List of finite simple groups
Encyclopedia
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 group
s.
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
Bn(q) has the same order as Cn(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).
Cyclic group
Simplicity: Always simple.
Order: p, where p is a prime number
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
.
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: Altn.
There is an unfortunate conflict with the notation for the (unrelated) groups An(q), and some authors use various different fonts for An to distinguish them. In particular,
in this article we make the distinction by setting the alternating groups An in Roman font and the Lie-type groups An(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.
solvable, the others are simple.
Order:
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 = pf.
Other names: Projective special linear groups, PSLn+1(q),
Ln+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 group
s GLn+1(q) by
taking the elements of determinant 1 (giving the special linear group
s SLn+1(q)) and then quotienting out by the center.
Order:
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 = pf.
Other names: O2n+1(q),
Ω2n+1(q) (for q odd).
Isomorphisms: 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).
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.
Cn(q) n > 2 Chevalley groups, symplectic group
Simplicity: All simple.
Order:
Schur multiplier: (2,q − 1) except for C3(2)
(order 2).
Outer automorphism group:
(2, q − 1) ·f·1 where q = pf.
Other names: Projective symplectic group, PSp2n(q), PSpn(q) (not recommended), S2n(q), Abelian group (archaic).
Isomorphisms: Cn(2m) is isomorphic to Bn(2m)
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).
Dn(q) n > 3 Chevalley groups, orthogonal group
Simplicity: All simple.
Order:
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 = pf, and S3 is the symmetric group on 3 points of order 6.
Other names: O2n+(q), PΩ2n+(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).
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 = pf.
Other names: Exceptional Chevalley group.
Remarks: Has two representations of dimension 27, and acts on the Lie algebra of dimension 78.
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 = pf.
Other names: Exceptional Chevalley group.
Remarks: Has a representations of dimension 56, and acts on the corresponding Lie algebra of dimension 133.
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 = pf.
Other names: Exceptional Chevalley group.
Remarks: It acts on the corresponding Lie algebra of dimension 248. E8(3) contains the Thompson simple group.
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 = pf.
Other names: Exceptional Chevalley group.
Remarks: These groups act on 27 dimensional exceptional Jordan algebra
s, 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.
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 = pf.
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.
2An(q2) n > 1 Steinberg groups, unitary group
Simplicity: 2A2(22) is solvable, the others are simple.
Order:
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 = 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)
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.
2Dn(q2) n > 3 Steinberg groups, orthogonal group
Simplicity: All simple.
Order:
Schur multiplier: Cyclic of order (4, qn + 1).
Outer automorphism group:
(4, qn + 1) ·f·1
where q2 = pf.
Other names: 2Dn(q),
O2n−(q), PΩ2n−(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).
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 = pf.
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.
Order:
q12
(q8+q4+1)
(q6−1)
(q2−1)
Schur multiplier: Trivial.
Outer automorphism group:
1·f·1
where q3 = pf.
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(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 group
s 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.
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 33(2n+1)+1 points and acts on a 7 dimensional vector space over the field with 32n+1 elements.
Mathieu group
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
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
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
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
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
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.
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
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
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
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
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
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
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
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
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.
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
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
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
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 integer
s.
Suzuki sporadic group
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 integer
s. It is not related to the Suzuki groups of Lie type.
O'Nan group
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.
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
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
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
Order:
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
Order:
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
.
(Complete for orders less than 100000)
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the classification of finite simple groups
Classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...
states that
every finite simple group
Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
is cyclic
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
, or alternating, or in one of 16 families of groups of Lie type (including the Tits group
Tits group
In mathematics, the Tits group 2F4′ is a finite simple group of order 17971200 = 211 · 33 · 52 · 13 found by ....
, which strictly speaking is not of Lie type),
or one of 26 sporadic group
Sporadic group
In the mathematical field of group theory, a sporadic group is one of the 26 exceptional groups in the classification of finite simple groups. A simple group is a group G that does not have any normal subgroups except for the subgroup consisting only of the identity element, and G itself...
s.
The list below gives all finite simple groups, together with their order
Order (group theory)
In group theory, a branch of mathematics, the term order is used in two closely related senses:* The order of a group is its cardinality, i.e., the number of its elements....
, the size of the Schur multiplier
Schur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2 of a group G.It was introduced by in his work on projective representations.-Examples and properties:...
, the size of the outer automorphism group
Outer automorphism group
In mathematics, the outer automorphism group of a group Gis the quotient Aut / Inn, where Aut is the automorphism group of G and Inn is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out...
