List of finite simple groups

In mathematics, the classification of finite simple groups states thatevery 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 = "A"₃(2) and "A"₂(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: Always simple.

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.

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 "A"₁(3) (solvable). A₅ is isomorphic to "A"₁(4) and to "A"₁(5). A₆ is isomorphic to "A"₁(9) and to the derived group "B"₂(2)'. A₈ is isomorphic to "A"₃(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: "A"₁(2) and "A"₁(3) aresolvable, the others are simple.

Order: :$\{1over\; (n+1,q-1)\}q^\{n(n+1)/2\}prod\_\{i=1\}^n(q^\{i+1\}-1)$

Schur multiplier: For the simple groups it is cyclic of order ("n"+1, "q" − 1) except for "A"₁(4) (order 2), "A"₁(9) (order 6), "A"₂(2) (order 2), "A"₂(4) (order 48, product of cyclic groups of orders 3, 4, 4), "A"₃(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: "A"₁(2) is isomorphic to the symmetric group on 3 points of order 6. "A"₁(3) is isomorphic to the alternating group A₄ (solvable). "A"₁(4) and "A"₁(5)are isomorphic, and are both isomorphic to the alternating group A₅."A"₁(7) and "A"₂(2) are isomorphic. "A"₁(8) is isomorphic to the derived group ²"G"₂(3)′. "A"₁(9) is isomorphic to A₆ and to the derived group "B"₂(2)′. "A"₃(2) is isomorphic to A₈.

Remarks: These groups are obtained from the general linear groups "GL"_"n"+1("q") bytaking 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 group

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

Order: :$\{1over\; (2,q-1)\}q^\{n^2\}prod\_\{i=1\}^n(q^\{2i\}-1)$

Schur multiplier: (2,"q" − 1) except for "B"₂(2) = S₆ (order 2 for "B"₂(2), order 6 for "B"₂(2)&prime;) and "B"₃(2)(order 2) and "B"₃(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"_2"n"+1("q"),Ω_2"n"+1("q") (for "q" odd).

Isomorphisms: "B_n"(2^"m") is isomorphic to "C_n"(2^"m"). "B"₂(2) is isomorphic to the symmetric group on 6 points, and the derived group "B"₂(2)′ is isomorphic to "A"₁(9) and toA₆. "B"₂(3) is isomorphic to ²"A"₃(2²).

Remarks: This is the group obtained from the orthogonal group in dimension 2"n"+1 bytaking 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: :$\{1over\; (2,q-1)\}q^\{n^2\}prod\_\{i=1\}^n(q^\{2i\}-1)$

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"_2"n"("q"), "PSp"_"n"("q") (not recommended), "S"_2"n"("q").

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

Remarks: This group is obtained from the symplectic group in 2"n" dimensions by "quotienting out" the center. "C"₁("q") also exists, but is the same as"A"₁("q"). "C"₂("q") also exists, but is the same as "B"₂("q").

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

Simplicity: All simple.

Order: :$\{1over\; (4,q^n-1)\}q^\{n(n-1)\}(q^n-1)prod\_\{i=1\}^\{n-1\}(q^\{2i\}-1)$

Schur multiplier: The order is (4, "qⁿ"-1) (cyclic for "n" odd, elementary abelian for "n" even) except for "D"₄(2) (order 4, elementary abelian).

Outer automorphism group: (2, "q" − 1) ²·"f"·"S"₃ for "n"=4, (2, "q" − 1) ²·"f"·2 for "n>4" even, (4, "qⁿ" − 1) ²·"f"·2 for "n" odd, where "q" = "p^f", and "S"₃ is the symmetric group on 3 points of order 6.

Other names: "O"_2"n"⁺("q"), "PΩ"_2"n"⁺("q").

Remarks: This is the group obtained from the split orthogonal group in dimension 2"n" bytaking the kernel of the determinant (or Dickson invariant in characteristic 2) and spinor norm maps and then killing the center. The groups of type "D"₄ have an unusually large diagram automorphism group of order 6, containing the triality automorphism. "D"₂("q") also exists, but is the same as"A"₁("q")×"A"₁("q"). "D"₃("q") also exists, but is the same as "A"₃("q").

"E"₆("q") Chevalley groups

Simplicity: All simple.

Order: "q"³⁶("q"¹²−1)("q"⁹−1)("q"⁸−1)("q"⁶−1)("q"⁵−1)("q"²−1)/(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.

"E"₇("q") Chevalley groups

Simplicity: All simple.

Order: "q"⁶³("q"¹⁸−1)("q"¹⁴−1)("q"¹²−1)("q"¹⁰−1)("q"⁸−1)("q"⁶−1)("q"²−1)/(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.

"E"₈("q") Chevalley groups

Simplicity: All simple.

