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CayleyGraph[group]

returns a Cayley graph representation of group.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Possible Issues  
Neat Examples  
See Also
Tech Notes
Related Guides
Related Demonstrations
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CayleyGraph

CayleyGraph[group]

returns a Cayley graph representation of group.

Details and Options

  • CayleyGraph[group] returns a graph object with head Graph.
  • A Cayley graph is both a description of a group and of the generators used to describe that group. The generators are those returned by the function GroupGenerators.
  • Group elements are represented as vertices, and generators are represented as directed edges. An edge from a group element g1 to an element g2 means that the product of g1 with the generator of the edge gives g2.
  • Vertices are numbered as ordered by GroupElements and GroupElementPosition. The identity element is always numbered 1.
  • Generators are represented by default using different colors, following the sequence of colors used by Plot with several curves.

Examples

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Basic Examples  (1)

This is the Cayley graph connecting the 24 elements of a permutation group defined by two generators, the first represented in blue and the second in green:

Wolfram Language code: CayleyGraph[PermutationGroup[{Cycles[{{1, 5, 4}}], Cycles[{{3, 4}}]}], VertexLabels -> Placed["Name", Center], VertexSize -> 0.4]

Scope  (3)

Cayley graph of the symmetric group of degree four defined by three transpositions:

Wolfram Language code: CayleyGraph[PermutationGroup[{Cycles[{{1, 2}}], Cycles[{{2, 3}}], Cycles[{{3, 4}}]}]]

Cayley graph of the symmetric group of degree four with the default generating set:

Wolfram Language code: CayleyGraph[SymmetricGroup[4]]

The identity permutation is removed from the list of generators:

Wolfram Language code: CayleyGraph[PermutationGroup[{Cycles[{}], Cycles[{{1, 2, 3}}]}]]

Possible Issues  (1)

This is only a useful representation for small groups. For groups with a few hundred elements, the graph is generally already too complex:

Wolfram Language code: GroupOrder[G = PermutationGroup[{Cycles[{{1, 4, 7, 3, 5}}], Cycles[{{1, 2, 3}}]}]]
Wolfram Language code: CayleyGraph[G]

Neat Examples  (1)

A point:

Wolfram Language code: CayleyGraph[AbelianGroup[{}]]//UndirectedGraph

A line:

Wolfram Language code: CayleyGraph[AbelianGroup[{2}]]//UndirectedGraph

A square:

Wolfram Language code: CayleyGraph[AbelianGroup[{2, 2}]]//UndirectedGraph

A cube:

Wolfram Language code: CayleyGraph[AbelianGroup[{2, 2, 2}]]//UndirectedGraph

A 4D cube:

Wolfram Language code: CayleyGraph[AbelianGroup[{2, 2, 2, 2}]]//UndirectedGraph

A 5D cube:

Wolfram Language code: CayleyGraph[AbelianGroup[{2, 2, 2, 2, 2}]]//UndirectedGraph
Wolfram Research (2010), CayleyGraph, Wolfram Language function, https://reference.wolfram.com/language/ref/CayleyGraph.html.

Text

Wolfram Research (2010), CayleyGraph, Wolfram Language function, https://reference.wolfram.com/language/ref/CayleyGraph.html.

CMS

Wolfram Language. 2010. "CayleyGraph." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/CayleyGraph.html.

APA

Wolfram Language. (2010). CayleyGraph. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/CayleyGraph.html

BibTeX

@misc{reference.wolfram_2026_cayleygraph, author="Wolfram Research", title="{CayleyGraph}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/CayleyGraph.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_cayleygraph, organization={Wolfram Research}, title={CayleyGraph}, year={2010}, url={https://reference.wolfram.com/language/ref/CayleyGraph.html}, note=[Accessed: 12-June-2026]}