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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the
classification of finite simple groups In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
states that every
finite Finite is the opposite of infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music *Infinite (band), a South Korean boy band *''Infin ...
simple group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
is
cyclic Cycle or cyclic may refer to: Anthropology and social sciences * Cyclic history, a theory of history * Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr. * Social cycle, various cycles in social scienc ...

cyclic
, or alternating, or in one of 16 families of
groups of Lie type In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a Reductive group, reductive linear algebraic group with values in a finite ...
, or one of 26
sporadic group In group theory, a sporadic group is one of the 26 exceptional Group (mathematics), groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group a ...
s. The list below gives all finite simple groups, together with their
order Order, ORDER or Orders may refer to: * Orderliness Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...
, the size of the
Schur multiplier In mathematical group theory, the Schur multiplier or Schur multiplicator is the second group homology, homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations. Examples and properties The Schur ...
, the size of the
outer automorphism group In mathematics, the outer automorphism group of a group (mathematics), group, , is the quotient group, quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually ...
, usually some small
representations ''Representations'' is an interdisciplinary journal in the humanities Humanities are academic disciplines An academic discipline or academic field is a subdivision of knowledge Knowledge is a familiarity or awareness, of someone o ...
, and lists of all duplicates.


Summary

The following table is a complete list of the 18 families of finite simple groups and the 26 sporadic simple groups, along with their orders. Any non-simple members of each family are listed, as well as any members duplicated within a family or between families. (In removing duplicates it is useful to note that no two finite simple groups have the same order, except that the group A8 = ''A''3(2) and ''A''2(4) both have order 20160, and that the group ''Bn''(''q'') has the same order as ''Cn''(''q'') for ''q'' odd, ''n'' > 2. The smallest of the latter pairs of groups are ''B''3(3) and ''C''3(3) which both have order 4585351680.) There is an unfortunate conflict between the notations for the alternating groups A''n'' and the groups of Lie type ''An''(''q''). 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 ''An''(''q'') in italic. In what follows, ''n'' is a positive integer, and ''q'' is a positive power of a prime number ''p'', with the restrictions noted. The notation (''a'',''b'') represents the greatest common divisor of the integers ''a'' and ''b''.


Cyclic group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

Cyclic group
s, 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/''p''Z, C''p'' Remarks: These are the only simple groups that are not
perfect Perfect commonly refers to: * Perfection, a philosophical concept * Perfect (grammar), a grammatical category in certain languages Perfect may also refer to: Film * Perfect (1985 film), ''Perfect'' (1985 film), a romantic drama * Perfect (2018 ...
.


Alternating group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s, A''n'', ''n'' > 4

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 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 group, alternating and symmetric groups. The covering gro ...
'' 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''. 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 Index may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastructure in the ''Halo'' series ...
2 subgroup of the
symmetric group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
of permutations of ''n'' points when ''n'' > 1.


Groups of Lie type

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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. 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 In commutative algebra and field theory (mathematics), field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative Ring (mathematics), rings with prime characteristic (algebra), characteristi ...
), and ''g'' is the order of the group of "graph automorphisms" (coming from automorphisms of the
Dynkin diagram In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
). The outer automorphism group is isomorphic to the semidirect product D \rtimes (F \times G) where all these groups D, F, G are cyclic of the respective orders ''d, f, g'', except for type D_n(q), q odd, where the group of order d=4 is C_2 \times C_2, and (only when n=4) G = S_3, the symmetric group on three elements. The notation (''a'',''b'') represents the greatest common divisor of the integers ''a'' and ''b''.


Chevalley group In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a Reductive group, reductive linear algebraic group with values in a finite ...
s, ''An''(''q''), ''Bn''(''q'') ''n'' > 1, ''Cn''(''q'') ''n'' > 2, ''Dn''(''q'') ''n'' > 3


Chevalley group In mathematics, specifically in group theory, the phrase ''group of Lie type'' usually refers to finite groups that are closely related to the group of rational points of a Reductive group, reductive linear algebraic group with values in a finite ...
s, ''E''6(''q''), ''E''7(''q''), ''E''8(''q''), ''F''4(''q''), ''G''2(''q'')


Steinberg groups, 2''An''(''q''2) ''n'' > 1, 2''Dn''(''q''2) ''n'' > 3, 2''E''6(''q''2), 3''D''4(''q''3)


