Fischer group Fi₂₃

Fi₂₃ is one of the 26 sporadic groups and is one the three Fischer groups introduced by Bernd Fischer (1971, 1976) while investigating 3-transposition groups.

The Schur multiplier and the outer automorphism group are both trivial.

The Fischer group Fi₂₃ has a rank 3 action on a graph of 31671 vertices corresponding to 3-transpositions, with point stabilizer the double cover of the Fischer group Fi22. It has a second rank-3 action on 137632 points

The smallest faithful complex representation has dimension 782. The group has an irreducible representation of dimension 253 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 Fi₂₃, the relevant McKay-Thompson series is ${\displaystyle T_{3A}(\tau )}$ where one can set the constant term a(0) = 42 (OEIS: A030197),

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

and η(τ) is the Dedekind eta function.

Kleidman, Parker & Wilson (1989) found the 14 conjugacy classes of maximal subgroups of Fi₂₃ as follows:

• 2.Fi₂₂
• O₈⁺(3):S₃
• 2².U₆(2).2
• S₈(2)
• O₇(3) × S₃
• 2¹¹.M₂₃
• 3¹⁺⁸.2¹⁺⁶.3¹⁺².2S₄
• [3¹⁰].(L₃(3) × 2)
• S₁₂
• (2² × 2¹⁺⁸).(3 × U₄(2)).2
• 2⁶⁺⁸:(A₇ × S₃)
• S₆(2) × S₄
• S₄(4):4
• L₂(23)

• 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 The remainder of the paper is unpublished (as of 2010).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
• Kleidman, Peter B.; Parker, Richard A.; Wilson, Robert A. (1989), "The maximal subgroups of the Fischer group Fi₂₃", Journal of the London Mathematical Society, Second Series, 39 (1): 89–101, doi:10.1112/jlms/s2-39.1.89, ISSN 0024-6107, MR 0989922
• 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 Representations.

• MathWorld: Fischer Groups
• Atlas of Finite Group Representations: Fi23