Order: "q"¹²⁰("q"³⁰−1)("q"²⁴−1)("q"²⁰−1)("q"¹⁸−1)("q"¹⁴−1)("q"¹²−1)("q"⁸−1)("q"²−1)

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. "E"₈(3) contains the Thompson simple group.

"F"₄("q") Chevalley groups

Simplicity: All simple.

Order: "q"²⁴("q"¹²−1)("q"⁸−1)("q"⁶−1)("q"²−1)

Schur multiplier: Trivial except for "F"₄(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. "F"₄("q") has a non-trivial graph automorphism when "q" is a power of 2.

"G"₂("q") Chevalley groups

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

Order: "q"⁶("q"⁶−1)("q"²−1)

Schur multiplier: Trivial for the simple groups except for "G"₂(3) (order 3) and "G"₂(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 "G"₂(2)′ is isomorphic to ²"A"₂(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. "G"₂("q") has a non-trivial graph automorphism when "q" is a power of 3.

²"A_n"("q"²) "n" > 1 Steinberg groups, unitary groups

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

Order: :$\{1over\; (n+1,q+1)\}q^\{n(n+1)/2\}prod\_\{i=1\}^n(q^\{i+1\}-(-1)^\{i+1\})$

Schur multiplier: Cyclic of order ("n" + 1, "q" + 1) for the simple groups, except for ²"A"₃(2²) (order 2),²"A"₃(3²) (order 36, product of cyclic groups of orders 3,3,4),²"A"₅(2²) (order 12, product of cyclic groups of orders 2,2,3)

Outer automorphism group: ("n"+1, "q" + 1) ·"f"·1where "q"² = "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"),²"A_n"("q", "q"²)

Isomorphisms: The solvable group ²"A"₂(2²) is isomorphic toan extension of the order 8 quaternion group by an elementary abelian group of order 9. ²"A"₂(3²) is isomorphic to the derived group "G"₂(2)′. ²"A"₃(2²) is isomorphic to "B"₂(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.

²"D_n"("q"²) "n" > 3 Steinberg groups, orthogonal groups

Simplicity: All simple.

Order: :$\{1over\; (4,q^n+1)\}q^\{n(n-1)\}(q^n+1)prod\_\{i=1\}^\{n-1\}(q^\{2i\}-1)$

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

Outer automorphism group: (4, "qⁿ" + 1) ·"f"·1where "q"² = "p^f".

Other names: ²"D_n"("q"),"O"_2"n"⁻("q"), "PΩ"_2"n"⁻("q"), twisted chevalley group.

Remarks: This is the group obtained from the non-split orthogonal group in dimension 2"n" bytaking the kernel of the determinant (or Dickson invariant in characteristic 2) and spinor norm maps and then killing the center. ²"D"₂("q"²) also exists, but is the same as"A"₁("q"). ²"D"₃("q"²) also exists, but is the same as ²"A"₃("q"²).

"²E₆"("q"²) Steinberg groups

Simplicity: All simple.

Order: "q"³⁶("q"¹²−1)("q"⁹+1)("q"⁸−1)("q"⁶−1)("q"⁵+1)("q"²−1)/(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"·1where "q"² = "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"("q"³) Steinberg groups

Simplicity: All simple.

Order: "q"¹²("q"⁸+"q"⁴+1)("q"⁶−1)("q"²−1)

Schur multiplier: Trivial.

Outer automorphism group: 1·"f"·1where "q"³ = "p^f".

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

Remarks: "³D₄"(2³) acts on the unique even 26 dimensional lattice of determinant 3 with no roots.

²"B"₂(2^2"n"+1) Suzuki groups

Simplicity: Simple for "n">1. The group ²"B"₂(2) is solvable.

Order: "q"²("q"²+1)("q"−1)where "q" = 2^2"n"+1.

Schur multiplier: Trivial for "n">2, elementary abelian of order 4for ²"B"₂(8).

Outer automorphism group: 1·"f"·1where "f" = 2"n"+1.

Other names: Suz(2^2"n"+1), Sz(2^2"n"+1).

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

Remarks: Suzuki group are Zassenhaus groups acting on sets of size (2^2"n"+1)²+1, and have 4 dimensional representations over the field with 2^2"n"+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.

²"F"₄(2^2"n"+1) Ree groups, Tits group

Simplicity: Simple for "n">1. The derived group ²"F"₄(2)′ is simple of index 2 in ²"F"₄(2), and is called the Tits group,named for the Belgian mathematician Jacques Tits.

Order: "q"¹²("q"⁶+1)("q"⁴−1)("q"³+1)("q"−1)where "q" = 2^2"n"+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"·1where "f" = 2"n"+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.

²"G"₂(3^2"n"+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: "q"³("q"³+1)("q"−1)where "q" = 3^2"n"+1

Schur multiplier: Trivial for "n">1 and for ²"G"₂(3)′.

