List of finite simple groups

In mathematics, the classification of finite simple groups states that every finite simple group is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of 26 sporadic groups.

The list below gives all finite simple groups, together with their order, the size of the Schur multiplier, the size of the outer automorphism group, usually some small representations, and lists of all duplicates.

Summary

The following table is a complete list of the 18 families of finite simple groups and the 26 sporadic simple groups, along with their orders. Any non-simple members of each family are listed, as well as any members duplicated within a family or between families. (In removing duplicates it is useful to note that no two finite simple groups have the same order, except that the group A₈ = A₃(2) and A₂(4) both have order 20160, and that the group B_n(q) has the same order as C_n(q) for q odd, n > 2. The smallest of the latter pairs of groups are B₃(3) and C₃(3) which both have order 4585351680.)

There is an unfortunate conflict between the notations for the alternating groups A_n and the groups of Lie type A_n(q). Some authors use various different fonts for A_n to distinguish them. In particular, in this article we make the distinction by setting the alternating groups A_n in Roman font and the Lie-type groups A_n(q) in italic.

In what follows, n is a positive integer, and q is a positive power of a prime number p, with the restrictions noted. The notation (a,b) represents the greatest common divisor of the integers a and b.

Class	Family	Order	Exclusions	Duplicates
Cyclic groups	Zp	p prime	None	None
Alternating groups	An n > 4	${\displaystyle {\frac {n!}{2}}}$	None	A5 ≃ A1(4) ≃ A1(5) A6 ≃ A1(9) A8 ≃ A3(2)
Classical Chevalley groups	An(q)	${\displaystyle {\frac {q^{{\frac {1}{2}}n(n+1)}}{(n+1,q-1)}}\prod _{i=1}^{n}\left(q^{i+1}-1\right)}$	A1(2), A1(3)	A1(4) ≃ A1(5) ≃ A5 A1(7) ≃ A2(2) A1(9) ≃ A6 A3(2) ≃ A8
	Bn(q) n > 1	${\displaystyle {\frac {q^{n^{2}}}{(2,q-1)}}\prod _{i=1}^{n}\left(q^{2i}-1\right)}$	B2(2)	Bn(2m) ≃ Cn(2m) B2(3) ≃ 2A3(22)
	Cn(q) n > 2	${\displaystyle {\frac {q^{n^{2}}}{(2,q-1)}}\prod _{i=1}^{n}\left(q^{2i}-1\right)}$	None	Cn(2m) ≃ Bn(2m)
	Dn(q) n > 3	${\displaystyle {\frac {q^{n(n-1)}(q^{n}-1)}{(4,q^{n}-1)}}\prod _{i=1}^{n-1}\left(q^{2i}-1\right)}$	None	None
Exceptional Chevalley groups	E6(q)	${\displaystyle {\frac {q^{36}}{(3,q-1)}}\prod _{i\in \{2,5,6,8,9,12\}}\left(q^{i}-1\right)}$	None	None
	E7(q)	${\displaystyle {\frac {q^{63}}{(2,q-1)}}\prod _{i\in \{2,6,8,10,12,14,18\}}\left(q^{i}-1\right)}$	None	None
	E8(q)	${\displaystyle q^{120}\prod _{i\in \{2,8,12,14,18,20,24,30\}}\left(q^{i}-1\right)}$	None	None
	F4(q)	${\displaystyle q^{24}\prod _{i\in \{2,6,8,12\}}\left(q^{i}-1\right)}$	None	None
	G2(q)	${\displaystyle q^{6}\prod _{i\in \{2,6\}}\left(q^{i}-1\right)}$	G2(2)	None
Classical Steinberg groups	2An(q2) n > 1	${\displaystyle {\frac {q^{{\frac {1}{2}}n(n+1)}}{(n+1,q+1)}}\prod _{i=1}^{n}\left(q^{i+1}-(-1)^{i+1}\right)}$	2A2(22)	2A3(22) ≃ B2(3)
	2Dn(q2) n > 3	${\displaystyle {\frac {q^{n(n-1)}}{(4,q^{n}+1)}}\left(q^{n}+1\right)\prod _{i=1}^{n-1}\left(q^{2i}-1\right)}$	None	None
Exceptional Steinberg groups	2E6(q2)	${\displaystyle {\frac {q^{36}}{(3,q+1)}}\prod _{i\in \{2,5,6,8,9,12\}}\left(q^{i}-(-1)^{i}\right)}$	None	None
	3D4(q3)	${\displaystyle q^{12}\left(q^{8}+q^{4}+1\right)\left(q^{6}-1\right)\left(q^{2}-1\right)}$	None	None
Suzuki groups	2B2(q) q = 22n+1 n ≥ 1	${\displaystyle q^{2}\left(q^{2}+1\right)\left(q-1\right)}$	None	None
Ree groups + Tits group	2F4(q) q = 22n+1 n ≥ 1	${\displaystyle q^{12}\left(q^{6}+1\right)\left(q^{4}-1\right)\left(q^{3}+1\right)\left(q-1\right)}$	None	None
	2F4(2)′	212(26 + 1)(24 − 1)(23 + 1)(2 − 1)/2 = 17971200
	2G2(q) q = 32n+1 n ≥ 1	${\displaystyle q^{3}\left(q^{3}+1\right)\left(q-1\right)}$	None	None
Mathieu groups	M11	7920
	M12	95040
	M22	443520
	M23	10200960
	M24	244823040
Janko groups	J1	175560
	J2	604800
	J3	50232960
	J4	86775571046077562880
Conway groups	Co3	495766656000
	Co2	42305421312000
	Co1	4157776806543360000
Fischer groups	Fi22	64561751654400
	Fi23	4089470473293004800
	Fi24′	1255205709190661721292800
Higman–Sims group	HS	44352000
McLaughlin group	McL	898128000
Held group	He	4030387200
Rudvalis group	Ru	145926144000
Suzuki sporadic group	Suz	448345497600
O'Nan group	O'N	460815505920
Harada–Norton group	HN	273030912000000
Lyons group	Ly	51765179004000000
Thompson group	Th	90745943887872000
Baby Monster group	B	4154781481226426191177580544000000
Monster group	M	808017424794512875886459904961710757005754368000000000