, usually some small representations
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
, 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
Bn(q) has the same order as Cn(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
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
.
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).
Cyclic groupCyclic groupIn group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
s Zp
Simplicity: Always simple.Order: p, where p is a prime number
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
Perfect group
In mathematics, more specifically in the area of modern algebra known as group theory, a group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian quotients...
.
An, 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
Covering groups of the alternating and symmetric groups
In the mathematical area of group theory, the covering groups of the alternating and symmetric groups are groups that are used to understand the projective representations 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: Altn.
There is an unfortunate conflict with the notation for the (unrelated) groups An(q), and some authors use various different fonts for An to distinguish them. In particular,
in this article we make the distinction by setting the alternating groups An in Roman font and the Lie-type groups An(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
Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
2 subgroup of the symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
of permutations of n points when n > 1.
An(q) Chevalley groups, linear groups
Simplicity: A1(2) and A1(3) aresolvable, the others are simple.
Order:
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 = pf.
Other names: Projective special linear groups, PSLn+1(q),
Ln+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 group
General linear group
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
s GLn+1(q) by
taking the elements of determinant 1 (giving the special linear group
Special linear group
In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
s SLn+1(q)) and then quotienting out by the center.
Bn(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:
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 = pf.
Other names: O2n+1(q),
Ω2n+1(q) (for q odd).
Isomorphisms: 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).
Remarks: This is the group obtained from the orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
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.
Cn(q) n > 2 Chevalley groups, symplectic groupSymplectic groupIn mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
s
Simplicity: All simple.Order:
Schur multiplier: (2,q − 1) except for C3(2)
(order 2).
Outer automorphism group:
(2, q − 1) ·f·1 where q = pf.
Other names: Projective symplectic group, PSp2n(q), PSpn(q) (not recommended), S2n(q), Abelian group (archaic).
Isomorphisms: Cn(2m) is isomorphic to Bn(2m)
Remarks: This group is obtained from the symplectic group
Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
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).
Dn(q) n > 3 Chevalley groups, orthogonal groupOrthogonal groupIn mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
s
Simplicity: All simple.Order:
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 = pf, and S3 is the symmetric group on 3 points of order 6.
Other names: O2n+(q), PΩ2n+(q). "Hypoabelian group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
" 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
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
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
Triality
In mathematics, triality is a relationship between three vector spaces, analogous to the duality relation between dual vector spaces. Most commonly, it describes those special features of the Dynkin diagram D4 and the associated Lie group Spin, the double cover of 8-dimensional rotation group SO,...
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 = pf.
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 = pf.
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 = pf.
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 = pf.
Other names: Exceptional Chevalley group.
Remarks: These groups act on 27 dimensional exceptional Jordan algebra
Jordan algebra
In abstract algebra, a Jordan algebra is an algebra over a field whose multiplication satisfies the following axioms:# xy = yx # = x ....
s, 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 = pf.
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.
2An(q2) n > 1 Steinberg groups, unitary groupUnitary groupIn mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...
s
Simplicity: 2A2(22) is solvable, the others are simple.Order:
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 = 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)
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
Unitary group
In mathematics, the unitary group of degree n, denoted U, is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL...
in n+1 dimensions by taking the subgroup of elements of determinant 1 and then quotienting out by the center.
2Dn(q2) n > 3 Steinberg groups, orthogonal groupOrthogonal groupIn mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
s
Simplicity: All simple.Order:
Schur multiplier: Cyclic of order (4, qn + 1).
Outer automorphism group:
(4, qn + 1) ·f·1
where q2 = pf.
Other names: 2Dn(q),
O2n−(q), PΩ2n−(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
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
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 = pf.
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 = pf.
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 group2B2(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 group
Zassenhaus group
In mathematics, a Zassenhaus group, named after Hans Julius Zassenhaus, is a certain sort of doubly transitive permutation group very closely related to rank-1 groups of Lie type.- Definition :...
s 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 groupTits groupIn mathematics, the Tits group 2F4′ is a finite simple group of order 17971200 = 211 · 33 · 52 · 13 found by ....
Simplicity: Simple for n≥1. The derived group 2F4(2)′ is simple of index 2in 2F4(2), and is called the Tits group
Tits group
In mathematics, the Tits group 2F4′ is a finite simple group of order 17971200 = 211 · 33 · 52 · 13 found by ....
,
named for the Belgian mathematician Jacques Tits
Jacques Tits
Jacques Tits is a Belgian and French mathematician who works on group theory and geometry and who introduced Tits buildings, the Tits alternative, and the Tits group.- Career :Tits received his doctorate in mathematics at the age of 20...