Outer automorphism group: 1·"f"·1where "f" = 2"n"+1.

Other names: Ree(3^2"n"+1), R(3^2"n"+1).

Isomorphisms: The derived group ²"G"₂(3)′ is isomorphic to "A"₁(8).

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

poradic groups

Mathieu group "M"₁₁

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 "M"₁₂. The subgroup fixing a point is sometimes called "M"₁₀, and has a subgroup of index 2 isomorphic to the alternating group A₆.

Mathieu group "M"₁₂

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 "M"₂₂

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 "M"₂₄ and the full covering group of "M"₂₂ as subgroups. J. Algebra 42 (1976), 564–596."giving evidence for the existence of the group "J"₄. At the time it was thought that the full covering group of"M"₂₂ was 6·"M"₂₂. In fact "J"₄ has no subgroup 12·"M"₂₂.)

Outer automorphism group: Order 2.

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

Mathieu group "M"₂₃

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 "M"₂₄.

Mathieu group "M"₂₄

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 "J"₁ =

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 "G"₂(11), and so has a 7 dimensional representation over the field with 11 elements.

Janko group "J"₂

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 "G"₂(4).

=Janko group "J"₃ =

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

Schur multiplier: Order 3.

Outer automorphism group: Order 2.

Other names: Higman-Janko-McKay group, HJM

Remarks: "J"₃ 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 "J"₄ =

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 "Co"₁

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 "Co"₁ is the automorphism group of the Leech lattice, and is sometimes denoted by ·0.

Conway group "Co"₂

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

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Other names: ·2

Remarks: Subgroup of "Co"₁; fixes a norm 4 vector in the Leech lattice.

Conway group "Co"₃

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

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Other names: ·3

Remarks: Subgroup of "Co"₁; fixes a norm 6 vector in the Leech lattice.

Fischer group "Fi"₂₂

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 "Fi"₂₃.

Fischer group "Fi"₂₃

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 "Fi"₂₄.

Fischer group "Fi"₂₄&prime;

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

Schur multiplier: Order 3.

Outer automorphism group: Order 2.

Other names: "M"(24)′, "F"₃₊.

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 "Co"₃.

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 McLauglin graph with 275 points, and is contained in "Co"₃.

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, "F"₇

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

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: "F"₅, "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: "F"₃, "E"

Remarks: Centralizes an element of order 3 in the monster, and is contained in "E"₈(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: "F"₂

Remarks: The double cover is contained in the monster group.

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: "F"₁, "M"₁, 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	Factorized order	Group	Schur multiplier	Outer automorphism group
60	22 · 3 · 5	A5 = "A"1(4) = "A"1(5)	2	2
168	23 · 3 · 7	"A"1(7) = "A"2(2)	2	2
360	23 · 32 · 5	A6 = "A"1(9) = "B"2(2)′	6	2×2
504	23 · 32 · 7	"A"1(8) = 2"G"2(3)′	1	3
660	22 · 3 · 5 · 11	"A"1(11)	2	2
1092	22 · 3 · 7 · 13	"A"1(13)	2	2
2448	24 · 32 · 17	"A"1(17)	2	2
2520	23 · 32 · 5 · 7	A7	6	2
3420	22 · 32 · 5 · 19	"A"1(19)	2	2
4080	24 · 3 · 5 · 17	"A"1(16)	1	4
5616	24 · 33 · 13	"A"2(3)	1	2
6048	25 · 33 · 7	2"A"2(9) = "G"2(2)′	1	2
6072	23 · 3 · 11 · 23	"A"1(23)	2	2
7800	23 · 3 · 52 · 13	"A"1(25)	2	2×2
7920	24 · 32 · 5 · 11	"M"11	1	1
9828	22 · 33 · 7 · 13	"A"1(27)	2	6

See also: List of small groups

* [http://www.eleves.ens.fr:8080/home/madore/math/simplegroups.html Orders of non abelian simple groups] up to order 10,000,000,000.

Further reading

* Daniel Gorenstein, Richard Lyons, Ronald Solomon "The Classification of the Finite Simple Groups" [http://www.ams.org/online_bks/surv401/ (volume 1)] , AMS, 1994 [http://www.ams.org/online_bks/surv402/ (volume 2)] , AMS,
* 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.
* [http://brauer.maths.qmul.ac.uk/Atlas/v3/ Atlas of Finite Group Representations] : contains representations and other data for many finite simple groups, including the sporadic groups.
*"Simple Groups of Lie Type" by Roger W. Carter, ISBN 0-471-50683-4
* Mark Ronan, "Symmetry and the Monster", Oxford University Press, 2006. (Concise introduction and history written for lay
*Marcus du Sautoy, "Finding Moonshine", Fourth Estate, 2008, ISBN 978-0-00-721461-7 (another introduction for the lay reader)