Fischer group Fi₂₄

Fi₂₄ is one of the 26 sporadic groups and is the largest of the three Fischer groups introduced by Bernd Fischer (1971, 1976) while investigating 3-transposition groups. It is the 3rd largest of the sporadic groups (after the Monster group and Baby Monster group).

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 sporadic simple group Fi₂₄, as a result of which the prime 3 plays a special role in its theory.

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.

Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster, but that similar phenomena may be found for other groups. Larissa Queen and others subsequently found that one can construct the expansions of many Hauptmoduln from simple combinations of dimensions of sporadic groups. For Fi₂₄ (as well as Fi₂₃), the relevant McKay-Thompson series is ${\displaystyle T_{3A}(\tau )}$ where one can set the constant term a(0) = 42 (OEIS: A030197),

${\displaystyle {\begin{aligned}j_{3A}(\tau )&=T_{3A}(\tau )+42\\&={\Big (}{\big (}{\tfrac {\eta (\tau )}{\eta (3\tau )}}{\big )}^{6}+3^{3}{\big (}{\tfrac {\eta (2\tau )}{\eta (\tau )}}{\big )}^{6}{\Big )}^{2}\\&={\frac {1}{q}}+42+783q+8672q^{2}+65367q^{3}+371520q^{4}+1741655q^{5}+\dots \end{aligned}}}$

Linton & Wilson (1991) found the 22 conjugacy classes of maximal subgroups of 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

• 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 1203.20012
• Wilson, R. A. ATLAS of Finite Group Representation.

• MathWorld: Fischer Groups
• Atlas of Finite Group Representations: Fi24