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FiniteGroupData

FiniteGroupData[name, "property"]
gives the value of the specified property for the finite group specified by name.
FiniteGroupData["class"]
gives a list of finite groups in the specified class.
  • Groups can be specified by names such as "C3", "Quaternion" and {"SymmetricGroup", 4}.
  • FiniteGroupData[name] gives the standard form of the name for the group specified by name.
  • FiniteGroupData[patt] gives a list of all group names that match the string pattern patt.
  • FiniteGroupData[] gives a list of available named finite groups, and small members of infinite families.
  • FiniteGroupData[{n, id}, ...] gives data for the finite group of order n with identifier id.
  • FiniteGroupData[n] gives a list of all available standard finite groups of order n.
  • FiniteGroupData[n1;;n2] gives a list of all standard finite groups of order greater than n1 and smaller than n2.
  • FiniteGroupData[{"type", id}, ...] gives data for the finite group of the specified type with identifier id. The identifier is typically an integer, or a list of integers.
  • Groups enumerated by integers include:
{"AlternatingGroup",n}alternating group An
{"CrystallographicPointGroup",n}crystallographic group
{"CyclicGroup",n}cyclic group DoubleStruckCapitalZn
{"CyclicGroupUnits",n}unit group DoubleStruckCapitalZn* of the cyclic group
{"DicyclicGroup",n}dicyclic group Dicn
{"DihedralGroup",n}dihedral group Dn
{"SymmetricGroup",n}symmetric group Sn
{"ProjectiveSpecialLinearGroup",{n,q}}projective special linear group PSL(n, q)
{"ProjectiveSymplecticGroup",{n,q}}projective symplectic group PSp(n, q)
{"ChevalleyGroupE",{n,q}}exceptional Chevalley group En(q)
{"ChevalleyGroupF",{4,q}}exceptional Chevalley group F4(q)
{"ChevalleyGroupG",{2,q}}exceptional Chevalley group G2(q)
{"ReeGroupF",q}Ree group 2F4(q), q=22n+1
{"ReeGroupG",q}Ree group 2G2(q), q=32n+1
{"SporadicGroup",n}sporadic group
{"SteinbergGroupA",{n,q}}Steinberg unitary group 2An(q2)
{"SteinbergGroupD",{n,q}}Steinberg orthogonal group 2Dn(q2)
{"SteinbergGroupD",q}Steinberg orthogonal group 3D4(q3)
{"SteinbergGroupE",q}Steinberg orthogonal group 2E6(q2)
{"SuzukiGroup",n}Suzuki group 2B2(22n+1)
  • Special group specifications include:
{"AbelianGroup",{m,n,...}}Abelian group DoubleStruckCapitalZm×DoubleStruckCapitalZn×...
{"DirectProduct",{group1,group2,...}}direct product of groups
{"SemiDirectProduct",{group1,...}}semi-direct product of groups
  • FiniteGroupData["Properties"] gives a list of properties available for groups.
  • Basic group properties include:
"Center"center of group (commuting elements)
"ClassNumber"class number
"CommutatorSubgroup"commutator subgroup
"ConjugacyClasses"conjugacy classes
"ElementNames"list of names of elements
"Generators"generator elements
"InverseGenerators"inverses of the generator elements
"Inverses"inverse elements
"MultiplicationTable"multiplication table
"NormalSubgroups"normal subgroups
"Order"total number of elements
"Subgroups"subgroups
"SylowSubgroups"p-Sylow subgroups
  • The elements of a group are specified by integers running from 1 to the order of the group, with 1 corresponding to the identity element.
  • Group structure properties include:
"AutomorphismGroup"automorphism group
"InnerAutomorphismGroup"inner automorphism group
"IsomorphicGroups"list of isomorphic groups
"OuterAutomorphismGroup"outer automorphism group
"QuotientGroups"list of quotient groups
"SchurCover"Schur cover
"SchurMultiplier"Schur multiplier
  • Permutation group properties include:
"CycleIndex"cycle index
"Cycles"cycles
"Permutations"representation as permutations
  • Other properties include:
"DefiningRelations"relations that describe the group algebra
"CayleyGraph"Cayley graph
"CayleyGraphImage"image of the Cayley graph
"CycleGraph"cycle graph
  • Group representation properties include:
"CharacterTable"group element characters
"MatrixRepresentation"representation as matrices
"SpaceRepresentation"representation as 3D Cartesian coordinate transformations
  • Additional properties for crystallographic point groups include:
"CrystalForm"crystal form
"CrystalSystem"crystal system
"HermannMauguin"Hermann-Mauguin notation
"Orbifold"orbifold
"PointGroupType"point group type
"Schoenflies"Schoenflies notation
"Shubnikov"Shubnikov notation
  • FiniteGroupData[name, "Classes"] gives a list of all the classes the specified group is in.
  • FiniteGroupData[name, "class"] gives True or False depending on whether a group corresponding to name is in the specified class.
  • FiniteGroupData["class", n] gives a list of finite groups of order n in the specified class.
  • FiniteGroupData["class", ;;n] gives a list of finite groups of order less than or equal to n in the specified class.
  • FiniteGroupData["class", n;;] gives a list of all standard finite groups of order greater than or equal to n in the specified class.
  • FiniteGroupData["class", n1;;n2] gives a list of all standard finite groups of order greater than n1 and less than n2 in the specified class.
  • Classes of groups include:
"Abelian"Abelian
"Cyclic"cyclic
"Perfect"perfect
"Simple"simple
"Solvable"solvable
"Sporadic"sporadic
"Symmetric"symmetric
"Transitive"transitive
  • Negative classes of groups include:
"Nonabelian"not Abelian
"Noncyclic"not cyclic
"Nonperfect"not perfect
"Nonsimple"not simple
"Nonsolvable"not solvable
"Nonsporadic"not sporadic
"Nonsymmetric"not symmetric
"Nontransitive"not transitive
  • Naming-related properties include:
"AlternateNames"alternate English names, as strings
"AlternateStandardNames"alternate standard Mathematica names
"Name"English name as a string
"Notation"group notation
"ShortName"short name as a string
"StandardName"standard Mathematica name
  • FiniteGroupData[name, "Information"] gives a hyperlink to additional information on the specified group.
The quaternion group:
Multiplication table of the quaternion group:
Named finite groups:
All finite groups available:
Cyclic groups:
Infinite family of alternating groups given as {"AlternatingGroup", n}:
Abelian groups:
Get a list of possible properties:
Get more information about a group:
Find the English name:
Standard typeset notations:
A property value can be any valid Mathematica expression:
Symbolic expressions:
FiniteGroupCount[n] gives the number of finite groups of order n:
Name of finite groups of order 1225:
FiniteAbelianGroupCount[n] gives the number of Abelian groups of order n:
All finite groups of order 1225 are Abelian:
Results from FiniteGroupData may contain isomorphic groups:
There is only one group of order 1:
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