title: Fischer group Fi24
text: # Fischer group Fi24
  
  Algebraic structure → Group theory Group theory :
  
  Basic notions
  
  Subgroup
  
  Normal subgroup
  
  Quotient group
  
  (Semi-)direct product
  
  Group homomorphisms
  
  kernel
  
  image
  
  direct sum
  
  wreath product
  
  simple
  
  finite
  
  infinite
  
  continuous
  
  multiplicative
  
  additive
  
  cyclic
  
  abelian
  
  dihedral
  
  nilpotent
  
  solvable
  
  action
  
  Glossary of group theory
  
  List of group theory topics :
  
  Finite groups
  
  Classification of finite simple groups
  
  cyclic
  
  alternating
  
  Lie type
  
  sporadic
  
  Cauchy's theorem
  
  Lagrange's theorem
  
  Sylow theorems
  
  Hall's theorem
  
  p-group
  
  Elementary abelian group
  
  Frobenius group
  
  Schur multiplier
  
  Symmetric group $S_n$
  
  Klein four-group V
  
  Dihedral group $D_n$
  
  Quaternion group Q
  
  Dicyclic group $Dic_n$ :
  
  Discrete groupsLattices
  
  Integers ($\mathbb {Z}$)
  
  Free group
  
  Modular groups PSL(2, $\mathbb {Z}$)SL(2, $\mathbb {Z}$)
  
  Arithmetic group
  
  Lattice
  
  Hyperbolic group :
  
  Topological and Lie groups
  
  Solenoid
  
  Circle
  
  General linear GL(n)
  
  Special linear SL(n)
  
  Orthogonal O(n)
  
  Euclidean E(n)
  
  Special orthogonal SO(n)
  
  Unitary U(n)
  
  Special unitary SU(n)
  
  Symplectic Sp(n)
  
  $G_2$
  
  $F_4$
  
  $E_6$
  
  $E_7$
  
  $E_8$
  
  Lorentz
  
  Poincaré
  
  Conformal
  
  Diffeomorphism
  
  Loop
  
  Infinite dimensional Lie group O(∞)SU(∞)Sp(∞) :
  
  Algebraic groups
  
  Linear algebraic group
  
  Reductive group
  
  Abelian variety
  
  Elliptic curve :
  
  vte :
  
  In the area of modern algebra known as group theory, the Fischer group $Fi_{24}$ or $F_{24}′$ is a sporadic simple group of order
  
      $2^{21}$ · $3^{16}$ · $5^2$ · $7^3$ · 11 · 13 · 17 · 23 · 29
      = 1255205709190661721292800
      ≈ $1×10^{24}.$
  
  ## History and properties
  
  $Fi_{24}$ 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 $Fi_{24},$ 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 $Fi_{24},$ as a result of which the prime 3 plays a special role in its theory.
  
  ## Representations
  
  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 $(=2^3.3^3.7^2.29)$ vertices corresponding to the 3-transpositions of $Fi_{24},$ 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.
  
  ## 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 $Fi_{24}$ (as well as $Fi_{23}),$ the relevant McKay-Thompson series is $T_{3A}(\tau )$ where one can set the constant term a(0) = 42 (OEIS: A030197),
  
      ${\begin{aligned}j_{3A}(\tau )&=T_{3A}(\tau )+42\\&=\left(\left({\tfrac {\eta (\tau )}{\eta (3\tau )}}\right)^{6}+3^{3}\left({\tfrac {\eta (2\tau )}{\eta (\tau )}}\right)^{6}\right)^{2}\\&={\frac {1}{q}}+42+783q+8672q^{2}+65367q^{3}+371520q^{4}+1741655q^{5}+\dots \end{aligned}}$
  
  ## Maximal subgroups
  
  Linton & Wilson (1991) found the 22 conjugacy classes of maximal subgroups of $Fi_{24}$ as follows:
  
  * $Fi_{23}$ Centralizes a 3-transposition in the automorphism group $Fi_{24}.$
  * $2.Fi_{22}:2$
  * (3 x $O^+$ $_8(3):3):2$
  * $O^–$ $_{10}(2)$
  * $3^7.O_7(3)$
  * $3^{1+10}:U_5(2):2$
  * $2^{11}.M_{24}$
  * $2^2.U_6(2):S_3$
  * $2^{1+12}:3.U_4(3).2$
  * $3^{2+4+8}.(A_5$ x $2A_4).2$
  * $(A_4$ x $O^+$ $_8(2):3):2$
  * He:2 (Two classes, fused by an outer automorphism)
  * $2^{3+12}.(L_3(2)$ x $A_6)$
  * $2^{6+8}.(S_3$ x $A_8)$
  * $(G_2(3)$ x $3^2:2).2$
  * $(A_9$ x $A_5):2$
  * $A_7$ x 7:6
  * $[3^{13}]:(L_3(3)$ x 2)
  * $L_2(8):3$ x $A_6$
  * $U_3(3):2$ (Two classes, fused by an outer automorphism)
  * $L_2(13):2$ (Two classes, fused by an outer automorphism)
  * 29:14
  