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Sporadic Group

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The sporadic groups are the 26 finite simple groups that do not fit into any of the four infinite families of finite simple groups (i.e., the cyclic groups of prime order, alternating groups of degree at least five, Lie-type Chevalley groups, and Lie-type groups). The smallest sporadic group is the Mathieu group M_(11), which has order 7920, and the largest is the monster group, which has order 808017424794512875886459904961710757005754368000000000.

The 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.

nameorderfactorization
Mathieu group M_(11)79202^4·3^2·5·11
Mathieu group M_(12)950402^6·3^3·5·11
Janko group J_11755602^3·3·5·7·11·19
Mathieu group M_(22)4435202^7·3^2·5·7·11
Janko group J_2=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 J_3502329602^7·3^5·5·17·19
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 J_4867755710460775628802^(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

See also

Baby Monster Group, Classification Theorem of Finite Groups, Conway Groups, Finite Group, Fischer Groups, Harada-Norton Group, Held Group, Higman-Sims Group, Janko Groups, Lyons Group, 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.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."Wilson, R. A. "ATLAS of Finite Group Representation." http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/.

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Sporadic Group

Cite this as:

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

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