁₂. This presentation is on the same generators as the CMY-presentation 13.2, but we have omitted the redundant relation (xyxyxy^-1xy^-1)⁵ = 1. (The omitted relation is clearly equivalent to [x, yxy]⁵ = 1.) The presentation is available in MAGMA code here. (This applies to the subgroups below too.)

Some subgroups:

• M1 = < x, yxyxyxy^-1 > = M₁₁ of order 7 920 and index 12 (the point stabiliser).
• M2 = < x, y^-1xyxyxy > = M₁₁ of order 7 920 and index 12 (the total stabiliser).

Realisation: x = (\infty, 0)(1, X)(2, 8)(3, 6) and y = (\infty, 1, 0)(2, 9, X)(3, 7, 8)(4, 5, 6). This gives xy = (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, X).
The above permutations are available in MAGMA format here. We have used 10 for X, 11 for 0 and 12 for \infty in the MAGMA file.

M₁₂: Length 79, 2-generator, 4-relator.

Remark: This presentation is on the same generators as the previous one and is available in MAGMA format here. This presentation is a little difficult for coset enumeration. I have included some redundant relators that have been commented out in the presentation file.

M₁₂:2: Length 73, 2-generator, 4-relator.

Remark: x and y are R.A.Wilson's standard generators for M₁₂:2. The presentation is available in MAGMA code here. (This applies to the subgroups below too.)

Some subgroups:

• M0 = < y, y^x > = M₁₁ of order 95 040 and index 2.
• M1 = < x, y^xy > = PGL₂(11) of order 1 320 and index 144 (the novelty).
• H1 = < y, xyxyxyxyxy^-1x > = M₁₁ of order 7 920 and index 24.

Realisation: As permutations on 24 points in MAGMA format.

John N. Bray