ATLAS: Fischer group Fi₂₃

The following information is available for Fi₂₃:

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

Standard generators

Black box algorithms

Finding generators

• Find any element of order 20, 28 or 60. It powers up to a 2B-element x.
• Find any element of order 3, y say, (probably as a power of an element of order 9 or 18).
• Find a conjugate a of x and a conjugate b of y whose product has order 28.
• If you succeed, standard generators for Fi23 have been obtained.
• If you fail, then y is probably in the wrong conjugacy class.

Checking generators

To check that elements x and y of Fi₂₃ are standard generators:

• Check o(x) = 2
• Check o(y) = 3
• Check o(xy) = 28
• Check o(x^yy(xy)¹⁴) = 5

Representations

• Permutations on 31671 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
• Permutations on 137632 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
• Permutations on 275264 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma). — imprimitive.
• Dimension 782 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
• Dimension 1494 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 253 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
• Dimension 528 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
• Dimension 782 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
• Dimension 782 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
• Dimension 782 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
• Dimension 782 over GF(13): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
• Dimension 782 over GF(17): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
• Dimension 782 over GF(23): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).

Maximal subgroups

• 2.Fi₂₂, with standard generators here and (non-standard) generators a, abab(abb)⁴ab.
• O₈⁺(3):S₃, with generators [a, babab]², (ab)⁸(abb)³(ab)¹¹.
• 2².U₆(2).2, with generators a, (b(ab)¹²)².
• S₈(2), with generators a, abab(abb)³ab.
• O₇(3) × S₃, with generators [a, babab]², abababb(ababbabb)²abababb.
• 2¹¹.M₂₃, with generators a, (ab)⁷(ba)².
• 3¹⁺⁸.2¹⁺⁶.3¹⁺².2S₄.
• [3¹⁰].(L₃(3) × 2).
• S₁₂, with generators a, (ab)⁹(abb)¹²(ab)⁹.
• (2² × 2¹⁺⁸).(3 × U₄(2)).2, with generators [a, babab]², ab(abb)¹³(ab)¹¹(abb)¹³.
• 2⁶⁺⁸:(A₇ × S₃).
• S₆(2) × S₄.
• S₄(4):4, with generators a, ab(ba)¹²(ab)¹⁶(ba)¹³.
• L₂(23), with standard generators [a, babab]², b^((ab)³(abb)¹⁰ab).

Conjugacy classes

Go to main ATLAS (version 2.0) page

Go to sporadic groups page

Go to old Fi23 page - ATLAS version 1