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.
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 (=220.127.116.11) 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.
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.
(3 x O+
32+4+8.(A5 x 2A4).2
(A4 x O+
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
:(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)
- Aschbacher, Michael (1997), 3-transposition groups, Cambridge Tracts in Mathematics 124, Cambridge University Press, doi:10.1017/CBO9780511759413, ISBN 978-0-521-57196-8, MR 1423599 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 The remainder of the paper is unpublished (as of 2010).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 251 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 05622792
- Wilson, R. A. ATLAS of Finite Group Representation.