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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)
7920👁 2^4·3^2·5·11
Mathieu group 👁 M_(12)
95040👁 2^6·3^3·5·11
Janko group 👁 J_1
175560👁 2^3·3·5·7·11·19
Mathieu group 👁 M_(22)
443520👁 2^7·3^2·5·7·11
Janko group 👁 J_2=HJ
604800👁 2^7·3^3·5^2·7
Mathieu group 👁 M_(23)
10200960👁 2^7·3^2·5·7·11·23
Higman-Sims group HS44352000👁 2^9·3^2·5^3·7·11
Janko group 👁 J_3
50232960👁 2^7·3^5·5·17·19
Mathieu group 👁 M_(24)
244823040👁 2^(10)·3^3·5·7·11·23
McLaughlin group McL898128000👁 2^7·3^6·5^3·7·11
Held group He4030387200👁 2^(10)·3^3·5^2·7^3·17
Rudvalis Group Ru145926144000👁 2^(14)·3^3·5^3·7·13·29
Suzuki group Suz448345497600👁 2^(13)·3^7·5^2·7·11·13
O'Nan group O'N460815505920👁 2^9·3^4·5·7^3·11·19·31
Conway group 👁 Co_3
495766656000👁 2^(10)·3^7·5^3·7·11·23
Conway group 👁 Co_2
42305421312000👁 2^(18)·3^6·5^3·7·11·23
Fischer group 👁 Fi_(22)
64561751654400👁 2^(17)·3^9·5^2·7·11·13
Harada-Norton group HN273030912000000👁 2^(14)·3^6·5^6·7·11·19
Lyons Group Ly51765179004000000👁 2^8·3^7·5^6·7·11·31·37·67
Thompson Group Th90745943887872000👁 2^(15)·3^(10)·5^3·7^2·13·19·31
Fischer group 👁 Fi_(23)
4089470473293004800👁 2^(18)·3^(13)·5^2·7·11·13·17·23
Conway group 👁 Co_1
4157776806543360000👁 2^(21)·3^9·5^4·7^2·11·13·23
Janko group 👁 J_4
86775571046077562880👁 2^(21)·3^3·5·7·11^3·23·29·31·37·43
Fischer group 👁 Fi_(24)^'
1255205709190661721292800👁 2^(21)·3^(16)·5^2·7^3·11·13·17·23·29
baby monster group 👁 B
4154781481226426191177580544000000👁 2^(41)·3^(13)·5^6·7^2·11·13·17·19·23·31·47
monster group 👁 M
808017424794512875886459904961710757005754368000000000👁 2^(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 Resource. https://mathworld.wolfram.com/SporadicGroup.html

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