Cyclic groups, Z_p

Simplicity: Simple for p a prime number.

Order:p

Schur multiplier: Trivial.

Outer automorphism group: Cyclic of order p − 1.

Other names: Z/pZ, C_p

Remarks: These are the only simple groups that are not perfect.

Alternating groups, A_n, n > 4

Simplicity: Solvable for n < 5, otherwise simple.

Order:n!/2 when n > 1.

Schur multiplier: 2 for n = 5 or n > 7, 6 for n = 6 or 7; see Covering groups of the alternating and symmetric groups

Outer automorphism group: In general 2. Exceptions: for n = 1, n = 2, it is trivial, and for n = 6, it has order 4 (elementary abelian).

Other names: Alt_n.

Isomorphisms: A₁ and A₂ are trivial. A₃ is cyclic of order 3. A₄ is isomorphic to A₁(3) (solvable). A₅ is isomorphic to A₁(4) and to A₁(5). A₆ is isomorphic to A₁(9) and to the derived group B₂(2)′. A₈ is isomorphic to A₃(2).

Remarks: An index 2 subgroup of the symmetric group of permutations of n points when n > 1.

Groups of Lie type

Notation:n is a positive integer, q > 1 is a power of a prime number p, and is the order of some underlying finite field. The order of the outer automorphism group is written as d⋅f⋅g, where d is the order of the group of "diagonal automorphisms", f is the order of the (cyclic) group of "field automorphisms" (generated by a Frobenius automorphism), and g is the order of the group of "graph automorphisms" (coming from automorphisms of the Dynkin diagram). The outer automorphism group is often, but not always, isomorphic to the semidirect product ${\displaystyle D\rtimes (F\times G)}$ where all these groups ${\displaystyle D,F,G}$ are cyclic of the respective orders d, f, g, except for type ${\displaystyle D_{n}(q)}$, ${\displaystyle q}$ odd, where the group of order ${\displaystyle d=4}$ is ${\displaystyle C_{2}\times C_{2}}$, and (only when ${\displaystyle n=4}$) ${\displaystyle G=S_{3}}$, the symmetric group on three elements. The notation (a,b) represents the greatest common divisor of the integers a and b.

Chevalley groups, A_n(q), B_n(q) n > 1, C_n(q) n > 2, D_n(q) n > 3

	Chevalley groups, An(q) linear groups	Chevalley groups, Bn(q) n > 1 orthogonal groups	Chevalley groups, Cn(q) n > 2 symplectic groups	Chevalley groups, Dn(q) n > 3 orthogonal groups
Simplicity	A1(2) and A1(3) are solvable, the others are simple.	B2(2) is not simple but its derived group B2(2)′ is a simple subgroup of index 2; the others are simple.	All simple	All simple
Order	${\displaystyle {\frac {q^{{\frac {1}{2}}n(n+1)}}{(n+1,q-1)}}\prod _{i=1}^{n}\left(q^{i+1}-1\right)}$	${\displaystyle {\frac {q^{n^{2}}}{(2,q-1)}}\prod _{i=1}^{n}\left(q^{2i}-1\right)}$	${\displaystyle {\frac {q^{n^{2}}}{(2,q-1)}}\prod _{i=1}^{n}\left(q^{2i}-1\right)}$	${\displaystyle {\frac {q^{n(n-1)}(q^{n}-1)}{(4,q^{n}-1)}}\prod _{i=1}^{n-1}\left(q^{2i}-1\right)}$
Schur multiplier	For the simple groups it is cyclic of order (n+1,q−1) except for A1(4) (order 2), A1(9) (order 6), A2(2) (order 2), A2(4) (order 48, product of cyclic groups of orders 3, 4, 4), A3(2) (order 2).	(2,q−1) except for B2(2) = S6 (order 2 for B2(2), order 6 for B2(2)′) and B3(2) (order 2) and B3(3) (order 6).	(2,q−1) except for C3(2) (order 2).	The order is (4,qn−1) (cyclic for n odd, elementary abelian for n even) except for D4(2) (order 4, elementary abelian).
Outer automorphism group	(2,q−1)⋅f⋅1 for n = 1; (n+1,q−1)⋅f⋅2 for n > 1, where q = pf	(2,q−1)⋅f⋅1 for q odd or n > 2; (2,q−1)⋅f⋅2 for q even and n = 2, where q = pf	(2,q−1)⋅f⋅1, where q = pf	(2,q−1)2⋅f⋅S3 for n = 4, (2,q−1)2⋅f⋅2 for n > 4 even, (4,qn−1)⋅f⋅2 for n odd, where q = pf, and S3 is the symmetric group of order 3! on 3 points.
Other names	Projective special linear groups, PSLn+1(q), Ln+1(q), PSL(n + 1,q)	O2n+1(q), Ω2n+1(q) (for q odd).	Projective symplectic group, PSp2n(q), PSpn(q) (not recommended), S2n(q), Abelian group (archaic).	O2n+(q), PΩ2n+(q). "Hypoabelian group" is an archaic name for this group in characteristic 2.
Isomorphisms	A1(2) is isomorphic to the symmetric group on 3 points of order 6. A1(3) is isomorphic to the alternating group A4 (solvable). A1(4) and A1(5) are both isomorphic to the alternating group A5. A1(7) and A2(2) are isomorphic. A1(8) is isomorphic to the derived group 2G2(3)′. A1(9) is isomorphic to A6 and to the derived group B2(2)′. A3(2) is isomorphic to A8.	Bn(2m) is isomorphic to Cn(2m). B2(2) is isomorphic to the symmetric group on 6 points, and the derived group B2(2)′ is isomorphic to A1(9) and to A6. B2(3) is isomorphic to 2A3(22).	Cn(2m) is isomorphic to Bn(2m)
Remarks	These groups are obtained from the general linear groups GLn+1(q) by taking the elements of determinant 1 (giving the special linear groups SLn+1(q)) and then quotienting out by the center.	This is the group obtained from the orthogonal group in dimension 2n + 1 by taking the kernel of the determinant and spinor norm maps. B1(q) also exists, but is the same as A1(q). B2(q) has a non-trivial graph automorphism when q is a power of 2.	This group is obtained from the symplectic group in 2n dimensions by quotienting out the center. C1(q) also exists, but is the same as A1(q). C2(q) also exists, but is the same as B2(q).	This is the group obtained from the split orthogonal group in dimension 2n by taking the kernel of the determinant (or Dickson invariant in characteristic 2) and spinor norm maps and then killing the center. The groups of type D4 have an unusually large diagram automorphism group of order 6, containing the triality automorphism. D2(q) also exists, but is the same as A1(q)×A1(q). D3(q) also exists, but is the same as A3(q).

