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Sporadic Group

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The sporadic groups are the 26 finite simple groups of group order greater than 1 that are outside three broad classes: the cyclic groups of prime group order, the alternating groups of degree at least five, and the Lie-type groups (including the Tits group). The smallest sporadic group is the Mathieu group M_(11), which has group order 7920, and the largest is the monster group, which has group order 808017424794512875886459904961710757005754368000000000.

Constructions published between 1984 and 1989 purported to realize 25 of the 26 sporadic groups as Galois groups over Q. There is a caveat to this usual summary. The original construction for the baby monster group was later found to contain a calculation error, although a correct realization was subsequently obtained. The Mathieu group M23, the sixth-smallest sporadic group with group order 10200960, remained the sole exception until Huang et al. (2026) constructed a regular Galois extension of Q(t) with Galois group M_(23). Consequently, every sporadic group occurs as a Galois group over Q.

The group orders of the sporadic groups given in increasing order are 7920, 95040, 175560, 443520, 604800, 10200960, 44352000, 50232960, ... (OEIS A001228). A summary of sporadic groups, as given by Conway et al. (1985), is given below.

namegroup orderprime factorization
Mathieu group M_(11)79202^4·3^2·5·11
Mathieu group M_(12)950402^6·3^3·5·11
Janko group J11755602^3·3·5·7·11·19
Mathieu group M_(22)4435202^7·3^2·5·7·11
Janko group J2=HJ6048002^7·3^3·5^2·7
Mathieu group M_(23)102009602^7·3^2·5·7·11·23
Higman-Sims group HS443520002^9·3^2·5^3·7·11
Janko group J3502329602^7·3^5·5·17·19
Mathieu group M_(24)2448230402^(10)·3^3·5·7·11·23
McLaughlin group McL8981280002^7·3^6·5^3·7·11
Held group He40303872002^(10)·3^3·5^2·7^3·17
Rudvalis Group Ru1459261440002^(14)·3^3·5^3·7·13·29
Suzuki group Suz4483454976002^(13)·3^7·5^2·7·11·13
O'Nan group O'N4608155059202^9·3^4·5·7^3·11·19·31
Conway group Co_34957666560002^(10)·3^7·5^3·7·11·23
Conway group Co_2423054213120002^(18)·3^6·5^3·7·11·23
Fischer group Fi_(22)645617516544002^(17)·3^9·5^2·7·11·13
Harada-Norton group HN2730309120000002^(14)·3^6·5^6·7·11·19
Lyons Group Ly517651790040000002^8·3^7·5^6·7·11·31·37·67
Thompson Group Th907459438878720002^(15)·3^(10)·5^3·7^2·13·19·31
Fischer group Fi_(23)40894704732930048002^(18)·3^(13)·5^2·7·11·13·17·23
Conway group Co_141577768065433600002^(21)·3^9·5^4·7^2·11·13·23
Janko group J4867755710460775628802^(21)·3^3·5·7·11^3·23·29·31·37·43
Fischer group Fi_(24)^'12552057091906617212928002^(21)·3^(16)·5^2·7^3·11·13·17·23·29
baby monster group B41547814812264261911775805440000002^(41)·3^(13)·5^6·7^2·11·13·17·19·23·31·47
monster group M8080174247945128758864599049617107570057543680000000002^(46)·3^(20)·5^9·7^6·11^2·13^3·17·19·23·29·31·41·47·59·71

Pegg (2016) gives an interactive visualization of words in two group generators represented by matrices over a finite field for 23 of the sporadic groups, including the Mathieu group M23. It omits the three largest sporadic groups.

See also

Baby Monster Group, Classification Theorem of Finite Groups, Conway Groups, Finite Group, Fischer Groups, Galois Group, Harada-Norton Group, Held Group, Higman-Sims Group, Inverse Galois Problem, Janko Groups, Lyons Group, Mathieu Group M11, Mathieu Group M12, Mathieu Group M22, Mathieu Group M23, Mathieu Group M24, Mathieu Groups, McLaughlin Group, Monster Group, O'Nan Group, Rudvalis Group, Simple Group, Suzuki Group, Thompson Group

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References

--. Cover of Math. Intell. 2, 1980.Aschbacher, M. Sporadic Groups. New York: Cambridge University Press, 1994.Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups. Oxford, England: Clarendon Press, p. viii, 1985.Huang, X.; Jackson, B.; Lee, K.-H.; Poonen, B.; Pries, R.; and Zhang, S. "The Mathieu Group M_(23) Is a Galois Group over Q." 8 Aug 2026. https://arxiv.org/abs/2608.08538.Ivanov, A. A. Geometry of Sporadic Groups I: Petersen and Tilde Geometries. Cambridge, England: Cambridge University Press, 1999.Sloane, N. J. A. Sequence A001228 in "The On-Line Encyclopedia of Integer Sequences." Pegg, E. Jr. "Sporadic Groups." Wolfram Demonstrations Project. 2016. https://demonstrations.wolfram.com/SporadicGroups/.Wilson, R. A.; Parker, R. A.; and Bray, J. N. "ATLAS: Sporadic Groups." https://brauer.maths.qmul.ac.uk/Atlas/spor/.

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Sporadic Group

Cite this as:

Weisstein, Eric W. "Sporadic Group." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SporadicGroup.html

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