This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# FiniteGroupData

 FiniteGroupData gives the value of the specified property for the finite group specified by name. FiniteGroupDatagives a list of finite groups in the specified class.
• 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.
• FiniteGroupData gives a list of available named finite groups, and small members of infinite families.
• FiniteGroupData 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 gives a list of all standard finite groups of order greater than n1 and smaller than n2.
• FiniteGroupData 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} crystallographic group {"CyclicGroup",n} cyclic group {"CyclicGroupUnits",n} unit 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 {"ChevalleyGroupE",{n,q}} exceptional Chevalley group {"ChevalleyGroupF",{4,q}} exceptional Chevalley group {"ChevalleyGroupG",{2,q}} exceptional Chevalley group {"ReeGroupF",q} Ree group 2F4(q),q=22n+1 {"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,...}} semi-direct product of 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 "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 gives a list of finite groups of order n in the specified class.
• FiniteGroupData gives a list of finite groups of order less than or equal to n in the specified class.
• FiniteGroupData gives a list of all standard finite groups of order greater than or equal to n in the specified class.
• FiniteGroupData 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 gives a hyperlink to additional information on the specified group.
The quaternion group:
Multiplication table of the quaternion group:
The quaternion group:
 Out[1]=
 Out[2]=

Multiplication table of the quaternion group:
 Out[1]=
 Scope   (11)
Named finite groups:
All finite groups available:
Cyclic groups:
Infinite family of alternating groups given as :
Abelian groups:
Get a list of possible properties: