Fischer group Fi₂₂

Fi₂₂ is one of the 26 sporadic groups and is the smallest of the three Fischer groups. It was introduced by Bernd Fischer (1971, 1976) while investigating 3-transposition groups.

The outer automorphism group has order 2, and the Schur multiplier has order 6.

The Fischer group Fi₂₂ has a rank 3 action on a graph of 3510 vertices corresponding to its 3-transpositions, with point stabilizer the double cover of the group PSU₆(2). It also has two rank 3 actions on 14080 points, exchanged by an outer automorphism.

Fi₂₂ has an irreducible real representation of dimension 78. Reducing an integral form of this mod 3 gives a representation of Fi₂₂ over the field with 3 elements, whose quotient by the 1-dimensional space of fixed vectors is a 77-dimensional irreducible representation.

The perfect triple cover of Fi₂₂ has an irreducible representation of dimension 27 over the field with 4 elements. This arises from the fact that Fi₂₂ is a subgroup of ²E₆(2²). All the ordinary and modular character tables of Fi₂₂ have been computed. Hiss & White (1994) found the 5-modular character table, and Noeske (2007) found the 2- and 3-modular character tables.

The automorphism group of Fi₂₂ centralizes an element of order 3 in the baby monster group.

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 Fi₂₂, the McKay-Thompson series is ${\displaystyle T_{6A}(\tau )}$ where one can set a(0) = 10 (OEIS: A007254),

${\displaystyle {\begin{aligned}j_{6A}(\tau )&=T_{6A}(\tau )+10\\&=\left(\left({\tfrac {\eta (\tau )\,\eta (3\tau )}{\eta (2\tau )\,\eta (6\tau )}}\right)^{3}+2^{3}\left({\tfrac {\eta (2\tau )\,\eta (6\tau )}{\eta (\tau )\,\eta (3\tau )}}\right)^{3}\right)^{2}\\&=\left(\left({\tfrac {\eta (\tau )\,\eta (2\tau )}{\eta (3\tau )\,\eta (6\tau )}}\right)^{2}+3^{2}\left({\tfrac {\eta (3\tau )\,\eta (6\tau )}{\eta (\tau )\,\eta (2\tau )}}\right)^{2}\right)^{2}-4\\&={\frac {1}{q}}+10+79q+352q^{2}+1431q^{3}+4160q^{4}+13015q^{5}+\dots \end{aligned}}}$

and η(τ) is the Dedekind eta function.

Wilson (1984) found the 12 conjugacy classes of maximal subgroups of Fi₂₂ as follows:

• 2·U₆(2)
• O₇(3) (Two classes, fused by an outer automorphism)
• O⁺
  ₈(2):S₃
• 2¹⁰:M₂₂
• 2⁶:S₆(2)
• (2 × 2¹⁺⁸):(U₄(2):2)
• U₄(3):2 × S₃
• ²F₄(2)' (This is the Tits group)
• 2⁵⁺⁸:(S₃ × A₆)
• 3¹⁺⁶:2³⁺⁴:3²:2
• S₁₀ (Two classes, fused by an outer automorphism)
• M₁₂

• Aschbacher, Michael (1997), 3-transposition groups, Cambridge Tracts in Mathematics, vol. 124, Cambridge University Press, doi:10.1017/CBO9780511759413, ISBN 978-0-521-57196-8, MR 1423599 contains a complete proof of Fischer's theorem.
• Conway, John Horton (1973), "A construction for the smallest Fischer group F₂₂", in Shult, and Ernest E.; Hale, Mark P.; Gagen, Terrence (eds.), Finite groups '72 (Proceedings of the Gainesville Conference on Finite Groups, University of Florida, Gainesville, Fla., March 23–24, 1972.), North-Holland Mathematics Studies, vol. 7, Amsterdam: North-Holland, pp. 27–35, MR 0372016
• 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
• Hiss, Gerhard; White, Donald L. (1994), "The 5-modular characters of the covering group of the sporadic simple Fischer group Fi₂₂ and its automorphism group", Communications in Algebra, 22 (9): 3591–3611, doi:10.1080/00927879408825043, ISSN 0092-7872, MR 1278807
• Noeske, Felix (2007), "The 2- and 3-modular characters of the sporadic simple Fischer group Fi₂₂ and its cover", Journal of Algebra, 309 (2): 723–743, doi:10.1016/j.jalgebra.2006.06.020, ISSN 0021-8693, MR 2303203
• Wilson, Robert A. (1984), "On maximal subgroups of the Fischer group Fi₂₂", Mathematical Proceedings of the Cambridge Philosophical Society, 95 (2): 197–222, doi:10.1017/S0305004100061491, ISSN 0305-0041, MR 0735364
• Wilson, Robert A. (2009), The finite simple groups, Graduate Texts in Mathematics 251, vol. 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 Representations.

• MathWorld: Fischer Groups
• Atlas of Finite Group Representations: Fi22