# Fischer group Fi24

In mathematics, the Fischer group Fi24 or M(24)′ or F24 or F3+ of order

221 · 316 · 52 · 73 · 11 · 13 · 17 · 23 · 29 (= 1255205709190661721292800) is the largest 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 3. The automorphism group is a 3-transposition group Fi24, 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 automorphism group Fi24, 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 (=23.33.72.29) vertices corresponding to the 3-transpositions of Fi24, 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 Fi24 (as well as Fi23), the relevant McKay-Thompson series is $T_{3A}(\tau)$ where one can set the constant term a(0) = 42 (),

\begin{align}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+1741655q^5+\dots \end{align}

Let $r = 782$, and $s_n$ = 1, 8671, 57477, 249458, 555611, 1603525, 1603525, 1666833,...() be the degrees of irreducible representations of Fi24. Then,

\begin{align} 1 &= s_1\\ 783 &= r + s_1\\ 8672 &= s_1 + s_2\\ 65367 &= -r + s_1 + s_2 + s_3\\ 371520 &= -2r + s_1 + s_2 + 2s_3 + s_4\\ 1741655 &= 3s_1 + 2s_2 + s_3 + s_8 \end{align}

and so on. Like the j-function, the function j3A(τ) can also assume an integer value for appropriate arguments. For example,

$j_{3A}\Big(\tfrac{3+\sqrt{-267}}{6}\Big) = -300^3$

implying,

$e^{\pi/3\sqrt{267}} = 300^3 + 41.99997\dots$

## Maximal subgroups

Linton & Wilson (1991) found the classes of maximal subgroups of the simple group Fi24' as follows:

Fi23 Centralizes a 3-transposition in the automorphism group Fi24.

2.Fi22:2

(3 x O+
8
(3):3):2

O
10
(2)

37.O7(3)

31+10:U5(2):2

211.M24

22.U6(2):S3

21+12:3.U4(3).2

32+4+8.(A5 x 2A4).2

(A4 x O+
8
(2):3):2

He:2 (Two classes, fused by an outer automorphism)

23+12.(L3(2) x A6)

26+8.(S3 x A8)

(G2(3) x 32:2).2

(A9 x A5):2

A7 x 7:6

[313]:(L3(3) x 2)

L2(8):3 x A6

U3(3):2 (Two classes, fused by an outer automorphism)

L2(13):2 (Two classes, fused by an outer automorphism)

29:14