Chevalley groups, E₆(q), E₇(q), E₈(q), F₄(q), G₂(q)

	Chevalley groups, E6(q)	Chevalley groups, E7(q)	Chevalley groups, E8(q)	Chevalley groups, F4(q)	Chevalley groups, G2(q)
Simplicity	All simple	All simple	All simple	All simple	G2(2) is not simple but its derived group G2(2)′ is a simple subgroup of index 2; the others are simple.
Order	q36(q12−1)(q9−1)(q8−1)(q6−1)(q5−1)(q2−1)/(3,q−1)	q63(q18−1)(q14−1)(q12−1)(q10−1)(q8−1)(q6−1)(q2−1)/(2,q−1)	q120(q30−1)(q24−1)(q20−1)(q18−1)(q14−1)(q12−1)(q8−1)(q2−1)	q24(q12−1)(q8−1)(q6−1)(q2−1)	q6(q6−1)(q2−1)
Schur multiplier	(3,q−1)	(2,q−1)	Trivial	Trivial except for F4(2) (order 2)	Trivial for the simple groups except for G2(3) (order 3) and G2(4) (order 2)
Outer automorphism group	(3,q−1)⋅f⋅2, where q = pf	(2,q−1)⋅f⋅1, where q = pf	1⋅f⋅1, where q = pf	1⋅f⋅1 for q odd, 1⋅f⋅2 for q even, where q = pf	1⋅f⋅1 for q not a power of 3, 1⋅f⋅2 for q a power of 3, where q = pf
Other names	Exceptional Chevalley group	Exceptional Chevalley group	Exceptional Chevalley group	Exceptional Chevalley group	Exceptional Chevalley group
Isomorphisms					The derived group G2(2)′ is isomorphic to 2A2(32).
Remarks	Has two representations of dimension 27, and acts on the Lie algebra of dimension 78.	Has a representations of dimension 56, and acts on the corresponding Lie algebra of dimension 133.	It acts on the corresponding Lie algebra of dimension 248. E8(3) contains the Thompson simple group.	These groups act on 27-dimensional exceptional Jordan algebras, which gives them 26-dimensional representations. They also act on the corresponding Lie algebras of dimension 52. F4(q) has a non-trivial graph automorphism when q is a power of 2.	These groups are the automorphism groups of 8-dimensional Cayley algebras over finite fields, which gives them 7-dimensional representations. They also act on the corresponding Lie algebras of dimension 14. G2(q) has a non-trivial graph automorphism when q is a power of 3. Moreover, they appear as automorphism groups of certain point-line geometries called split Cayley generalized hexagons.

Steinberg groups, ²A_n(q²) n > 1, ²D_n(q²) n > 3, ²E₆(q²), ³D₄(q³)

	Steinberg groups, 2An(q2) n > 1 unitary groups	Steinberg groups, 2Dn(q2) n > 3 orthogonal groups	Steinberg groups, 2E6(q2)	Steinberg groups, 3D4(q3)
Simplicity	2A2(22) is solvable, the others are simple.	All simple	All simple	All simple
Order	${\displaystyle {1 \over (n+1,q+1)}q^{{\frac {1}{2}}n(n+1)}\prod _{i=1}^{n}\left(q^{i+1}-(-1)^{i+1}\right)}$	${\displaystyle {1 \over (4,q^{n}+1)}q^{n(n-1)}(q^{n}+1)\prod _{i=1}^{n-1}\left(q^{2i}-1\right)}$	q36(q12−1)(q9+1)(q8−1)(q6−1)(q5+1)(q2−1)/(3,q+1)	q12(q8+q4+1)(q6−1)(q2−1)
Schur multiplier	Cyclic of order (n+1,q+1) for the simple groups, except for 2A3(22) (order 2), 2A3(32) (order 36, product of cyclic groups of orders 3,3,4), 2A5(22) (order 12, product of cyclic groups of orders 2,2,3)	Cyclic of order (4,qn+1)	(3,q+1) except for 2E6(22) (order 12, product of cyclic groups of orders 2,2,3).	Trivial
Outer automorphism group	(n+1,q+1)⋅f⋅1, where q2 = pf	(4,qn+1)⋅f⋅1, where q2 = pf	(3,q+1)⋅f⋅1, where q2 = pf	1⋅f⋅1, where q3 = pf
Other names	Twisted Chevalley group, projective special unitary group, PSUn+1(q), PSU(n + 1, q), Un+1(q), 2An(q), 2An(q, q2)	2Dn(q), O2n−(q), PΩ2n−(q), twisted Chevalley group. "Hypoabelian group" is an archaic name for this group in characteristic 2.	2E6(q), twisted Chevalley group	3D4(q), D42(q3), Twisted Chevalley groups
Isomorphisms	The solvable group 2A2(22) is isomorphic to an extension of the order 8 quaternion group by an elementary abelian group of order 9. 2A2(32) is isomorphic to the derived group G2(2)′. 2A3(22) is isomorphic to B2(3).
Remarks	This is obtained from the unitary group in n + 1 dimensions by taking the subgroup of elements of determinant 1 and then quotienting out by the center.	This is the group obtained from the non-split orthogonal group in dimension 2n by taking the kernel of the determinant (or Dickson invariant in characteristic 2) and spinor norm maps and then killing the center. 2D2(q2) also exists, but is the same as A1(q2). 2D3(q2) also exists, but is the same as 2A3(q2).	One of the exceptional double covers of 2E6(22) is a subgroup of the baby monster group, and the exceptional central extension by the elementary abelian group of order 4 is a subgroup of the monster group.	3D4(23) acts on the unique even 26-dimensional lattice of determinant 3 with no roots.

