# Fischer group Fi24

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vte :
In the area of modern algebra known as group theory, the Fischer group Fi$_{24}$ or F$_{24}$′ or F$_{3+}$ is a sporadic simple group of order
    1,255,205,709,190,661,721,292,800
    = 2$^{21}$ · 3$^{16}$ · 5$^2$ · 7$^3$ · 11 · 13 · 17 · 23 · 29
    ≈ 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 Fi$_{23}$.
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 25 conjugacy classes of maximal subgroups of Fi$_{24}$' as follows:
Maximal subgroups of Fi$_{24}$' :
| No. | Structure | Order | Index | Comments |
| 1 | Fi$_{23}$ | 4,089,470,473,293,004,800 = 2$^{18}$·3$^{13}$·5$^2$·7·11·13·17·23 | 306,936 = 2$^3$·3$^3$·7$^2$·29 | centralizer of a 3-transposition in the automorphism group Fi$_{24}$ |
| 2 | 2$^·$Fi$_{22}$:2 | 258,247,006,617,600 = 2$^{19}$·3$^9$·5$^2$·7·11·13 | 4,860,485,028 = 2$^2$·3$^7$·7$^2$·17·23·29 | centralizer of an involution (bi-transposition) |
| 3 | (3xO$^+$ $_8$(3):3):2 | 89,139,236,659,200 = 2$^{13}$·3$^{14}$·5$^2$·7·13 | 14,081,405,184 = 2$^8$·3$^2$·7$^2$·11·17·23·29 | normalizer of a subgroup of order 3 |
| 4 | O$^–$ $_{10}$(2) | 25,015,379,558,400 = 2$^{20}$·3$^6$·5$^2$·7·11·17 | 50,177,360,142 = 2·3$^{10}$·7$^2$·13·23·29 | |
| 5 | 3$^{7·}$O$_7$(3) | 10,028,164,124,160 = 2$^9$·3$^{16}$·5·7·13 | 125,168,046,080 = 2$^{12}$·5·7$^2$·11·17·23·29 | |
| 6 | 3$^{1+10}$:U$_5$(2):2 | 4,848,782,653,440 = 2$^{11}$·3$^{16}$·5·11 | 258,870,277,120 = 2$^{10}$·5·7$^3$·13·17·23·29 | normalizer of a subgroup of order 3 |
| 7 | 2$^{11·}$M$_{24}$ | 501,397,585,920 = 2$^{21}$·3$^3$·5·7·11·23 | 2,503,413,946,215 = 3$^{13}$·5·7$^2$·13·17·29 | |
| 8 | 2$^{2·}$U$_6$(2):S$_3$ | 220,723,937,280 = 2$^{18}$·3$^7$·5·7·11 | 5,686,767,482,760 = 2$^3$·3$^9$·5·7$^2$·13·17·23·29 | centralizer of an involution in the automorphism group Fi$_{24}$ (tri-transposition) |
| 9 | 2$^{1+12·}$3$^·$U$_4$(3).2 | 160,526,499,840 = 2$^{21}$·3$^7$·5·7 | 7,819,305,288,795 = 3$^9$·5·7$^2$·11·13·17·23·29 | centralizer of an involution (tetra-transposition) |
| 10 | [3$^{13}$]:(L$_3$(3)x2) | 17,907,435,936 = 2$^5$·3$^{16}$·13 | 70,094,105,804,800 = 2$^{16}$·5$^2$·7$^3$·11·17·23·29 | |
| 11 | 3$^{2+4+8}$.(A$_5$x2A$_4$).2 | 13,774,950,720 = 2$^6$·3$^{16}$·5 | 91,122,337,546,240 = 2$^{15}$·5·7$^3$·11·13·17·23·29 | |
| 12 | (A$_4$xO$^+$ $_8$(2):3):2 | 12,541,132,800 = 2$^{15}$·3$^7$·5$^2$·7 | 100,087,107,696,576 = 2$^6$·3$^9$·7$^2$·11·13·17·23·29 | |
| 13, 14 | He:2 | 8,060,774,400 = 2$^{11}$·3$^3$·5$^2$·7$^3$·17 | 155,717,756,992,512 = 2$^{10}$·3$^{13}$·11·13·23·29 | two classes, fused by an outer automorphism |
| 15 | 2$^{3+12}$.(L$_3$(2)xA$_6$) | 1,981,808,640 = 2$^{21}$·3$^3$·5·7 | 633,363,728,392,395 = 3$^{13}$·5·7$^2$·11·13·17·23·29 | |
| 16 | 2$^{6+8}$.(S$_3$xA$_8$) | 1,981,808,640 = 2$^{21}$·3$^3$·5·7 | 633,363,728,392,395 = 3$^{13}$·5·7$^2$·11·13·17·23·29 | |
| 17 | (G$_2$(3)x3$^2$:2).2 | 152,845,056 = 2$^8$·3$^8$·7·13 | 8,212,275,503,308,800 = 2$^{13}$·3$^8$·5$^2$·7$^2$·11·17·23·29 | |
| 18 | (A$_9$xA$_5$):2 | 21,772,800 = 2$^9$·3$^5$·5$^2$·7 | 57,650,174,033,227,776 = 2$^{12}$·3$^{11}$·7$^2$·11·13·17·23·29 | |
| 19 | L$_2$(8):3xA$_6$ | 544,320 = 2$^6$·3$^5$·5·7 | 2,306,006,961,329,111,040 = 2$^{15}$·3$^{11}$·5·7$^2$·11·13·17·23·29 | |
| 20 | A$_7$x7:6 | 105,840 = 2$^4$·3$^3$·5·7$^2$ | 11,859,464,372,549,713,920 = 2$^{17}$·3$^{13}$·5·7·11·13·17·23·29 | normalizer of a cyclic subgroup of order 7 |
| 21, 22 | U$_3$(3):2 | 12,096 = 2$^6$·3$^3$·7 | 103,770,313,259,809,996,800 = 2$^{15}$·3$^{13}$·5$^2$·7$^2$·11·13·17·23·29 | two classes, fused by an outer automorphism |
| 23, 24 | L$_2$(13):2 | 2,184 = 2$^3$·3·7·13 | 574,727,888,823,563,059,200 = 2$^{18}$·3$^{15}$·5$^2$·7$^2$·11·17·23·29 | two classes, fused by an outer automorphism |
| 25 | 29:14 | 406 = 2·7·29 | 3,091,639,677,809,511,628,800 = 2$^{20}$·3$^{16}$·5$^2$·7$^2$·11·13·17·23 | normalizer of a Sylow 29-subgroup |