List of finite simple groups

Last updated August 04, 2024

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.

Summary
Cyclic groups, Zp
Alternating groups, An, n > 4
Groups of Lie type
Chevalley groups, An(q), Bn(q) n > 1, Cn(q) n > 2, Dn(q) n > 3
Chevalley groups, E6(q), E7(q), E8(q), F4(q), G2(q)
Steinberg groups, 2An(q2) n > 1, 2Dn(q2) n > 3, 2E6(q2), 3D4(q3)
Suzuki groups, 2B2(22n+1)
Ree groups and Tits group, 2F4(22n+1)
Ree groups, 2G2(32n+1)
Sporadic groups
Mathieu groups, M11, M12, M22, M23, M24
Janko groups, J1, J2, J3, J4
Conway groups, Co1, Co2, Co3
Fischer groups, Fi22, Fi23, Fi24′
Higman–Sims group, HS
McLaughlin group, McL
Held group, He
Rudvalis group, Ru
Suzuki sporadic group, Suz
O'Nan group, O'N
Harada–Norton group, HN
Lyons group, Ly
Thompson group, Th
Baby Monster group, B
Fischer–Griess Monster group, M
Non-cyclic simple groups of small order
See also
Notes
References
Further reading
External links

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	Z_p	p prime	None	None

Alternating groups	A_n n > 4	${\frac {n!}{2}}$	None	A₅ ≃ A₁(4) ≃ A₁(5) A₆ ≃ A₁(9) A₈ ≃ A₃(2)

Classical Chevalley groups	A_n(q)	${\frac {q^{{\frac {1}{2}}n(n+1)}}{(n+1,q-1)}}\prod _{i=1}^{n}\left(q^{i+1}-1\right)$	A₁(2), A₁(3)	A₁(4) ≃ A₁(5) ≃ A₅ A₁(7) ≃ A₂(2) A₁(9) ≃ A₆ A₃(2) ≃ A₈
	B_n(q) n > 1	${\frac {q^{n^{2}}}{(2,q-1)}}\prod _{i=1}^{n}\left(q^{2i}-1\right)$	B₂(2)	B_n(2^m) ≃ C_n(2^m) B₂(3) ≃ ²A₃(2²)
	C_n(q) n > 2	${\frac {q^{n^{2}}}{(2,q-1)}}\prod _{i=1}^{n}\left(q^{2i}-1\right)$	None	C_n(2^m) ≃ B_n(2^m)
	D_n(q) n > 3	${\frac {q^{n(n-1)}(q^{n}-1)}{(4,q^{n}-1)}}\prod _{i=1}^{n-1}\left(q^{2i}-1\right)$	None	None
Exceptional Chevalley groups	E₆(q)	${\frac {q^{36}}{(3,q-1)}}\prod _{i\in \{2,5,6,8,9,12\}}\left(q^{i}-1\right)$	None	None
	E₇(q)	${\frac {q^{63}}{(2,q-1)}}\prod _{i\in \{2,6,8,10,12,14,18\}}\left(q^{i}-1\right)$	None	None
	E₈(q)	$q^{120}\prod _{i\in \{2,8,12,14,18,20,24,30\}}\left(q^{i}-1\right)$	None	None
	F₄(q)	$q^{24}\prod _{i\in \{2,6,8,12\}}\left(q^{i}-1\right)$	None	None
	G₂(q)	$q^{6}\prod _{i\in \{2,6\}}\left(q^{i}-1\right)$	G₂(2)	None
Classical Steinberg groups	²A_n(q²) n > 1	${\frac {q^{{\frac {1}{2}}n(n+1)}}{(n+1,q+1)}}\prod _{i=1}^{n}\left(q^{i+1}-(-1)^{i+1}\right)$	²A₂(2²)	²A₃(2²) ≃ B₂(3)
Classical Steinberg groups	²D_n(q²) n > 3	${\frac {q^{n(n-1)}}{(4,q^{n}+1)}}\left(q^{n}+1\right)\prod _{i=1}^{n-1}\left(q^{2i}-1\right)$	None	None
Exceptional Steinberg groups	²E₆(q²)	${\frac {q^{36}}{(3,q+1)}}\prod _{i\in \{2,5,6,8,9,12\}}\left(q^{i}-(-1)^{i}\right)$	None	None
Exceptional Steinberg groups	³D₄(q³)	$q^{12}\left(q^{8}+q^{4}+1\right)\left(q^{6}-1\right)\left(q^{2}-1\right)$	None	None
Suzuki groups	²B₂(q) q = 2²ⁿ⁺¹	$q^{2}\left(q^{2}+1\right)\left(q-1\right)$	None	None
Ree groups + Tits group	²F₄(q) q = 2²ⁿ⁺¹	$q^{12}\left(q^{6}+1\right)\left(q^{4}-1\right)\left(q^{3}+1\right)\left(q-1\right)$	None	None
	²F₄(2)′	2¹²(2⁶ + 1)(2⁴ − 1)(2³ + 1)(2 − 1)/2 = 17971200
	²G₂(q) q = 3²ⁿ⁺¹	$q^{3}\left(q^{3}+1\right)\left(q-1\right)$	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 ≤ 4, 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 $D\rtimes (F\times G)$ where all these groups $D,F,G$ are cyclic of the respective orders d, f, g, except for type $D_{n}(q)$ , $q$ odd, where the group of order $d=4$ is $C_{2}\times C_{2}$ , and (only when $n=4$ ) $G=S_{3}$ , 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, A_n(q) linear groups	Chevalley groups, B_n(q) n > 1 orthogonal groups	Chevalley groups, C_n(q) n > 2 symplectic groups	Chevalley groups, D_n(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	${\frac {q^{{\frac {1}{2}}n(n+1)}}{(n+1,q-1)}}\prod _{i=1}^{n}\left(q^{i+1}-1\right)$	${\frac {q^{n^{2}}}{(2,q-1)}}\prod _{i=1}^{n}\left(q^{2i}-1\right)$	${\frac {q^{n^{2}}}{(2,q-1)}}\prod _{i=1}^{n}\left(q^{2i}-1\right)$	${\frac {q^{n(n-1)}(q^{n}-1)}{(4,q^{n}-1)}}\prod _{i=1}^{n-1}\left(q^{2i}-1\right)$
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 = p^f	(2,q−1)⋅f⋅1 for q odd or n > 2; (2,q−1)⋅f⋅2 for q even and n = 2, where q = p^f	(2,q−1)⋅f⋅1, where q = p^f	(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 = p^f, and S₃ is the symmetric group of order 3! on 3 points.
Other names	Projective special linear groups, PSL_n+1(q), L_n+1(q), PSL(n + 1,q)	O_2n+1(q), Ω_2n+1(q) (for q odd).	Projective symplectic group, PSp_2n(q), PSp_n(q) (not recommended), S_2n(q), Abelian group (archaic).	O_2n⁺(q), PΩ_2n⁺(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.	B_n(2^m) is isomorphic to C_n(2^m). 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).	C_n(2^m) is isomorphic to B_n(2^m)
Remarks	These groups are obtained from the general linear groups GL_n+1(q) by taking the elements of determinant 1 (giving the special linear groups SL_n+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 = p^f	(2,q−1)⋅f⋅1, where q = p^f	1⋅f⋅1, where q = p^f	1⋅f⋅1 for q odd, 1⋅f⋅2 for q even, where q = p^f	1⋅f⋅1 for q not a power of 3, 1⋅f⋅2 for q a power of 3, where q = p^f
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, ²A_n(q²) n > 1 unitary groups	Steinberg groups, ²D_n(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	${1 \over (n+1,q+1)}q^{{\frac {1}{2}}n(n+1)}\prod _{i=1}^{n}\left(q^{i+1}-(-1)^{i+1}\right)$	${1 \over (4,q^{n}+1)}q^{n(n-1)}(q^{n}+1)\prod _{i=1}^{n-1}\left(q^{2i}-1\right)$	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 = p^f	(4,qⁿ+1)⋅f⋅1, where q² = p^f	(3,q+1)⋅f⋅1, where q² = p^f	1⋅f⋅1, where q³ = p^f
Other names	Twisted Chevalley group, projective special unitary group, PSU_n+1(q), PSU(n + 1, q), U_n+1(q), ²A_n(q), ²A_n(q, q²)	²D_n(q), O_2n⁻(q), PΩ_2n⁻(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. Has a 248-dimensional representation which, when reduced modulo 3, leads to containment in E₈(3).