Suzuki groups, ²B₂(2²ⁿ⁺¹)

Simplicity: Simple for n ≥ 1. The group ²B₂(2) is solvable.

Order:q² (q² + 1) (q − 1), where q = 2²ⁿ⁺¹.

Schur multiplier: Trivial for n ≠ 1, elementary abelian of order 4 for ²B₂(8).

Outer automorphism group:

1⋅f⋅1,

where f = 2n + 1.

Other names: Suz(2²ⁿ⁺¹), Sz(2²ⁿ⁺¹).

Isomorphisms:²B₂(2) is the Frobenius group of order 20.

Remarks: Suzuki group are Zassenhaus groups acting on sets of size (2²ⁿ⁺¹)² + 1, and have 4-dimensional representations over the field with 2²ⁿ⁺¹ elements. They are the only non-cyclic simple groups whose order is not divisible by 3. They are not related to the sporadic Suzuki group.

Ree groups and Tits group, ²F₄(2²ⁿ⁺¹)

Simplicity: Simple for n ≥ 1. The derived group ²F₄(2)′ is simple of index 2 in ²F₄(2), and is called the Tits group, named for the Belgian mathematician Jacques Tits.

Order:q¹² (q⁶ + 1) (q⁴ − 1) (q³ + 1) (q − 1), where q = 2²ⁿ⁺¹.

The Tits group has order 17971200 = 2¹¹ ⋅ 3³ ⋅ 5² ⋅ 13.

Schur multiplier: Trivial for n ≥ 1 and for the Tits group.

Outer automorphism group:

1⋅f⋅1,

where f = 2n + 1. Order 2 for the Tits group.

Remarks: Unlike the other simple groups of Lie type, the Tits group does not have a BN pair, though its automorphism group does so most authors count it as a sort of honorary group of Lie type.

Ree groups, ²G₂(3²ⁿ⁺¹)

Simplicity: Simple for n ≥ 1. The group ²G₂(3) is not simple, but its derived group ²G₂(3)′ is a simple subgroup of index 3.

Order:q³ (q³ + 1) (q − 1), where q = 3²ⁿ⁺¹

Schur multiplier: Trivial for n ≥ 1 and for ²G₂(3)′.

Outer automorphism group:

1⋅f⋅1,

where f = 2n + 1.

Other names: Ree(3²ⁿ⁺¹), R(3²ⁿ⁺¹), E₂^∗(3²ⁿ⁺¹) .

Isomorphisms: The derived group ²G₂(3)′ is isomorphic to A₁(8).

Remarks:²G₂(3²ⁿ⁺¹) has a doubly transitive permutation representation on 3^3(2n+1) + 1 points and acts on a 7-dimensional vector space over the field with 3²ⁿ⁺¹ elements.

Sporadic groups

Mathieu groups, M₁₁, M₁₂, M₂₂, M₂₃, M₂₄

	Mathieu group, M11	Mathieu group, M12	Mathieu group, M22	Mathieu group, M23	Mathieu group, M24
Order	24 ⋅ 32 ⋅ 5 ⋅ 11 = 7920	26 ⋅ 33 ⋅ 5 ⋅ 11 = 95040	27 ⋅ 32 ⋅ 5 ⋅ 7 ⋅ 11 = 443520	27 ⋅ 32 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 23 = 10200960	210 ⋅ 33 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 23 = 244823040
Schur multiplier	Trivial	Order 2	Cyclic of order 12 [lower-alpha 1]	Trivial	Trivial
Outer automorphism group	Trivial	Order 2	Order 2	Trivial	Trivial
Remarks	A 4-transitive permutation group on 11 points, and is the point stabilizer of M12 (in the 5-transitive 12-point permutation representation of M12). The group M11 is also contained in M23. The subgroup of M11 fixing a point in the 4-transitive 11-point permutation representation is sometimes called M10, and has a subgroup of index 2 isomorphic to the alternating group A6.	A 5-transitive permutation group on 12 points, contained in M24.	A 3-transitive permutation group on 22 points, and is the point stabilizer of M23 (in the 4-transitive 23-point permutation representation of M23). The subgroup of M22 fixing a point in the 3-transitive 22-point permutation representation is sometimes called M21, and is isomorphic to PSL(3,4) (i.e. isomorphic to A2(4)).	A 4-transitive permutation group on 23 points, and is the point stabilizer of M24 (in the 5-transitive 24-point permutation representation of M24).	A 5-transitive permutation group on 24 points.

