Fischer group Fi24

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In mathematics, the Fischer group Fi₂₄′ or M(24)′ or F₂₄′ or F₃₊ of order

2²¹ · 3¹⁶ · 5² · 7³ · 11 · 13 · 17 · 23 · 29 (= 1255205709190661721292800) is the largest of the three Fischer groups, sporadic simple groups introduced by Bernd Fischer (1971, 1976) while investigating 3-transposition groups.

The outer automorphism group has order 2, and the Schur multiplier has order 3. The automorphism group is a 3-transposition group Fi₂₄, containing the simple group with index 2.

The centralizer of an element of order 3 in the monster group is a triple cover of the automorphism group Fi₂₄, as a result of which the prime 3 plays a special role in its theory.

Representations

The centralizer of an element of order 3 in the monster group is a triple cover of the Fischer group, as a result of which the prime 3 plays a special role in its theory. In particular it acts on a vertex operator algebra over the field with 3 elements.

The simple Fischer group has a rank 3 action on a graph of 306936 (=2³.3³.7².29) vertices corresponding to the 3-transpositions of Fi₂₄, with point stabilizer the Fischer group Fi23.

The triple cover has a complex representation of dimension 783. When reduced modulo 3 this has 1-dimensional invariant subspaces and quotient spaces, giving an irreducible representation of dimension 781 over the field with 3 elements.

Maximal subgroups

Linton & Wilson (1991) found the classes of maximal subgroups of the simple group Fi₂₄' as follows:

Fi₂₃ Centralizes a 3-transposition in the automorphism group Fi₂₄.

2.Fi₂₂:2

(3 x O+
8(3):3):2

O–
10(2)

3⁷.O₇(3)

3¹⁺¹⁰:U₅(2):2

2¹¹.M₂₄

2².U₆(2):S₃

2¹⁺¹²:3.U₄(3).2

3²⁺⁴⁺⁸.(A₅ x 2A₄).2

(A₄ x O+
8(2):3):2

He:2 (Two classes, fused by an outer automorphism)

2³⁺¹².(L₃(2) x A₆)

2⁶⁺⁸.(S₃ x A₈)

(G₂(3) x 3²:2).2

(A₉ x A₅):2

A₇ x 7:6

[3¹³]:(L₃(3) x 2)

L₂(8):3 x A₆

U₃(3):2 (Two classes, fused by an outer automorphism)

L₂(13):2 (Two classes, fused by an outer automorphism)

29:14

References

• Aschbacher, Michael (1997), 3-transposition groups, Cambridge Tracts in Mathematics 124, Cambridge University Press, doi:10.1017/CBO9780511759413, ISBN 978-0-521-57196-8, MR 1423599  contains a complete proof of Fischer's theorem.
• Fischer, Bernd (1971), "Finite groups generated by 3-transpositions. I", Inventiones Mathematicae 13 (3): 232–246, doi:10.1007/BF01404633, ISSN 0020-9910, MR 0294487  This is the first part of Fischer's preprint on the construction of his groups. The remainder of the paper is unpublished (as of 2010).
• Fischer, Bernd (1976), Finite Groups Generated by 3-transpositions, Preprint, Mathematics Institute, University of Warwick 
• Linton, Stephen A.; Wilson, Robert A. (1991), "The maximal subgroups of the Fischer groups Fi₂₄ and Fi₂₄'", Proceedings of the London Mathematical Society. Third Series 63 (1): 113–164, doi:10.1112/plms/s3-63.1.113, ISSN 0024-6115, MR 1105720 
• Wilson, Robert A. (2009), The finite simple groups, Graduate Texts in Mathematics 251 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 05622792 
• Wilson, R. A. ATLAS of Finite Group Representation.