BUILT-IN MATHEMATICA SYMBOL

# 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.

## DetailsDetails

• Groups can be specified by names such as , , and .
• 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.
• 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.
• gives a list of nonisomorphic finite groups of order n.
• FiniteGroupData[;;n] gives a list of nonisomorphic groups with orders up to n.
• FiniteGroupData[n1;;n2] gives a list of nonisomorphic 2F4(q),q=22n+1 finite groups of order greater than and smaller than .
• 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 {"CrystallographicPointGroup",n} 3-dimensional crystallographic point group, {"CyclicGroup",n} cyclic group {"CyclicGroupUnits",n} units group of the cyclic group {"DicyclicGroup",n} dicyclic group {"DihedralGroup",n} dihedral group {"SymmetricGroup",n} symmetric group {"ProjectiveSpecialLinearGroup",{n,q}} projective special linear group {"ProjectiveSymplecticGroup",{n,q}} projective symplectic group {"ChevalleyGroupB",{n,q}} exceptional Chevalley group {"ChevalleyGroupD",{n,q}} exceptional Chevalley group {"ChevalleyGroupE",{n,q}} exceptional Chevalley group , {"ChevalleyGroupF",{4,q}} exceptional Chevalley group {"ChevalleyGroupG",{2,q}} exceptional Chevalley group {"ReeGroupF",q} Ree group {"ReeGroupG",q} Ree group sporadic group, {"SteinbergGroupA",{n,q}} Steinberg unitary group {"SteinbergGroupD",{n,q}} Steinberg orthogonal group Steinberg orthogonal group {"SteinbergGroupE",q} Steinberg orthogonal group Suzuki group
• Special group specifications include:
•  {"AbelianGroup",{m,n,...}} Abelian group {"DirectProduct",{group1,group2,...}} direct product of groups {"SemidirectProduct",{group1,group2}} semi-direct product , with normal in the result
• 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" -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 "PermutationRepresentation" representation as permutation lists "PermutationGroupRepresentation" representation as a Mathematica permutation group "Transitivity" transitivity level
• Other properties include:
•  "DefiningRelations" relations that describe the group algebra "CayleyGraph" 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["Classes"] gives a list of all supported classes.
• 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"] gives the list of finite groups in belonging to 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 finite groups of order greater than or equal to n in the specified class.
• FiniteGroupData["class", n1;;n2] gives a list of finite groups of order greater than and less than in the specified class.
• Classes of groups include:
•  "Abelian" Abelian "Alternating" alternating "Cyclic" cyclic "Perfect" perfect "Simple" simple "Solvable" solvable "Sporadic" sporadic "Symmetric" symmetric "Transitive" transitive
• Negative classes of groups include:
•  "Nonabelian" not Abelian "Nonalternating" not alternating "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.
• Using FiniteGroupData may require internet connectivity.

## ExamplesExamplesopen allclose all

### Basic Examples (2)Basic Examples (2)

The quaternion group:

 Out[1]=
 Out[2]=

Multiplication table of the quaternion group:

 Out[1]=