Janko groups, J₁, J₂, J₃, J₄

	Janko group, J1	Janko group, J2	Janko group, J3	Janko group, J4
Order	23 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 19 = 175560	27 ⋅ 33 ⋅ 52 ⋅ 7 = 604800	27 ⋅ 35 ⋅ 5 ⋅ 17 ⋅ 19 = 50232960	221 ⋅ 33 ⋅ 5 ⋅ 7 ⋅ 113 ⋅ 23 ⋅ 29 ⋅ 31 ⋅ 37 ⋅ 43 = 86775571046077562880
Schur multiplier	Trivial	Order 2	Order 3	Trivial
Outer automorphism group	Trivial	Order 2	Order 2	Trivial
Other names	J(1), J(11)	Hall–Janko group, HJ	Higman–Janko–McKay group, HJM
Remarks	It is a subgroup of G2(11), and so has a 7-dimensional representation over the field with 11 elements.	The automorphism group J2:2 of J2 is the automorphism group of a rank 3 graph on 100 points called the Hall-Janko graph. It is also the automorphism group of a regular near octagon called the Hall-Janko near octagon. The group J2 is contained in G2(4).	J3 seems unrelated to any other sporadic groups (or to anything else). Its triple cover has a 9-dimensional unitary representation over the field with 4 elements.	Has a 112-dimensional representation over the field with 2 elements.

Conway groups, Co₁, Co₂, Co₃

	Conway group, Co1	Conway group, Co2	Conway group, Co3
Order	221 ⋅ 39 ⋅ 54 ⋅ 72 ⋅ 11 ⋅ 13 ⋅ 23 = 4157776806543360000	218 ⋅ 36 ⋅ 53 ⋅ 7 ⋅ 11 ⋅ 23 = 42305421312000	210 ⋅ 37 ⋅ 53 ⋅ 7 ⋅ 11 ⋅ 23 = 495766656000
Schur multiplier	Order 2	Trivial	Trivial
Outer automorphism group	Trivial	Trivial	Trivial
Other names	·1	·2	·3, C3
Remarks	The perfect double cover Co0 of Co1 is the automorphism group of the Leech lattice, and is sometimes denoted by ·0.	Subgroup of Co0; fixes a norm 4 vector in the Leech lattice.	Subgroup of Co0; fixes a norm 6 vector in the Leech lattice. It has a doubly transitive permutation representation on 276 points.

Fischer groups, Fi₂₂, Fi₂₃, Fi₂₄′

	Fischer group, Fi22	Fischer group, Fi23	Fischer group, Fi24′
Order	217 ⋅ 39 ⋅ 52 ⋅ 7 ⋅ 11 ⋅ 13 = 64561751654400	218 ⋅ 313 ⋅ 52 ⋅ 7 ⋅ 11 ⋅ 13 ⋅ 17 ⋅ 23 = 4089470473293004800	221 ⋅ 316 ⋅ 52 ⋅ 73 ⋅ 11 ⋅ 13 ⋅ 17 ⋅ 23 ⋅ 29 = 1255205709190661721292800
Schur multiplier	Order 6	Trivial	Order 3
Outer automorphism group	Order 2	Trivial	Order 2
Other names	M(22)	M(23)	M(24)′, F3+
Remarks	A 3-transposition group whose double cover is contained in Fi23.	A 3-transposition group contained in Fi24′.	The triple cover is contained in the monster group.

Higman–Sims group, HS

Order: 2⁹ ⋅ 3² ⋅ 5³ ⋅ 7 ⋅ 11 = 44352000

Schur multiplier: Order 2.

Outer automorphism group: Order 2.

Remarks: It acts as a rank 3 permutation group on the Higman Sims graph with 100 points, and is contained in Co₂ and in Co₃.

McLaughlin group, McL

Order: 2⁷ ⋅ 3⁶ ⋅ 5³ ⋅ 7 ⋅ 11 = 898128000

Schur multiplier: Order 3.

Outer automorphism group: Order 2.

