# Fischer group Fi23

Algebraic structure → Group theory Group theory :
Basic notions
Subgroup
Normal subgroup
Group action
Quotient group
(Semi-)direct product
Direct sum
Free product
Wreath product
Group homomorphisms
kernel
image
simple
finite
infinite
continuous
multiplicative
additive
cyclic
abelian
dihedral
nilpotent
solvable
Glossary of group theory
List of group theory topics :
Finite groups
Cyclic group Z$_n$
Symmetric group S$_n$
Alternating group A$_n$
Dihedral group D$_n$
Quaternion group Q
Cauchy's theorem
Lagrange's theorem
Sylow theorems
Hall's theorem
p-group
Elementary abelian group
Frobenius group
Schur multiplier
Classification of finite simple groups
cyclic
alternating
Lie type
sporadic :
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$_{23}$ is a sporadic simple group of order
    4,089,470,473,293,004,800
    = 2$^{18}$ · 3$^{13}$ · 5$^2$ · 7 · 11 · 13 · 17 · 23
    ≈ 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
Fi$_{23}$ is the centralizer of a transposition in the Fischer group Fi24. When realizing Fi$_{24}$ as a subgroup of the Monster group, the full centralizer of a transposition is the double cover of the Baby monster group. As a result, Fi$_{23}$ is a subgroup of the Baby monster, and is the normalizer of a certain S$_3$ group in the Monster.
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:
Maximal subgroups of Fi$_{23}$ :
| No. | Structure | Order | Index | Comments |
| 1 | 2.Fi$_{22}$ | 129,123,503,308,800 = 2$^{18}$·3$^9$·5$^2$·7·11·13 | 31,671 = 3$^4$·17·23 | centralizer of an involution of class 2A |
| 2 | O$^+$ $_8$(3):S$_3$ | 29,713,078,886,400 = 2$^{13}$·3$^{13}$·5$^2$·7·13 | 137,632 = 2$^5$·11·17·23 | |
| 3 | 2$^2$.U$_6$(2).2 | 73,574,645,760 = 2$^{18}$·3$^6$·5·7·11 | 55,582,605 = 3$^7$·5·13·17·23 | centralizer of an involution of class 2B |
| 4 | S$_8$(2) | 47,377,612,800 = 2$^{16}$·3$^5$·5$^2$·7·17 | 86,316,516 = 2$^2$·3$^8$·11·13·23 | |
| 5 | O$_7$(3) × S$_3$ | 27,512,110,080 = 2$^{10}$·3$^{10}$·5·7·13 | 148,642,560 = 2$^8$·3$^3$·5·11·17·23 | normalizer of a subgroup of order 3 (class 3A) |
| 6 | 2$^{11}$.M$_{23}$ | 20,891,566,080 = 2$^{18}$·3$^2$·5·7·11·23 | 195,747,435 = 3$^{11}$·5·13·17 | |
| 7 | 3$^{1+8}$.2$^{1+6}$.3$^{1+2}$.2S$_4$ | 3,265,173,504 = 2$^{11}$·3$^{13}$ | 1,252,451,200 = 2$^7$·5$^2$·7·11·13·17·23 | normalizer of a subgroup of order 3 (class 3B) |
| 8 | [3$^{10}$].(L$_3$(3) × 2) | 663,238,368 = 2$^5$·3$^{13}$·13 | 6,165,913,600 = 2$^{13}$·5$^2$·7·11·17·23 | |
| 9 | S$_{12}$ | 479,001,600 = 2$^{10}$·3$^5$·5$^2$·7·11 | 8,537,488,128 = 2$^8$·3$^8$·13·17·23 | |
| 10 | (2$^2$ × 2$^{1+8}$).(3 × U$_4$(2)).2 | 318,504,960 = 2$^{18}$·3$^5$·5 | 12,839,581,755 = 3$^8$·5·7·11·13·17·23 | centralizer of an involution of class 2C |
| 11 | 2$^{6+8}$:(A$_7$ × S$_3$) | 247,726,080 = 2$^{18}$·3$^3$·5·7 | 16,508,033,685 = 3$^{10}$·5·11·13·17·23 | |
| 12 | S$_6$(2) × S$_4$ | 34,836,480 = 2$^{12}$·3$^5$·5·7 | 117,390,461,760 = 2$^6$·3$^8$·5·11·13·17·23 | |
| 13 | S$_4$(4):4 | 3,916,800 = 2$^{10}$·3$^2$·5$^2$·17 | 1,044,084,577,536 = 2$^8$·3$^{11}$·7·11·13·23 | |
| 14 | L$_2$(23) | 6,072 = 2$^3$·3·11·23 | 673,496,454,758,400 = 2$^{15}$·3$^{12}$·5$^2$·7·13·17 | |