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Sporadic Group

One of the 26 finite Simple Groups. The most complicated is the Monster Group. A summary, as given by Conway et al. (1985), is given below.

Symbol Name Order $M$ $A$
$M_{11}$ Mathieu $2^4\cdot 3^2\cdot 5\cdot 11$ 1 1
$M_{12}$ Mathieu $2^6\cdot 3^3\cdot 5\cdot 11$ 2 2
$M_{22}$ Mathieu $2^7\cdot 3^2\cdot 5\cdot 7\cdot 11$ 12 2
$M_{23}$ Mathieu $2^7\cdot 3^2\cdot 5\cdot 7\cdot 11\cdot 23$ 1 1
$M_{24}$ Mathieu $2^{10}\cdot 3^3\cdot 5\cdot 7\cdot 11\cdot 23$ 1 1
$J_2={\it HJ}$ Janko $2^7\cdot 3^3\cdot 5^2\cdot 7$ 2 2
Suz Suzuki $2^{13}\cdot 3^7\cdot 5^2\cdot 7 \cdot 11\cdot 13$ 6 2
HS Higman-Sims $2^9\cdot 3^2\cdot 5^3\cdot 7\cdot 11$ 2 2
McL McLaughlin $2^7\cdot 3^6\cdot 5^3\cdot 7\cdot 11$ 3 2
${\it Co}_{3}$ Conway $2^{10}\cdot 3^7\cdot 5^3\cdot 7\cdot 11\cdot 23$ 1 1
${\it Co}_{2}$ Conway $2^{18}\cdot 3^6\cdot 5^3\cdot 7\cdot 11\cdot 23$ 1 1
${\it Co}_{1}$ Conway $2^{21}\cdot 3^9\cdot 5^4\cdot 7^2\cdot 11\cdot 13\cdot 23$ 2 1
He Held $2^{10}\cdot 3^3\cdot 5^2\cdot 7^3\cdot 17$ 1 2
${\it Fi}_{22}$ Fischer $2^{17}\cdot 3^9\cdot 5^2\cdot 7\cdot 11\cdot 13$ 6 2
${\it Fi}_{23}$ Fischer $2^{18}\cdot 3^{13}\cdot 5^2\cdot 7\cdot 11\cdot 13\cdot 17\cdot 23$ 1 1
${\it Fi}'_{24}$ Fischer $2^{21}\cdot 3^{16}\cdot 5^2\cdot 7^3\cdot 11\cdot 13\cdot 17\cdot 23\cdot 29$ 3 2
HN Harada-Norton $2^{14}\cdot 3^6\cdot 5^6\cdot 7\cdot 11\cdot 19$ 1 2
Th Thompson $2^{15}\cdot 3^{10}\cdot 5^3\cdot 7^2\cdot 13\cdot 19\cdot 31$ 1 1
$B$ Baby Monster $2^{41}\cdot 3^{13}\cdot 5^6\cdot 7^2\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23\cdot 31\cdot 47$ 2 1
$M$ Monster $2^{46}\cdot 3^{20}\cdot 5^9\cdot 7^6\cdot 11^2\cdot 13^3\cdot 17\cdot 19\cdot 23\cdot 29\cdot 31\cdot 41\cdot 47\cdot 59\cdot 71$ 1 1
$J_1$ Janko $2^3\cdot 3\cdot 5\cdot 7\cdot 11\cdot 19$ 1 1
O'N O'Nan $2^9\cdot 3^4\cdot 5\cdot 7^3\cdot 11\cdot 19\cdot 31$ 3 2
$J_3$ Janko $2^7\cdot 3^5\cdot 5\cdot 17\cdot 19$ 3 2
Ly Lyons $2^8\cdot 3^7\cdot 5^6\cdot 7\cdot 11\cdot 31\cdot 37\cdot 67$ 1 1
Ru Rudvalis $2^{14}\cdot 3^3\cdot 5^3\cdot 7\cdot 13\cdot 29$ 2 1
$J_4$ Janko $2^{21}\cdot 3^3\cdot 5\cdot 7\cdot 11^3\cdot 23\cdot 29\cdot 31\cdot 37\cdot 43$ 1 1

See also Baby Monster Group, Conway Groups, 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, Suzuki Group, Thompson Group


References

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.

Math. Intell. Cover of volume 2, 1980.

Wilson, R. A. ``ATLAS of Finite Group Representation.'' http://for.mat.bham.ac.uk/atlas/html/contents.html#spo.



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© 1996-9 Eric W. Weisstein
1999-05-26