Notable mathematical groups of finite order.

Entity["FiniteGroup",name] or

represents an entity of type "FiniteGroup".

[prop]

gives the value of a specified property.

[{propi,}]

gives the value of a list of properties.

EntityClass["FiniteGroup",{propispeci,}]

represents a class of entities with values of propi defined by speci.

Sample Entities

Sample Entity Classes

Properties

  • AbelianAbelian
    AlternateNamesalternate names
    Alternatingalternating
    AssociatedPeopleassociated people
    AutomorphismGroupautomorphism group
    BravaisLatticesBravais lattice
    CayleyGraphCayley graph
    Centercenter
    CenterElementscenter elements
    CharacterTablecharacter table
    Classesraw classes
    ClassNumberclass number
    CommutatorSubgroupcommutator subgroup
    CommutatorSubgroupElementscommutator subgroup elements
    ConjugacyClassesconjugacy classes
    ConjugacyClassNamesconjugacy class names
    ConjugacyClassSizesconjugacy class sizes
    CrystalFamilycrystal family
    CrystalFormcrystal form
    CrystalSystemcrystal system
    CycleGraphcycle graph
    CycleIndexcycle index
    Cyclescycles
    Cycliccyclic
    DefiningRelationsdefining relations
    Dihedraldihedral
    ElementNameselement names
    Elementselements
    EntityClassesentity classes
    Exponentexponent
    Generatorsindexed generators
    HermannMauguinHermannMauguin notation
    Informationinformation
    InnerAutomorphismGroupinner automorphism group
    InverseGeneratorsinverse generators
    Inversesinverses
    IsomorphicGroupsisomorphic groups
    LatticeSystemslattice system
    MatrixRepresentationmatrix representation
    MultiplicationTableindexed multiplication table
    Namename
    Nonabeliannonabelian
    Nonalternatingnonalternating
    Noncyclicnoncyclic
    Nondihedralnondihedral
    Nonperfectnonperfect
    Nonsimplenonsimple
    Nonsolvablenonsolvable
    Nonsporadicnonsporadic
    Nonsymmetricnonsymmetric
    Nontransitivenontransitive
    NormalSubgroupElementsnormal subgroup elements
    NormalSubgroupsnormal subgroups
    Notationnotation
    Orbifoldorbifold notation
    Orderorder
    OuterAutomorphismGroupouter automorphism group
    ParameterRangeparameter range
    Perfectperfect
    PermutationGroupRepresentationpermutation group representation
    PermutationRepresentationpermutation representation
    PointGroupTypetype
    QuotientGroupsquotient groups
    RepresentationDimensionsrepresentation dimensions
    RepresentationNamesrepresentation names
    SchoenfliesSchönflies notation
    SchurMultiplierSchur multiplier
    ShortNameshort name
    ShubnikovShubnikov magnetic group
    Simplesimple
    Solvablesolvable
    SpaceRepresentationspace representation
    Sporadicsporadic
    SubgroupElementssubgroup elements
    Subgroupssubgroups
    SylowSubgroupCountsSylow subgroup counts
    SylowSubgroupElementsSylow subgroup elements
    SylowSubgroupsSylow subgroups
    Symmetricsymmetric
    Transitivetransitive
    Transitivitytransitivity

Details

  • "FiniteGroup" entities include particular named finite groups, as well as members of parametrized families.
  • "FiniteGroup" entity classes include common mathematical types of groups such as "Abelian", "Cyclic" and "Sporadic", together with the negations of these.
  • Mathematical properties are available for most "FiniteGroup" entities that are concisely representable and either well known or straightforward to compute. Properties for some parametrized family members are computed on the fly and may be specified using symbolic parameters.
  • Some properties are available for the "FiniteGroup" entity type as a whole and can be given using the form EntityValue["FiniteGroup",property]. Such properties include:
  • "Properties"the list of available properties
    "PropertyCanonicalNames"the standard names of available properties
    "SampleEntities"a sample list of available entities (typically of length 10)
    "SampleEntityClasses"a sample list of available entity classes (typically of length 10)
    "EntityCount"number of entities available
    "Entities"the list of available entities
    "EntityCanonicalNames"the standard names of available entities
    "EntityClasses"the list of available entity classes
    "EntityClassCanonicalNames"the standard names of available entity classes
    "PropertyClasses"the list of available property classes
    "PropertyClassCanonicalNames"the standard names of available property classes
    "PropertyCount"number of properties available
  • The following annotations can be used in the third argument of EntityValue["FiniteGroup",property,annotation]:
  • "EntityAssociation"an association of entities and entity-property values
    "PropertyAssociation"an association of properties and entity-property values
    "EntityPropertyAssociation"an association in which the specified entities are keys, and values are a nested association of properties and entity-property values
    "PropertyEntityAssociation"an association in which the specified properties are keys, and values are a nested association of entities and entity-property values
    "Dataset"a dataset in which the specified entities are keys, and values are an association of property names and entity-property values
  • The following annotations can be used in the second argument of EntityValue[property,annotation]:
  • "Description"a brief textual description of the property

Examples

Basic Examples  (5)

Use for entity discovery:

Find a property value for an entity:

Retrieve a dataset of all available properties for an entity:

Find finite groups associated with John Conway:

Find tabulated non-Abelian finite groups of order 8:

Introduced in 2019
 (12.0)