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	M₁₁	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	M₁₂	2	2

(Complete for orders less than 100,000)

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

Notes

↑ 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.)

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References

↑ 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 .

External links

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

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[2] 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.)

[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 .

[lower-alpha 1]

[1]

List of finite simple groups

Contents

Summary

Cyclic groups, Zp

Alternating groups, An, n > 4

Groups of Lie type

Chevalley groups, An(q), Bn(q) n > 1, Cn(q) n > 2, Dn(q) n > 3

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

Steinberg groups, 2An(q2) n > 1, 2Dn(q2) n > 3, 2E6(q2), 3D4(q3)

Suzuki groups, 2B2(22n+1)

Ree groups and Tits group, 2F4(22n+1)

Ree groups, 2G2(32n+1)

Sporadic groups

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

Janko groups, J1, J2, J3, J4

Conway groups, Co1, Co2, Co3

Fischer groups, Fi22, Fi23, Fi24′

Higman–Sims group, HS

McLaughlin group, McL

Held group, He

Rudvalis group, Ru

Suzuki sporadic group, Suz

O'Nan group, O'N

Harada–Norton group, HN

Lyons group, Ly

Thompson group, Th

Baby Monster group, B

Fischer–Griess Monster group, M

Non-cyclic simple groups of small order

See also

Notes

Related Research Articles

References

Further reading

External links

Cyclic groups, Z_p

Alternating groups, A_n, n > 4

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

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

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

Suzuki groups, ²B₂(2²ⁿ⁺¹)

Ree groups and Tits group, ²F₄(2²ⁿ⁺¹)

Ree groups, ²G₂(3²ⁿ⁺¹)

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

Janko groups, J₁, J₂, J₃, J₄

Conway groups, Co₁, Co₂, Co₃

Fischer groups, Fi₂₂, Fi₂₃, Fi₂₄′