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 (including the Tits group, which strictly speaking is not 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).

1 Infinite families
2 Sporadic groups
3 Non-cyclic simple groups of small order
4 See also
5 References
6 External links

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.

A_n(q) Chevalley groups, linear groups

Simplicity: A₁(2) and A₁(3) are solvable, the others are simple.

Order:

${1\over (n%2B1,q-1)} q^{n(n%2B1)/2} \prod_{i=1}^n(q^{i%2B1}-1)$

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).

Outer automorphism group: (2, q − 1) ·f·1 for n = 1; (n+1, q − 1) ·f·2 for n > 1, where q = p^f.

Other names: Projective special linear groups, PSL_n+1(q), L_n+1(q), PSL(n+1,q)

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.

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.

B_n(q) n > 1 Chevalley groups, orthogonal groups

Simplicity: B₂(2) is not simple and has a simple subgroup of index 2; the others are simple.

Order:

${1\over (2,q-1)} q^{n^2} \prod_{i=1}^n(q^{2i}-1)$

Schur multiplier: (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).

Outer automorphism group: (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.

Other names: O_2n+1(q), Ω_2n+1(q) (for q odd).

Isomorphisms: 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).

Remarks: 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.

C_n(q) n > 2 Chevalley groups, symplectic groups

Simplicity: All simple.

Order:

${1\over (2,q-1)} q^{n^2} \prod_{i=1}^n(q^{2i}-1)$

Schur multiplier: (2,q − 1) except for C₃(2) (order 2).

Outer automorphism group: (2, q − 1) ·f·1 where q = p^f.

Other names: Projective symplectic group, PSp_2n(q), PSp_n(q) (not recommended), S_2n(q), Abelian group (archaic).

Isomorphisms: C_n(2^m) is isomorphic to B_n(2^m)

Remarks: 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).

D_n(q) n > 3 Chevalley groups, orthogonal groups

Simplicity: All simple.

Order:

${1\over (4,q^n-1)} q^{n(n-1)} (q^n-1) \prod_{i=1}^{n-1}(q^{2i}-1)$

Schur multiplier: 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) 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 on 3 points of order 6.

Other names: O_2n⁺(q), PΩ_2n⁺(q). "Hypoabelian group" is an archaic name for this group in characteristic 2.

Remarks: 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).

E₆(q) Chevalley groups

Simplicity: All simple.

Order: q³⁶ (q¹² − 1) (q⁹ − 1) (q⁸ − 1) (q⁶ − 1) (q⁵ − 1) (q² − 1) /(3,q − 1)

Schur multiplier: (3,q − 1)

Outer automorphism group: (3, q − 1) ·f·2 where q = p^f.

Other names: Exceptional Chevalley group.

Remarks: Has two representations of dimension 27, and acts on the Lie algebra of dimension 78.

E₇(q) Chevalley groups

Simplicity: All simple.

Order: q⁶³ (q¹⁸ − 1) (q¹⁴ − 1) (q¹² − 1) (q¹⁰ − 1) (q⁸ − 1) (q⁶ − 1) (q² − 1) /(2,q − 1)

Schur multiplier: (2,q − 1)

Outer automorphism group: (2, q − 1) ·f·1 where q = p^f.

Other names: Exceptional Chevalley group.

Remarks: Has a representations of dimension 56, and acts on the corresponding Lie algebra of dimension 133.

E₈(q) Chevalley groups

Simplicity: All simple.

Order: q¹²⁰ (q³⁰−1) (q²⁴−1) (q²⁰−1) (q¹⁸−1) (q¹⁴−1) (q¹²−1) (q⁸−1) (q²−1)

Schur multiplier: Trivial.

Outer automorphism group: 1·f·1 where q = p^f.

Other names: Exceptional Chevalley group.

Remarks: It acts on the corresponding Lie algebra of dimension 248. E₈(3) contains the Thompson simple group.

F₄(q) Chevalley groups

Simplicity: All simple.

Order: q²⁴ (q¹²−1) (q⁸−1) (q⁶−1) (q²−1)

Schur multiplier: Trivial except for F₄(2) (order 2).

Outer automorphism group: 1·f·1 for q odd, 1·f·2 for q even, where q = p^f.

Other names: Exceptional Chevalley group.

Remarks: 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.

G₂(q) Chevalley groups

Simplicity: G₂(2) is not simple but has a simple subgroup of index 2; the others are simple.

