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

More information , ...

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

Close

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)

More information , ...

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

Close

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.

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

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

Schur multiplier: Order 2.

Outer automorphism group: Order 2.

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

McLaughlin group, McL

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

Schur multiplier: Order 3.

Outer automorphism group: Order 2.

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

Held group, He

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

Schur multiplier: Trivial.

Outer automorphism group: Order 2.

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

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

Rudvalis group, Ru

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

Schur multiplier: Order 2.

Outer automorphism group: Trivial.

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

Suzuki sporadic group, Suz

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

Schur multiplier: Order 6.

Outer automorphism group: Order 2.

Other names: Sz

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

O'Nan group, O'N

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

Schur multiplier: Order 3.

Outer automorphism group: Order 2.

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

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

Harada–Norton group, HN

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

Schur multiplier: Trivial.

Outer automorphism group: Order 2.

Other names: F₅, D

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

Lyons group, Ly

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

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Other names: Lyons–Sims group, LyS

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

Thompson group, Th

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

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

Other names: F₃, E

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

Baby Monster group, B

Order:

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

= 4154781481226426191177580544000000

Schur multiplier: Order 2.

Outer automorphism group: Trivial.

Other names: F₂

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

Fischer–Griess Monster group, M

Order:

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

= 808017424794512875886459904961710757005754368000000000

Schur multiplier: Trivial.

Outer automorphism group: Trivial.

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

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