.
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 33(2n+1)+1 points and acts on a 7 dimensional vector space over the field with 32n+1 elements.
Mathieu groupMathieu groupIn the mathematical field of group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple groups he discovered and reported in papers in 1861 and 1873; these were the first sporadic simple groups discovered...
M11
Order: 24 · 32 · 5 · 11=7920Schur multiplier: Trivial.
Outer automorphism group: Trivial.
Remarks: A 4-transitive permutation group
Permutation group
In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ; the relationship is often written as...
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 groupMathieu groupIn the mathematical field of group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple groups he discovered and reported in papers in 1861 and 1873; these were the first sporadic simple groups discovered...
M12
Order: 26 · 33 · 5 · 11=95040Schur multiplier: Order 2.
Outer automorphism group: Order 2.
Remarks: A 5-transitive permutation group
Permutation group
In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ; the relationship is often written as...
on 12 points.
Mathieu groupMathieu groupIn the mathematical field of group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple groups he discovered and reported in papers in 1861 and 1873; these were the first sporadic simple groups discovered...
M22
Order: 27 · 32 · 5 · 7 · 11 = 443520Schur 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
Permutation group
In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ; the relationship is often written as...
on 22 points.
Mathieu groupMathieu groupIn the mathematical field of group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple groups he discovered and reported in papers in 1861 and 1873; these were the first sporadic simple groups discovered...
M23
Order: 27 · 32 · 5 · 7 · 11 · 23=10200960Schur multiplier: Trivial.
Outer automorphism group: Trivial.
Remarks: A 4-transitive permutation group
Permutation group
In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ; the relationship is often written as...
on 23 points, contained in M24.
Mathieu groupMathieu groupIn the mathematical field of group theory, the Mathieu groups, named after the French mathematician Émile Léonard Mathieu, are five finite simple groups he discovered and reported in papers in 1861 and 1873; these were the first sporadic simple groups discovered...
M24
Order: 210 · 33 · 5 · 7 · 11 · 23= 244823040Schur multiplier: Trivial.
Outer automorphism group: Trivial.
Remarks: A 5-transitive permutation group
Permutation group
In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ; the relationship is often written as...
on 24 points.
Janko groupJanko group J1In mathematics, the smallest Janko group, J1, is a simple sporadic group of order 175560. It was originally described by Zvonimir Janko and was the first sporadic group to be found since the discovery of the Mathieu groups in the 19th century...
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 = 604800Schur 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 groupJanko group J3In mathematics, the third Janko group J3, also known as the Higman-Janko-McKay group, is a finite simple sporadic group of order 50232960. Evidence for its existence was uncovered by , and it was shown to exist by...
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
Unitary representation
In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π is a unitary operator for every g ∈ G...
over the field with 4 elements.
Janko groupJanko group J4In mathematics, the fourth Janko group J4 is the sporadic finite simple group of order 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43 = 86775571046077562880 whose existence was suggested by Zvonimir Janko . Its existence and uniqueness was shown by Simon P. Norton and others in 1980...
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 groupConway groupIn mathematics, the Conway groups Co1, Co2, and Co3 are three sporadic groups discovered by John Horton Conway.The largest of the Conway groups, Co1, of order...
Co1
Order: 221 · 39 · 54 · 72 · 11 · 13 · 23 = 4157776806543360000Schur 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
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space E24 found by .-History:Many of the cross-sections of the Leech lattice, including the Coxeter–Todd lattice and Barnes–Wall lattice, in 12 and 16 dimensions, were found much earlier than...
, and is sometimes denoted by ·0.
Conway groupConway groupIn mathematics, the Conway groups Co1, Co2, and Co3 are three sporadic groups discovered by John Horton Conway.The largest of the Conway groups, Co1, of order...
Co2
Order: 218 · 36 · 53 · 7 · 11 · 23 = 42305421312000Schur multiplier: Trivial.
Outer automorphism group: Trivial.
Other names: ·2
Remarks: Subgroup of Co1; fixes a norm 4 vector in the Leech lattice
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space E24 found by .-History:Many of the cross-sections of the Leech lattice, including the Coxeter–Todd lattice and Barnes–Wall lattice, in 12 and 16 dimensions, were found much earlier than...
.
Conway groupConway groupIn mathematics, the Conway groups Co1, Co2, and Co3 are three sporadic groups discovered by John Horton Conway.The largest of the Conway groups, Co1, of order...