Order: q⁶ (q⁶−1) (q²−1)

Schur multiplier: Trivial for the simple groups except for G₂(3) (order 3) and G₂(4) (order 2).

Outer automorphism group: 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.

Isomorphisms: The derived group G₂(2)′ is isomorphic to ²A₂(3²).

Remarks: 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.

²A_n(q²) n > 1 Steinberg groups, unitary groups

Simplicity: ²A₂(2²) is solvable, the others are simple.

Order:

${1\over (n%2B1,q%2B1)} q^{n(n%2B1)/2} \prod_{i=1}^n(q^{i%2B1}-(-1)^{i%2B1})$

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)

Outer automorphism group: (n+1, q + 1) ·f·1 where q² = 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)

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.

²D_n(q²) n > 3 Steinberg groups, orthogonal groups

Simplicity: All simple.

Order:

${1\over (4,q^n%2B1)} q^{n(n-1)} (q^n%2B1) \prod_{i=1}^{n-1}(q^{2i}-1)$

Schur multiplier: Cyclic of order (4, qⁿ + 1).

Outer automorphism group: (4, qⁿ + 1) ·f·1 where q² = p^f.

Other names: ²D_n(q), O_2n⁻(q), PΩ_2n⁻(q), twisted Chevalley group. "Hypoabelian group" is an archaic name for this group in characteristic 2.

Remarks: 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).

²E₆(q²) Steinberg groups

Simplicity: All simple.

Order: q³⁶ (q¹²−1) (q⁹+1) (q⁸−1) (q⁶−1) (q⁵+1) (q²−1) /(3,q+1)

Schur multiplier: (3, q + 1) except for ²E₆(2²) (order 12, product of cyclic groups of orders 2,2,3).

Outer automorphism group: (3, q + 1) ·f·1 where q² = p^f.

Other names: ²E₆(q), twisted Chevalley group.

Remarks: 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(q³) Steinberg groups

Simplicity: All simple.

Order: q¹² (q⁸+q⁴+1) (q⁶−1) (q²−1)

Schur multiplier: Trivial.

Outer automorphism group: 1·f·1 where q³ = p^f.

Other names: ³D₄(q), D₄²(q³), Twisted Chevalley groups.

Remarks: 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: The Tits group is strictly speaking not a group of Lie type, and in particular it is not the group of points of a connected simple algebraic group with values in some field, nor does it have a BN pair. However 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 group M₁₁

Order: 2⁴ · 3² · 5 · 11=7920

Schur multiplier: Trivial.

Outer automorphism group: 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.

Mathieu group M₁₂

Order: 2⁶ · 3³ · 5 · 11=95040

Schur multiplier: Order 2.

Outer automorphism group: Order 2.

Remarks: A 5-transitive permutation group on 12 points.

Mathieu group M₂₂

Order: 2⁷ · 3² · 5 · 7 · 11 = 443520

Schur multiplier: Cyclic of order 12. 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 "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 (1976), 564–596." 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.)

Outer automorphism group: Order 2.

Remarks: A 3-transitive permutation group on 22 points.

Mathieu group M₂₃

Order: 2⁷ · 3² · 5 · 7 · 11 · 23=10200960

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Remarks: A 4-transitive permutation group on 23 points, contained in M₂₄.

Mathieu group M₂₄

Order: 2¹⁰ · 3³ · 5 · 7 · 11 · 23= 244823040

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Remarks: A 5-transitive permutation group on 24 points.

Janko group J₁

Order: 2³ · 3 · 5 · 7 · 11 · 19 = 175560

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Other names: J(1), J(11)

Remarks: It is a subgroup of G₂(11), and so has a 7 dimensional representation over the field with 11 elements.

Janko group J₂

Order: 2⁷ · 3³ · 5² · 7 = 604800

Schur multiplier: Order 2.

Outer automorphism group: Order 2.

Other names: Hall–Janko group, HJ

Remarks: It is the automorphism group of a rank 3 graph on 100 points, and is also contained in G₂(4).

Janko group J₃

Order: 2⁷ · 3⁵ · 5 · 17 · 19 = 50232960

Schur multiplier: Order 3.

Outer automorphism group: Order 2.

Other names: Higman–Janko–McKay group, HJM

Remarks: 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.

Janko group J₄

Order: 2²¹ · 3³ · 5 · 7 · 11³ · 23 · 29 · 31 · 37 · 43 = 86775571046077562880

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Remarks: Has a 112 dimensional representation over the field with 2 elements.

