BUILT-IN MATHEMATICA SYMBOL

FiniteGroupData

gives the value of the specified property for the finite group specified by name.

gives a list of finite groups in the specified class.

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] 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.

	{"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

	{"AbelianGroup",{m,n,...}}	Abelian group
	{"DirectProduct",{group₁,group₂,...}}	direct product of groups
	{"SemiDirectProduct",{group₁,...}}	semi-direct product of groups

	"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.

	"CycleIndex"	cycle index
	"Cycles"	cycles
	"Permutations"	representation as permutations

	"DefiningRelations"	relations that describe the group algebra
	"CayleyGraph"	Cayley graph
	"CayleyGraphImage"	image of the Cayley graph
	"CycleGraph"	cycle graph

	"CharacterTable"	group element characters
	"MatrixRepresentation"	representation as matrices
	"SpaceRepresentation"	representation as 3D Cartesian coordinate transformations

FiniteGroupData gives True or False depending on whether a group corresponding to name is in the specified class.

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.

	"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