Suzuki groups In the area of modern algebra known as group theory, the Suzuki groups, denoted by Sz(22''n''+1), 2''B''2(22''n''+1), Suz(22''n''+1), or ''G''(22''n''+1), form an infinite family of groups of Lie type found by , that are simple for ''n'' ≥ 1. T ...
, 2''B''2(22''n''+1)

Simplicity: Simple for ''n'' ≥ 1. The group 2''B''2(2) is solvable. Order: ''q''2 (''q''2 + 1) (''q'' − 1), where ''q'' = 22''n''+1. Schur multiplier: Trivial for ''n'' ≠ 1, elementary abelian of order 4 for 2''B''2(8). Outer automorphism group: : 1⋅''f''⋅1, where ''f'' = 2''n'' + 1. Other names: Suz(22''n''+1), Sz(22''n''+1). Isomorphisms: 2''B''2(2) is the Frobenius group of order 20. Remarks: Suzuki group are
Zassenhaus group In mathematics, a Zassenhaus group, named after Hans Zassenhaus, is a certain sort of doubly transitive permutation group very closely related to rank-1 groups of Lie type. Definition A Zassenhaus group is a permutation group ''G'' on a finite s ...
s acting on sets of size (22''n''+1)2 + 1, and have 4-dimensional representations over the field with 22''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.


Ree group In mathematics, a Ree group is a group of Lie type over a finite field constructed by from an exceptional automorphism of a Dynkin diagram that reverses the direction of the multiple bonds, generalizing the Suzuki groups found by Suzuki using a dif ...
s and
Tits group In group theory, the Tits group 2''F''4(2)′, named for Jacques Tits (), is a finite simple group of Order (group theory), order :   211 · 33 · 52 · 13 = 17,971,200. It is sometimes considered a ...
, 2''F''4(22''n''+1)

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 In group theory, the Tits group 2''F''4(2)′, named for Jacques Tits (), is a finite simple group of Order (group theory), order :   211 · 33 · 52 · 13 = 17,971,200. It is sometimes considered a ...
, named for the Belgian mathematician
Jacques Tits Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and r ...
. Order: ''q''12 (''q''6 + 1) (''q''4 − 1) (''q''3 + 1) (''q'' − 1), where ''q'' = 22''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''⋅1, where ''f'' = 2''n'' + 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 In mathematics, a (''B'', ''N'') pair is a structure on groups of Lie type that allows one to give uniform proofs of many results, instead of giving a large number of case-by-case proofs. Roughly speaking, it shows that all such groups are similar t ...
, though its automorphism group does so most authors count it as a sort of honorary group of Lie type.


Ree group In mathematics, a Ree group is a group of Lie type over a finite field constructed by from an exceptional automorphism of a Dynkin diagram that reverses the direction of the multiple bonds, generalizing the Suzuki groups found by Suzuki using a dif ...
s, 2''G''2(32''n''+1)

Simplicity: Simple for ''n'' ≥ 1. The group 2''G''2(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'' = 32''n''+1 Schur multiplier: Trivial for ''n'' ≥ 1 and for 2''G''2(3)′. Outer automorphism group: : 1⋅''f''⋅1, where ''f'' = 2''n'' + 1. Other names: Ree(32''n''+1), R(32''n''+1), E2(32''n''+1) . Isomorphisms: The derived group 2''G''2(3)′ is isomorphic to ''A''1(8). Remarks: 2''G''2(32''n''+1) has a doubly transitive permutation representation on 33(2''n''+1) + 1 points and acts on a 7-dimensional vector space over the field with 32''n''+1 elements.


Sporadic groups


Mathieu group In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( ...
s, M11, M12, M22, M23, M24


Janko group In the area of modern algebra known as group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik" ...
s, J1, J2, J3, J4


Conway group In the area of modern algebra known as group theory, the Conway groups are the three sporadic simple groups Conway group Co1, Co1, Conway group Co2, Co2 and Conway group Co3, Co3 along with the related finite group Leech lattice#Symmetries, Co0 in ...
s, Co1, Co2, Co3


Fischer group In the area of modern algebra known as group theory, the Fischer groups are the three sporadic simple groups Fischer group Fi22, Fi22, Fischer group Fi23, Fi23 and Fischer group Fi24, Fi24 introduced by . 3-transposition groups The Fischer g ...
s, Fi22, Fi23, Fi24


Higman–Sims group In the area of modern algebra known as group theory, the Higman–Sims group HS is a sporadic simple group of Order (group theory), order :   29⋅32⋅53⋅7⋅11 = 44352000 : ≈ 4. The Schur multiplier has order 2, t ...
, 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 In the area of modern algebra known as group theory, the Held group ''He'' is a sporadic simple group of Order (group theory), order :   21033527317 = 4030387200 : ≈ 4. History ''He'' is one of the 26 sporadic groups and was fou ...
, 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, HTH Remarks: Centralizes an element of order 7 in the monster group.


