Sporadic simple group

This article is about a term related to the Classification of finite simple groups

This article defines a group property that can be evaluated, or makes sense, for simple groups

Definition

Symbol-free definition

A sporadic simple group is a finite simple group that is not an alternating group, classical group, or exceptional group of Lie type.

There are 26 sporadic simple groups.

The list

Group name	Symbol	Order	Prime factorization of order	Number of conjugacy classes
Mathieu group:M11	$M_{11}$	7920	$2^4\cdot 3^2\cdot 5\cdot 11$	10
Mathieu group:M12	$M_{12}$	95040	$2^6\cdot 3^3\cdot 5\cdot 11$	15
Mathieu group:M22	$M_{22}$	443520	$2^7\cdot 3^2\cdot 5\cdot 7\cdot 11$	12
Mathieu group:M23	$M_{23}$	10200960	$2^7\cdot 3^2\cdot 5\cdot 7\cdot 11\cdot 23$	17
Mathieu group:M24	$M_{24}$	244823040	$2^{10}\cdot 3^3\cdot 5\cdot 7\cdot 11\cdot 23$	26
Janko group:J1	$J_1$	175560	$2^3\cdot 3\cdot 5\cdot 7\cdot 11\cdot 19$	15
Janko group:J2 (also called the Hall-Janko group)	$J_2$ or $HJ$	604800	$2^7\cdot 3^3\cdot 5^2\cdot 7$	21
Janko group:J3	$J_3$	50232960	$2^7\cdot 3^5\cdot 5\cdot 17\cdot 19$	21
Janko group:J4	$J_4$	86775571046077562880	$2^{21}\cdot 3^3\cdot 5\cdot 7\cdot 11^3\cdot 23\cdot 29\cdot 31\cdot 37\cdot 43$	62
Conway group:Co1	${\mathrm{C}}_{\mathrm{o}}$	4157776806543360000	$2^{21}\cdot 3^9\cdot 5^4\cdot 7^2\cdot 11\cdot 13\cdot 23$	101
Conway group:Co2	${\mathrm{C}}_{\mathrm{o}}$	42305421312000	$2^{18}\cdot 3^6\cdot 5^3\cdot 7\cdot 11\cdot 23$	60
Conway group:Co3	${\mathrm{C}}_{\mathrm{o}}$	495766656000	$2^{10}\cdot 3^7\cdot 5^3\cdot 7\cdot 11\cdot 23$	42
Fischer group:Fi22	${\mathrm{F}}_{\mathrm{i}}$	64561751654400	$2^{17}\cdot 3^9\cdot 5^2\cdot 7\cdot 11\cdot 13$	65
Fischer group:Fi23	${\mathrm{F}}_{\mathrm{i}}$	4089470473293004800	$2^{18}\cdot 3^{13}\cdot 5^2\cdot 7\cdot 11\cdot 13\cdot 17\cdot 23$	98
derived subgroup of Fischer group:Fi24	$F_{24}$	1255205709190661721292800	$2^{21}\cdot 3^{16}\cdot 5^2\cdot 7^3\cdot 11\cdot 13\cdot 17\cdot 23\cdot 29$	183
Higman-Sims group	$HS$	44352000	$2^9\cdot 3^2\cdot 5^3\cdot 7\cdot 11$	24
McLaughlin group	$McL$	898128000	$2^7\cdot 3^6\cdot 5\cdot 7\cdot 11$	24
Held group	$He$	4030387200	$2^{10}\cdot 3^3\cdot 5^2\cdot 7^3\cdot 17$	33
Rudvalis group	$Ru$	145926144000	$2^{14}\cdot 3^3\cdot 5^3\cdot 7\cdot 13\cdot 29$	36
Suzuki sporadic group	$Suz$	448345497600	$2^{13}\cdot 3^7\cdot 5^2\cdot 7\cdot 11\cdot 13$	43
O'Nan group	$ON$	460815505920	$2^9\cdot 3^4\cdot 5\cdot 7^3\cdot 11\cdot 19\cdot 31$	30
Harada-Norton group	$HN$	273030912000000	$2^{14}\cdot 3^6\cdot 5^6\cdot 7\cdot 11\cdot 19$	54
Lyons group	$Ly$	51765179004000000	$2^8\cdot 3^7\cdot 5^6\cdot 7\cdot 11\cdot 31\cdot 37\cdot 67$	53
Thompson group	$Th$	90745943887872000	$2^{15}\cdot 3^{10}\cdot 5^3\cdot 7^2\cdot 13\cdot 19\cdot 31$	48
baby monster group	$B$	4154781481226426191177580544000000	[SHOW MORE] $2^{41}\cdot 3^{13}\cdot 5^6\cdot 7^2\cdot 11\cdot 13\cdot 17\cdot 19\cdot 23\cdot 31\cdot 47$	184
monster group	$M$	808017424794512875886459904961710757005754368000000000	[SHOW MORE] $2^{46}\cdot 3^{20}\cdot 5^9\cdot 7^6\cdot 11^2\cdot 13\cdot 17\cdot 19\cdot 23\cdot 29\cdot 31\cdot 41\cdot 47\cdot 59\cdot 71$	194

See also

Category: Sporadic simple groups