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.