Co3
Order: 210 · 37 · 53 · 7 · 11 · 23 = 495766656000Schur multiplier: Trivial.
Outer automorphism group: Trivial.
Other names: ·3
Remarks: Subgroup of Co1; fixes a norm 6 vector in the Leech lattice
Leech lattice
In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space E24 found by .-History:Many of the cross-sections of the Leech lattice, including the Coxeter–Todd lattice and Barnes–Wall lattice, in 12 and 16 dimensions, were found much earlier than...
.
Fischer groupFischer groupIn mathematics, the Fischer groups are the three sporadic simple groups Fi22, Fi23,Fi24' introduced by .- 3-transposition groups :...
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 groupFischer groupIn mathematics, the Fischer groups are the three sporadic simple groups Fi22, Fi23,Fi24' introduced by .- 3-transposition groups :...
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 groupFischer groupIn mathematics, the Fischer groups are the three sporadic simple groups Fi22, Fi23,Fi24' introduced by .- 3-transposition groups :...
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 = 44352000Schur 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 groupMcLaughlin group (mathematics)In the mathematical discipline known as group theory, the McLaughlin group McL is a sporadic simple group of order 27 · 36 · 53· 7 · 11 = 898,128,000, discovered by as an index 2 subgroup of a rank 3 permutation group acting on the McLaughlin graph with 275 =1+112+162 vertices...
McL
Order: 27 · 36 · 53· 7 · 11 = 898128000Schur 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 groupHeld groupIn the mathematical field of group theory, the Held group He is one of the 26 sporadic simple groups, and has order...
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 groupRudvalis groupIn the mathematical field of group theory, the Rudvalis group Ru is a sporadic simple group of order-Properties:The Rudvalis group acts as a rank 3 permutation group on 4060 points, with one point stabilizer the Ree 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 integer
Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z[i]. The Gaussian integers are a special case of the quadratic...
s.
Suzuki sporadic groupSuzuki sporadic groupIn mathematical group theory, the Suzuki group Suz or Sz is a sporadic simple group of order 213 · 37 · 52· 7 · 11 · 13 = 448345497600 discovered by . It is not related to the Suzuki groups of Lie type....
Suz
Order: 213 · 37 · 52· 7 · 11 · 13 = 448345497600Schur 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 integer
Eisenstein integer
In mathematics, Eisenstein integers , also known as Eulerian integers , are complex numbers of the formz = a + b\omega \,\!where a and b are integers and...
s. It is not related to the Suzuki groups of Lie type.
O'Nan groupO'Nan groupIn the mathematical field of group theory, the O'Nan group O'N is a sporadic simple group of orderThe Schur multiplier has order 3, and its outer automorphism group has order 2...
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 groupLyons groupIn the mathematical field of group theory, the Lyons group Ly , is a sporadic simple group of order...
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 groupThompson group (finite)In the mathematical field of group theory, the Thompson group Th, found by and constructed by , is a sporadic simple group of orderThe centralizer of an element of order 3 of type 3C in the Monster group is a product of the Thompson group and a group of order 3, as a result of which the Thompson...
Th
Order: 215 · 310 · 53 · 72 · 13 · 19 · 31 = 90745943887872000Schur 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 groupBaby Monster groupIn the mathematical field of group theory, the Baby Monster group B is a group of orderThe Baby Monster group is one of the sporadic groups, and has the second highest order of these, with the highest order being that of the 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 groupMonster groupIn the mathematical field of group theory, the Monster group M or F1 is a group of finite order:...
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
Monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is a term devised by John Horton Conway and Simon P. Norton in 1979, used to describe the connection between the monster group M and modular functions .- History :Specifically, Conway and Norton, following an initial observationby John...
. The monster is the automorphism group of the 196884 dimensional Griess algebra
Griess algebra
In mathematics, the Griess algebra is a commutative non-associative algebra on a real vector space of dimension 196884 that has the Monster group M as its automorphism group. It is named after mathematician R. L. Griess, who constructed it in 1980 and subsequently used it in 1982 to construct M...
and the infinite dimensional monster vertex operator algebra
Vertex operator algebra
In mathematics, a vertex operator algebra is an algebraic structure that plays an important role in conformal field theory and related areas of physics...
, and acts naturally on the monster Lie algebra
Monster Lie algebra
In mathematics, the monster Lie algebra is an infinitedimensional generalized Kac–Moody algebra acted on by the monster group, which was used to prove the monstrous moonshine conjectures.- Structure :...
.
Non-cyclic simple groups of small order
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)
External links
- Orders of non abelian simple groups up to order 10,000,000,000.