Remarks: Acts as a rank 3 permutation group on the McLaughlin graph with 275 points, and is contained in Co₂ and in Co₃.

Held group, He

Order: 2¹⁰ ⋅ 3³ ⋅ 5² ⋅ 7³ ⋅ 17 = 4030387200

Schur multiplier: Trivial.

Outer automorphism group: Order 2.

Other names: Held–Higman–McKay group, HHM, F₇, HTH

Remarks: Centralizes an element of order 7 in the monster group.

Rudvalis group, Ru

Order: 2¹⁴ ⋅ 3³ ⋅ 5³ ⋅ 7 ⋅ 13 ⋅ 29 = 145926144000

Schur multiplier: Order 2.

Outer automorphism group: Trivial.

Remarks: The double cover acts on a 28-dimensional lattice over the Gaussian integers.

Suzuki sporadic group, Suz

Order: 2¹³ ⋅ 3⁷ ⋅ 5² ⋅ 7 ⋅ 11 ⋅ 13 = 448345497600

Schur multiplier: Order 6.

Outer automorphism group: Order 2.

Other names: Sz

Remarks: The 6 fold cover acts on a 12-dimensional lattice over the Eisenstein integers. It is not related to the Suzuki groups of Lie type.

O'Nan group, O'N

Order: 2⁹ ⋅ 3⁴ ⋅ 5 ⋅ 7³ ⋅ 11 ⋅ 19 ⋅ 31 = 460815505920

Schur multiplier: Order 3.

Outer automorphism group: Order 2.

Other names: O'Nan–Sims group, O'NS, O–S

Remarks: The triple cover has two 45-dimensional representations over the field with 7 elements, exchanged by an outer automorphism.

Harada–Norton group, HN

Order: 2¹⁴ ⋅ 3⁶ ⋅ 5⁶ ⋅ 7 ⋅ 11 ⋅ 19 = 273030912000000

Schur multiplier: Trivial.

Outer automorphism group: Order 2.

Other names:F₅, D

Remarks: Centralizes an element of order 5 in the monster group.

Lyons group, Ly

Order: 2⁸ ⋅ 3⁷ ⋅ 5⁶ ⋅ 7 ⋅ 11 ⋅ 31 ⋅ 37 ⋅ 67 = 51765179004000000

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Other names: Lyons–Sims group, LyS

Remarks: Has a 111-dimensional representation over the field with 5 elements.

Thompson group, Th

Order: 2¹⁵ ⋅ 3¹⁰ ⋅ 5³ ⋅ 7² ⋅ 13 ⋅ 19 ⋅ 31 = 90745943887872000

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Other names:F₃, E

Remarks: Centralizes an element of order 3 in the monster, and is contained in E₈(3), so has a 248-dimensional representation over the field with 3 elements.

Baby Monster group, B

Order:

   2⁴¹ ⋅ 3¹³ ⋅ 5⁶ ⋅ 7² ⋅ 11 ⋅ 13 ⋅ 17 ⋅ 19 ⋅ 23 ⋅ 31 ⋅ 47

= 4154781481226426191177580544000000

Schur multiplier: Order 2.

Outer automorphism group: Trivial.

Other names:F₂

Remarks: The double cover is contained in the monster group. It has a representation of dimension 4371 over the complex numbers (with no nontrivial invariant product), and a representation of dimension 4370 over the field with 2 elements preserving a commutative but non-associative product.

Fischer–Griess Monster group, M

Order:

   2⁴⁶ ⋅ 3²⁰ ⋅ 5⁹ ⋅ 7⁶ ⋅ 11² ⋅ 13³ ⋅ 17 ⋅ 19 ⋅ 23 ⋅ 29 ⋅ 31 ⋅ 41 ⋅ 47 ⋅ 59 ⋅ 71

= 808017424794512875886459904961710757005754368000000000

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Other names:F₁, M₁, Monster group, Friendly giant, Fischer's monster.

Remarks: Contains all but 6 of the other sporadic groups as subquotients. Related to monstrous moonshine. The monster is the automorphism group of the 196,883-dimensional Griess algebra and the infinite-dimensional monster vertex operator algebra, and acts naturally on the monster Lie algebra.

