title: Fischer group Fi23
text: # Fischer group Fi23
  
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  In the area of modern algebra known as group theory, the Fischer group $Fi_{23}$ is a sporadic simple group of order
  
      $2^{18}$ · $3^{13}$ · $5^2$ · 7 · 11 · 13 · 17 · 23
      = 4089470473293004800
      ≈ $4×10^{18}.$
  
  ## History
  
  $Fi_{23}$ is one of the 26 sporadic groups and is one of 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.
  
  ## Representations
  
  The Fischer group $Fi_{23}$ 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.
  
  ## Generalized Monstrous Moonshine
  
  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 $Fi_{23},$ the relevant McKay-Thompson series is $T_{3A}(\tau )$ where one can set the constant term a(0) = 42 (OEIS: A030197),
  
      ${\begin{aligned}j_{3A}(\tau )&=T_{3A}(\tau )+42\\&=\left(\left({\tfrac {\eta (\tau )}{\eta (3\tau )}}\right)^{6}+3^{3}\left({\tfrac {\eta (2\tau )}{\eta (\tau )}}\right)^{6}\right)^{2}\\&={\frac {1}{q}}+42+783q+8672q^{2}+65367q^{3}+371520q^{4}+\dots \end{aligned}}$
  
  and η(τ) is the Dedekind eta function.
  
  ## Maximal subgroups
  
  Kleidman, Parker & Wilson (1989) found the 14 conjugacy classes of maximal subgroups of $Fi_{23}$ as follows:
  
  * $2.Fi_{22}$
  * $O_8^+(3):S_3$
  * $2^2.U_6(2).2$
  * $S_8(2)$
  * $O_7(3)$ × $S_3$
  * $2^{11}.M_{23}$
  * $3^{1+8}.2^{1+6}.3^{1+2}.2S_4$
  * $[3^{10}].(L_3(3)$ × 2)
  * $S_{12}$
  * $(2^2$ × $2^{1+8}).(3$ × $U_4(2)).2$
  * $2^{6+8}:(A_7$ × $S_3)$
  * $S_6(2)$ × $S_4$
  * $S_4(4):4$
  * $L_2(23)$
  