Fischer group Fi22

In mathematics, the Fischer group Fi₂₂ or M(22) or F₂₂, of order

2¹⁷ · 3⁹ · 5² · 7 · 11 · 13 (= 64561751654400)

is the smallest 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 6.

Representations

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.

Maximal subgroups

Wilson (1984) found the 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)'

2⁵⁺⁸:(S₃ × A₆)

3¹⁺⁶:2³⁺⁴:3²:2

S₁₀ (Two classes, fused by an outer automorphism)

M₁₂

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.
• Conway, John Horton (1973), "A construction for the smallest Fischer group F₂₂", in Shult, and Ernest E.; Hale, Mark P.; Gagen, Terrence, Finite groups '72 (Proceedings of the Gainesville Conference on Finite Groups, University of Florida, Gainesville, Fla., March 23–24, 1972.), North-Holland Mathematics Studies 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 735364 
• 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 Representations.