Non-cyclic simple groups of small order

Order	Factored order	Group	Schur multiplier	Outer automorphism group
60	22 ⋅ 3 ⋅ 5	A5 ≃ A1(4) ≃ A1(5)	2	2
168	23 ⋅ 3 ⋅ 7	A1(7) ≃ A2(2)	2	2
360	23 ⋅ 32 ⋅ 5	A6 ≃ A1(9) ≃ B2(2)′	6	2×2
504	23 ⋅ 32 ⋅ 7	A1(8) ≃ 2G2(3)′	1	3
660	22 ⋅ 3 ⋅ 5 ⋅ 11	A1(11)	2	2
1092	22 ⋅ 3 ⋅ 7 ⋅ 13	A1(13)	2	2
2448	24 ⋅ 32 ⋅ 17	A1(17)	2	2
2520	23 ⋅ 32 ⋅ 5 ⋅ 7	A7	6	2
3420	22 ⋅ 32 ⋅ 5 ⋅ 19	A1(19)	2	2
4080	24 ⋅ 3 ⋅ 5 ⋅ 17	A1(16)	1	4
5616	24 ⋅ 33 ⋅ 13	A2(3)	1	2
6048	25 ⋅ 33 ⋅ 7	2A2(9) ≃ G2(2)′	1	2
6072	23 ⋅ 3 ⋅ 11 ⋅ 23	A1(23)	2	2
7800	23 ⋅ 3 ⋅ 52 ⋅ 13	A1(25)	2	2×2
7920	24 ⋅ 32 ⋅ 5 ⋅ 11	M 11	1	1
9828	22 ⋅ 33 ⋅ 7 ⋅ 13	A1(27)	2	6
12180	22 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 29	A1(29)	2	2
14880	25 ⋅ 3 ⋅ 5 ⋅ 31	A1(31)	2	2
20160	26 ⋅ 32 ⋅ 5 ⋅ 7	A3(2) ≃ A8	2	2
20160	26 ⋅ 32 ⋅ 5 ⋅ 7	A2(4)	3×42	D12
25308	22 ⋅ 32 ⋅ 19 ⋅ 37	A1(37)	2	2
25920	26 ⋅ 34 ⋅ 5	2A3(4) ≃ B2(3)	2	2
29120	26 ⋅ 5 ⋅ 7 ⋅ 13	2B2(8)	22	3
32736	25 ⋅ 3 ⋅ 11 ⋅ 31	A1(32)	1	5
34440	23 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 41	A1(41)	2	2
39732	22 ⋅ 3 ⋅ 7 ⋅ 11 ⋅ 43	A1(43)	2	2
51888	24 ⋅ 3 ⋅ 23 ⋅ 47	A1(47)	2	2
58800	24 ⋅ 3 ⋅ 52 ⋅ 72	A1(49)	2	22
62400	26 ⋅ 3 ⋅ 52 ⋅ 13	2A2(16)	1	4
74412	22 ⋅ 33 ⋅ 13 ⋅ 53	A1(53)	2	2
95040	26 ⋅ 33 ⋅ 5 ⋅ 11	M 12	2	2

(Complete for orders less than 100,000)

Hall (1972) lists the 56 non-cyclic simple groups of order less than a million.

See also

• List of small groups

Notes

1. There were several mistakes made in the initial calculations of the Schur multiplier, so some older books and papers list incorrect values. (This caused an error in the title of Janko's original 1976 paper ^[1] giving evidence for the existence of the group J₄. At the time it was thought that the full covering group of M₂₂ was 6⋅M₂₂. In fact J₄ has no subgroup 12⋅M₂₂.)

References

1. Z. Janko (1976). "A new finite simple group of order 86,775,571,046,077,562,880 which possesses M₂₄ and the full covering group of M₂₂ as subgroups". J. Algebra. 42: 564–596. doi: 10.1016/0021-8693(76)90115-0 .

Further reading

• Simple Groups of Lie Type by Roger W. Carter, ISBN   0-471-50683-4
• 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 1985.
• Daniel Gorenstein, Richard Lyons, Ronald Solomon The Classification of the Finite Simple Groups (volume 1), AMS, 1994 (volume 3), AMS, 1998
• Hall, Marshall Jr. (1972), "Simple groups of order less than one million", Journal of Algebra , 20: 98–102, doi: 10.1016/0021-8693(72)90090-7 , ISSN   0021-8693, MR   0285603
• Wilson, Robert A. (2009), The finite simple groups, Graduate Texts in Mathematics 251, vol. 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN   978-1-84800-987-5, Zbl   1203.20012
• Atlas of Finite Group Representations: contains representations and other data for many finite simple groups, including the sporadic groups.
• Orders of non abelian simple groups up to 10¹⁰, and on to 10⁴⁸ with restrictions on rank.

External links

• Orders of non abelian simple groups up to order 10,000,000,000.