ATLAS: Fischer group Fi₂₄'

The following information is available for Fi₂₄':

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

Standard generators

Standard generators of the automorphism group Fi₂₄ = Fi₂₄':2 are c and d where c is in class 2C, d is in class 8D and cd has order 29.
Standard generators of 3.Fi₂₄':2 are preimages C and D where D has order 29.

A pair of generators conjugate to a, b can be obtained as
a' = (cdd)¹⁰, b' = (cdcdd)⁻¹⁰(cdcdcddcdcdd)⁸(cdcdd)¹⁰.

Black box algorithms

Finding generators

To find standard generators for Fi₂₄':

• Find an element of order 22, 26, 28 or 60. It powers up to a 2A-element x.
• Find an element of order 39. It has a 50-50 chance of powering up to a 3E-element y.
• Find conjugates a of x and b of y whose product has order 29 [probability about 1 in 127].
• If you fail, you probably have a 3A-element, so go back to step 2.
• If you succeed, you definitely have a 3E-element, so keep going until abababb has order 33.

To find standard generators of Fi₂₄ = Fi₂₄':2:

• Find an element of order 34, 46, 54, 70, or 78. It powers to an element x in class 2C.
• Find an element of order 40. It powers to an element y in class 8D.
• Find conjugates c of x and d of y such that cd has order 29. These are standard generators.

Checking generators

To check that elements x and y of Fi₂₄' are standard generators:

• Check o(x) = 2
• Check o(y) = 3
• Check o(xy) = 29
• Check o(xyxyxyy) = 23
• Let z = (xy)⁶y
• Check o(z) = 60
• Check o(x(z³⁰)^xyxy) = 5

To check that elements x and y of Fi₂₄':2 are standard generators:

• Check o(x) = 2
• Check o(y) = 8
• Check o(xy) = 29
• Let z = xyxy⁶
• Check o(z) = 54
• Check o(xz²⁷) = 3
• Check o(xyy) = 20

Representations

• Dimension 3774 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma). — kindly donated by Jürgen Müller.
• Dimension 781 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).

• Dimension 783 over GF(4): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP), A and B (Magma).
• Permutations on 920808 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP), A and B (Magma).

• Dimension 781 over GF(3): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP), c and d (Magma).
• Permutations on 306936 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP), c and d (Magma).

• Dimension 1566 over GF(2): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP), C and D (Magma).
• Permutations on 920808 points: C and D (Meataxe), C and D (Meataxe binary), C and D (GAP), C and D (Magma).

Maximal subgroups

• Fi₂₃.
• 2.Fi₂₂:2.
• (3 × O₈⁺(3):3):2.
• O₁₀⁻(2).
• 3⁷.O₇(3).
• 3¹⁺¹⁰:U₅(2):2.
• 2¹¹.M₂₄.
• 2².U₆(2):S₃.
• 2¹⁺¹²:3₁.U₄(3).2.
• 3²⁺⁴⁺⁸.(A₅ × 2A₄).2.
• [3¹³]:(L₃(3) × 2).
• (A₄ × O₈⁺(2):3):2.
• He:2.
• He:2.
• 2³⁺¹².(L₃(2) × A₆).
• 2⁶⁺⁸.(S₃ × A₈).
• (G₂(3) × 3²:2).2.
• (A₉ × A₅):2.
• L₂(8):3 × A₆.
• A₇ × 7:6.
• U₃(3):2.
• U₃(3):2.
• L₂(13):2.
• L₂(13):2.
• F₄₀₆ = 29:14.

• Fi₂₄', with standard generators here.
• Fi₂₃ × 2, with generators here.
• (2 × 2.Fi₂₂):2, with generators here.
• S₃ × O₈⁺(3):S₃, with generators here.
• O₁₀⁻(2):2, with generators here.
• 3⁷.O₇(3):2, with generators here.
• 3¹⁺¹⁰:(U₅(2):2 × 2), with generators here.
• 2¹².M₂₄, with generators here.
• (2 × 2².U₆(2)):S₃, with generators here.
• 2¹⁺¹²:3₁.U₄(3).(2²)₁₂₂, with generators here.
• 3²⁺⁴⁺⁸.(S₅ × 2S₄), with generators here.
• [3¹³]:(L₃(3) × 2 × 2), with generators here.
• S₄ × O₈⁺(2):S₃, with generators here.
• 2³⁺¹².(L₃(2) × S₆), with generators here.
• 2⁷⁺⁸.(S₃ × A₈), with generators here.
• (G₂(3) × S₃ × S₃).2, with generators here.
• (S₉ × S₅), with generators here.
• L₂(8):3 × S₆, with generators here.
• S₇ × 7:6, with generators here.
• 7¹⁺²:(6 × S₃).2, with generators here.
• F₈₁₂ = 29:28, with generators here.

Conjugacy classes

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