Fischer group Fi24

For general background and history of the Fischer sporadic groups, see Fischer group.

In the area of modern algebra known as group theory, the Fischer group Fi₂₄ or F₂₄′ or F₃₊ 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

Fi₂₄ 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 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 sporadic simple 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 (=2^3.3^3.7^2.29) 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 ${\displaystyle T_{3A}(\tau )}$ where one can set the constant term a(0) = 42 (OEIS: A030197),

${\displaystyle {\begin{aligned}j_{3A}(\tau )&=T_{3A}(\tau )+42\\&=\left(\left({\tfrac {\eta (\tau )}{\eta (3\tau )}}\right)^{6}+3^{3}\left({\tfrac {\eta (2\tau )}{\eta (\tau )}}\right)^{6}\right)^{2}\\&={\frac {1}{q}}+42+783q+8672q^{2}+65367q^{3}+371520q^{4}+1741655q^{5}+\dots \end{aligned}}}$

Maximal subgroups

Linton & Wilson (1991) found the 25 conjugacy classes of maximal subgroups of Fi₂₄' as follows:

Maximal subgroups of Fi₂₄'

No.	Structure	Order	Index	Comments
1	Fi23	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 Fi24
2	2^·Fi22:2	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 ·O7(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:U5(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·M24	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 ·U6(2):S3	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 Fi24 (tri-transposition)
9	2^1+12 ·3^·U4(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]:(L3(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.(A5x2A4).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	(A4xO^+ 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.(L3(2)xA6)	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.(S3xA8)	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	(G2(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	(A9xA5):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	L2(8):3xA6	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	A7x7:6	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	U3(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	L2(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

References

• Aschbacher, Michael (1997), 3-transposition groups, Cambridge Tracts in Mathematics, vol. 124, Cambridge University Press, doi:10.1017/CBO9780511759413, ISBN 978-0-521-57196-8, MR 1423599, archived from the original on 2016-03-04, retrieved 2012-06-21 contains a complete proof of Fischer's theorem.
• Fischer, Bernd (1971), "Finite groups generated by 3-transpositions. I", Inventiones Mathematicae, 13 (3): 232–246, doi:10.1007/BF01404633, ISSN 0020-9910, MR 0294487 This is the first part of Fischer's preprint on the construction of his groups. The remainder of the paper is unpublished (as of 2010).
• Fischer, Bernd (1976), Finite Groups Generated by 3-transpositions, Preprint, Mathematics Institute, University of Warwick
• Linton, Stephen A.; Wilson, Robert A. (1991), "The maximal subgroups of the Fischer groups Fi₂₄ and Fi₂₄'", Proceedings of the London Mathematical Society, Third Series, 63 (1): 113–164, doi:10.1112/plms/s3-63.1.113, ISSN 0024-6115, MR 1105720
• Wilson, Robert A. (2009), The finite simple groups, Graduate Texts in Mathematics 251, vol. 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 1203.20012
• Wilson, R. A. ATLAS of Finite Group Representation.

External links

• MathWorld: Fischer Groups
• Atlas of Finite Group Representations: Fi24