Equianharmonic

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. (In removing duplicates it is useful to note that finite simple groups are determined by their orders, except that the group B_n(q) has the same order as C_n(q) for q odd, n > 2; and the groups A₈ = A₃(2) and A₂(4) both have orders 20160.)

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).

Infinite families

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

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

A_n, n > 4, Alternating groups

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.

There is an unfortunate conflict with the notation for the (unrelated) groups A_n(q), and 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.

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.

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:	$\frac{1}{(n + 1, q - 1)} q^{n (n + 1) / 2} \prod_{i = 1}^{n} (q^{i + 1} - 1)$	$\frac{1}{(2, q - 1)} q^{n^{2}} \prod_{i = 1}^{n} (q^{2 i} - 1)$	$\frac{1}{(2, q - 1)} q^{n^{2}} \prod_{i = 1}^{n} (q^{2 i} - 1)$	$\frac{1}{(4, q^{n} - 1)} q^{n (n - 1)} (q^{n} - 1) \prod_{i = 1}^{n - 1} (q^{2 i} - 1)$
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 if q is 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 isomorphic, and 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.

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:	$\frac{1}{(n + 1, q + 1)} q^{n (n + 1) / 2} \prod_{i = 1}^{n} (q^{i + 1} - (- 1)^{i + 1})$	$\frac{1}{(4, q^{n} + 1)} q^{n (n - 1)} (q^{n} + 1) \prod_{i = 1}^{n - 1} (q^{2 i} - 1)$	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), ² A_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.

²B₂(2²ⁿ⁺¹) Suzuki groups

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.

²F₄(2²ⁿ⁺¹) Ree groups, Tits group

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.

²G₂(3²ⁿ⁺¹) Ree groups

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 12Template:Efn	Trivial	Trivial
Outer automorphism group:	Trivial	Order 2	Order 2	Trivial	Trivial
Remarks:	A 4-transitive permutation group on 11 points, and the point stabilizer in M₁₂. The subgroup fixing a point 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.	A 3-transitive permutation group on 22 points.	A 4-transitive permutation group on 23 points, contained in 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
Remarks:	It is a subgroup of G₂(11), and so has a 7 dimensional representation over the field with 11 elements.	It is the automorphism group of a rank 3 graph on 100 points, and is also 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 of Co₁ is the automorphism group of the Leech lattice, and is sometimes denoted by ·0.	Subgroup of Co₁; fixes a norm 4 vector in the Leech lattice.	Subgroup of Co₁; 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 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 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 196884 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)

Template:Harvtxt lists the 56 non-cyclic simple groups of order less than a million.

Notes

Template:Notelist

References

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External links

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

Equianharmonic

Contents

Infinite families

Cyclic groups Z_p

A_n, n > 4, Alternating groups

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)

²B₂(2²ⁿ⁺¹) Suzuki groups

²F₄(2²ⁿ⁺¹) Ree groups, Tits group

²G₂(3²ⁿ⁺¹) Ree groups

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

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Equianharmonic

Infinite families

Cyclic groups Zp

An, n > 4, Alternating groups

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)

2B2(22n+1) Suzuki groups

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

2G2(32n+1) Ree groups

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

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Cyclic groups Z_p

A_n, n > 4, Alternating groups

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)

²B₂(2²ⁿ⁺¹) Suzuki groups

²F₄(2²ⁿ⁺¹) Ree groups, Tits group

²G₂(3²ⁿ⁺¹) Ree groups

Mathieu groups M₁₁, M₁₂, M₂₂, M₂₃, M₂₄

Janko groups J₁, J₂, J₃, J₄

Conway groups Co₁, Co₂, Co₃

Fischer groups Fi₂₂, Fi₂₃, Fi₂₄'