Conway group Co₁

Order: 2²¹ · 3⁹ · 5⁴ · 7² · 11 · 13 · 23 = 4157776806543360000

Schur multiplier: Order 2.

Outer automorphism group: Trivial.

Other names: ·1

Remarks: The perfect double cover of Co₁ is the automorphism group of the Leech lattice, and is sometimes denoted by ·0.

Conway group Co₂

Order: 2¹⁸ · 3⁶ · 5³ · 7 · 11 · 23 = 42305421312000

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Other names: ·2

Remarks: Subgroup of Co₁; fixes a norm 4 vector in the Leech lattice.

Conway group Co₃

Order: 2¹⁰ · 3⁷ · 5³ · 7 · 11 · 23 = 495766656000

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Other names: ·3

Remarks: Subgroup of Co₁; fixes a norm 6 vector in the Leech lattice.

Fischer group Fi₂₂

Order: 2¹⁷ · 3⁹ · 5² · 7 · 11 · 13 = 64561751654400.

Schur multiplier: Order 6.

Outer automorphism group: Order 2.

Other names: M(22)

Remarks: A 3-transposition group whose double cover is contained in Fi₂₃.

Fischer group Fi₂₃

Order: 2¹⁸ · 3¹³ · 5² · 7 · 11 · 13 · 17 · 23 = 4089470473293004800.

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Other names: M(23)

Remarks: A 3-transposition group contained in Fi₂₄.

Fischer group Fi₂₄′

Order: 2²¹ · 3¹⁶ · 5² · 7³ · 11 · 13 · 17 · 23 · 29 = 1255205709190661721292800.

Schur multiplier: Order 3.

Outer automorphism group: Order 2.

Other names: M(24)′, F₃₊.

Remarks: 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 representations 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

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

Order	Factorized 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 100000)

References

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 2), AMS,
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, 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 05622792
Atlas of Finite Group Representations: contains representations and other data for many finite simple groups, including the sporadic groups.

External links

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

List of finite simple groups

Contents

Infinite families

Cyclic groups Zp

An, n > 4, Alternating groups

An(q) Chevalley groups, linear groups

Bn(q) n > 1 Chevalley groups, orthogonal groups

Cn(q) n > 2 Chevalley groups, symplectic groups

Dn(q) n > 3 Chevalley groups, orthogonal groups

E6(q) Chevalley groups

E7(q) Chevalley groups

E8(q) Chevalley groups

F4(q) Chevalley groups

G2(q) Chevalley groups

2An(q2) n > 1 Steinberg groups, unitary groups

2Dn(q2) n > 3 Steinberg groups, orthogonal groups

2E6(q2) Steinberg groups

3D4(q3) Steinberg groups

2B2(22n+1) Suzuki groups

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

2G2(32n+1) Ree groups

Sporadic groups

Mathieu group M11

Mathieu group M12

Mathieu group M22

Mathieu group M23

Mathieu group M24

Janko group J1

Janko group J2

Janko group J3

Janko group J4

Conway group Co1

Conway group Co2

Conway group Co3

Fischer group Fi22

Fischer group Fi23

Fischer group 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

References

External links

Cyclic groups Z_p

A_n, n > 4, Alternating groups

A_n(q) Chevalley groups, linear groups

B_n(q) n > 1 Chevalley groups, orthogonal groups

C_n(q) n > 2 Chevalley groups, symplectic groups

D_n(q) n > 3 Chevalley groups, orthogonal groups

E₆(q) Chevalley groups

E₇(q) Chevalley groups

E₈(q) Chevalley groups

F₄(q) Chevalley groups

G₂(q) Chevalley groups

²A_n(q²) n > 1 Steinberg groups, unitary groups

²D_n(q²) n > 3 Steinberg groups, orthogonal groups

²E₆(q²) Steinberg groups

³D₄(q³) Steinberg groups

²B₂(2²ⁿ⁺¹) Suzuki groups

²F₄(2²ⁿ⁺¹) Ree groups, Tits group

²G₂(3²ⁿ⁺¹) Ree groups

Mathieu group M₁₁

Mathieu group M₁₂

Mathieu group M₂₂

Mathieu group M₂₃

Mathieu group M₂₄

Janko group J₁

Janko group J₂

Janko group J₃

Janko group J₄

Conway group Co₁

Conway group Co₂

Conway group Co₃

Fischer group Fi₂₂

Fischer group Fi₂₃

Fischer group Fi₂₄′