FiniteGroupData—Wolfram Language Documentation (original) (raw)

BUILT-IN 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.

Details

Examples

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Basic Examples (2)

The quaternion group:

Multiplication table of the quaternion group:

Scope (34)

Names and Classes (6)

A list of all named finite groups and small members of infinite families, many of them mutually isomorphic:

Finite groups whose standard name is a string:

Find the English name of a finite group:

A list of alternate names can also be found:

Some groups can be given in shorter form:

Classes of finite groups:

List of the groups in FiniteGroupData[] belonging to a given class:

Test membership of a group in the class:

Restrict the members of a class to a given order. These are all mutually isomorphic groups:

The list of 26 simple sporadic groups:

Short forms of the 3-dimensional crystallographic point groups:

FiniteGroupData[n] returns a list of non-isomorphic groups of order n, first having the Abelian groups and then the non-Abelian groups:

When FiniteGroupData does not have a named version of a given group, it is specified as {order,index}:

Properties and Annotations (4)

Get a list of possible properties:

Get more information about a group:

Find the English name:

Standard typeset notations:

Property Values (3)

A property value can be any valid Wolfram Language expression:

A property that is not available for a group has the value Missing["NotAvailable"]:

A property whose value is too large to include has the value Missing["TooLarge"]:

Detailed Properties (21)

Names & Notations (3)

In general, groups are referred to by standard name:

That is equivalent to this simpler syntax:

Other acceptable forms of input are called alternate standard names:

This is the English name for a group, always a string:

Some groups have alternate names:

There is also a short string form of the name:

Every group has a short notation form:

The default notation for the 32 three-dimensional crystallographic point groups is Schoenflies notation, but other notations are possible:

Those are only applicable to the crystallographic groups:

Basic Group Properties (4)

Two important properties of a group are its order and multiplication table:

The multiplication table defines the group as an abstract group (that is, up to isomorphism). For example, the group is Abelian if and only if its multiplication table is symmetric:

The elements of the group are abstractly identified by their position in the multiplication table:

These are their inverses:

This allows identification of which elements form the various subgroups. For example, the fourth element commutes with all other group elements:

More information on the commutativity of a group is given by the group of all commutators in the group. This group is not Abelian because the commutator (or derived) subgroup is not trivial:

However, upon further commutation we get the trivial group:

Hence, the original group is solvable:

These are the cycles obtained by repeated multiplication of each element with itself:

They have a natural representation as a graph. The identity element is highlighted:

Group Representations (4)

Presentations in terms of generators and relations:

The group product is represented as a small circle, also used to denote its associated power:

With those two generators (of respective orders 4 and 2) we have this Cayley graph:

Naming the generators, it is possible to give all elements of the groups as words:

Permutation group representations as the complete set of permutation lists:

Or as a permutation group that can be used for further computation:

It is a transitive representation, because it has a single orbit:

However, any stabilizer is trivial:

Hence, this is not a multiply-transitive representation:

The cyclic structure of a permutation representation gives the so-called cycle index polynomial:

This is a matrix representation of a cyclic group:

Group Isomorphism (1)

Obtain groups isomorphic to a given one:

They all share the same properties as abstract groups:

Group Structure (8)

Conjugacy classes of a group, given as respective lists of elements:

The class number is defined as the number of conjugacy classes:

Those elements that are self-conjugate, hence commuting with all other elements in the group, form the center of that group:

The Klein 4-group only has one type of proper subgroup:

But that type is realized three times:

For n≥5 the alternating groups are equal to their commutator subgroups:

Groups with such property are called perfect:

All simple groups are perfect:

Normal subgroups of the "Tetrahedral" group:

It has nontrivial normal subgroups, and therefore it is not simple:

Sylow subgroups are those whose order is a maximal power of a prime:

Possible quotient groups, obtained by finding the quotient with respect to normal subgroups:

Schur multiplier group:

Automorphism groups:

Crystallographic-Specific Properties (1)

These are various properties only applicable to the crystallographic groups:

Generalizations & Extensions (2)

Find the list of group names matching a pattern:

Some properties are available for symbolic parameters in infinite families of groups:

This is the valid range of values for the parameter:

However, most properties require integer values for such parameters. In those cases Missing["NotApplicable"] is returned:

Applications (1)

Infinite family of alternating groups given as {"AlternatingGroup",n}:

Properties & Relations (3)

FiniteGroupCount[n] gives the number of finite groups of order n:

Consistently, when address by order, FiniteGroupData returns a list of non-isomorphic groups:

Obtain other groups isomorphic to those:

FiniteAbelianGroupCount[n] gives the number of Abelian groups of order n:

All finite groups of order 1225 are Abelian:

The group of units of the ring of integers modulo n is always Abelian, but not always cyclic:

A group is cyclic if there is an element whose order is the group order. The order of {"CyclicGroupUnits", n} is EulerPhi[n], while CarmichaelLambda[n] gives the maximum order amongst its elements.

For example, the following group is cyclic because there are elements whose order is the group order:

The condition EulerPhi[n]CarmichaelLambda[n] is only obeyed for positive integers n of the form 2, 4, pk, 2pk with prime p≠2 and k≥1.

Possible Issues (1)

Results from FiniteGroupData may contain isomorphic groups:

There is only one group of order 1:

Using the order as first argument guarantees non-isomorphic results:

Wolfram Research (2008), FiniteGroupData, Wolfram Language function, https://reference.wolfram.com/language/ref/FiniteGroupData.html (updated 2020).

Text

Wolfram Research (2008), FiniteGroupData, Wolfram Language function, https://reference.wolfram.com/language/ref/FiniteGroupData.html (updated 2020).

CMS

Wolfram Language. 2008. "FiniteGroupData." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2020. https://reference.wolfram.com/language/ref/FiniteGroupData.html.

APA

Wolfram Language. (2008). FiniteGroupData. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FiniteGroupData.html

BibTeX

@misc{reference.wolfram_2025_finitegroupdata, author="Wolfram Research", title="{FiniteGroupData}", year="2020", howpublished="\url{https://reference.wolfram.com/language/ref/FiniteGroupData.html}", note=[Accessed: 16-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_finitegroupdata, organization={Wolfram Research}, title={FiniteGroupData}, year={2020}, url={https://reference.wolfram.com/language/ref/FiniteGroupData.html}, note=[Accessed: 16-June-2025 ]}