Rudvalis group In the area of modern algebra known as group theory, the Rudvalis group ''Ru'' is a sporadic simple group of Order (group theory), order :   214335371329 : = 145926144000 : ≈ 1. History ''Ru'' is one of the 26 sporadic groups and ...
, 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 In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as . This inte ...
s.


Suzuki sporadic group In the area of modern algebra known as group theory, the Suzuki group ''Suz'' or ''Sz'' is a sporadic simple group of Order (group theory), order :   213 · 37 · 52 · 7 · 11 · 13 = 448345497600 : ≈ 4. History ''Suz'' is one ...
, 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 integer In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s. It is not related to the Suzuki groups of Lie type.


O'Nan group In the area of abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (m ...
, 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 In the area of modern algebra known as group theory, the Harada–Norton group ''HN'' is a sporadic simple group of Order (group theory), order :   214365671119 : = 273030912000000 : ≈ 3. History and properties ''HN'' is one of t ...
, 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 In the area of modern algebra known as group theory, the Lyons group ''Ly'' or Lyons-Sims group ''LyS'' is a sporadic simple group of Order (group theory), order :    283756711313767 : = 51765179004000000 : ≈ 5. History ''Ly'' i ...
, 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), 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: ''F''2 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 In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having Order (group theory), order    2463205976 ...
, 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, M1, Monster group, Friendly giant, Fischer's monster. Remarks: Contains all but 6 of the other sporadic groups as subquotients. Related to
monstrous moonshine In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the j-invariant, ''j'' function. The term was coined by John Horton Conway, John Conway ...
. The monster is the automorphism group of the 196,883-dimensional
Griess algebra In mathematics, the Griess algebra is a commutative Algebra over a field#Non-associative algebras, non-associative algebra on a real number, real vector space of dimension 196884 that has the Monster group ''M'' as its automorphism group. It is name ...
and the infinite-dimensional monster
vertex operator algebra #REDIRECT Vertex operator algebra In mathematics, a vertex operator algebra (VOA) is an algebraic structure that plays an important role in two-dimensional conformal field theory and string theory. In addition to physical applications, vertex ope ...
, and acts naturally on the monster Lie algebra.


Non-cyclic simple groups of small order

(Complete for orders less than 100,000) lists the 56 non-cyclic simple groups of order less than a million.


See also

*
List of small groups The following list in mathematics contains the finite groups of small order of a group, order up to group isomorphism. Counts For ''n'' = 1, 2, … the number of nonisomorphic groups of order ''n'' is : 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, ...


Notes


References


Further reading

*''Simple Groups of Lie Type'' by Roger W. Carter, * 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 Daniel E. Gorenstein (January 1, 1923 – August 26, 1992) was an American mathematician. He earned his undergraduate and graduate degrees at Harvard University, where he earned his Ph.D. in 1950 under Oscar Zariski, introducing in his dissertati ...

Daniel Gorenstein
, Richard Lyons, Ronald Solomon ''The Classification of the Finite Simple Groups'
(volume 1)
AMS, 199
(volume 3)
AMS, 1998 * *{{Citation , last1=Wilson , first1=Robert A. , authorlink = Robert Arnott Wilson , title=The finite simple groups , publisher=
Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing Publishing is the activity of making information, literature, music, software and other content available to the public for sale or for free. ...
, location=Berlin, New York , series=
Graduate Texts in Mathematics Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level File:CCMDonation49.JPG, Student receives degree from the Monterrey Institute of Technology and Higher Education, Mexico City, 2013 A graduate school (sometimes sh ...
251 , isbn=978-1-84800-987-5 , doi=10.1007/978-1-84800-988-2 , year=2009 , zbl=1203.20012 , volume=251
Atlas of Finite Group Representations
contains
representations ''Representations'' is an interdisciplinary journal in the humanities Humanities are academic disciplines An academic discipline or academic field is a subdivision of knowledge Knowledge is a familiarity or awareness, of someone o ...
and other data for many finite simple groups, including the sporadic groups.
Orders of non abelian simple groups
up to 1010, and on to 1048 with restrictions on rank.


External links



up to order 10,000,000,000. Finite simple groups Group theory Sporadic groups