PermutationGroup
✖
PermutationGroup
represents the group generated by multiplication of the permutations perm1,…,permn.
Details

- The generating permutations permi must be given in disjoint cyclic form, with head Cycles.
- Properties of a permutation group are typically computed by constructing a strong generating set representation of the group using the Schreier–Sims algorithm.
Examples
open allclose allBasic Examples (1)Summary of the most common use cases
Scope (3)Survey of the scope of standard use cases
An empty list of generators represents the identity (or trivial, or neutral) group:

https://wolfram.com/xid/0b0kvcg79u6-tjxnvo

Find the order of a group generated by two permutations:

https://wolfram.com/xid/0b0kvcg79u6-rzues1

Test the equality of permutation groups with the same support but possibly generated by different permutations:

https://wolfram.com/xid/0b0kvcg79u6-ys8g72

They are different as Wolfram Language expressions:

https://wolfram.com/xid/0b0kvcg79u6-j8yb7r

Applications (2)Sample problems that can be solved with this function
This is the group of all rotations and reflections of a regular -sided polygon, the dihedral
group, for
. It can be generated by a rotation of an angle
and a reflection along an axis through a vertex:

https://wolfram.com/xid/0b0kvcg79u6-0i60t2

Construct the octagon corresponding to each group element:

https://wolfram.com/xid/0b0kvcg79u6-y6gxgz
This is the original polygon and its seven rotations. Numbers increase counterclockwise:

https://wolfram.com/xid/0b0kvcg79u6-ulosfc

This is the polygon reflected along the bisection 1–5 and its seven rotations. Numbers increase clockwise:

https://wolfram.com/xid/0b0kvcg79u6-o4x2m2

The group of automorphisms of a graph is represented using PermutationGroup:

https://wolfram.com/xid/0b0kvcg79u6-4ab4gc


https://wolfram.com/xid/0b0kvcg79u6-tapanv

This is the number of automorphisms of the graph:

https://wolfram.com/xid/0b0kvcg79u6-xcl1m0

Properties & Relations (3)Properties of the function, and connections to other functions
Explicit representation of a named group:

https://wolfram.com/xid/0b0kvcg79u6-flcalc

Generate the symmetric group of degree using
transpositions:

https://wolfram.com/xid/0b0kvcg79u6-zugfvy

Generate the alternating group of degree using
generators:

https://wolfram.com/xid/0b0kvcg79u6-mi49r7

Neat Examples (1)Surprising or curious use cases
The moves of a Rubik's cube form a group. Number the moving facelets from 1 to 48:

These are the six basic rotations:

https://wolfram.com/xid/0b0kvcg79u6-hg7msy

https://wolfram.com/xid/0b0kvcg79u6-yghw70

Swapping two neighbor edge facelets is not allowed:

https://wolfram.com/xid/0b0kvcg79u6-xcu83f

Simultaneous swaps of two edge pairs is allowed:

https://wolfram.com/xid/0b0kvcg79u6-hx5r7j

This is the superflip move, which switches all edge pairs simultaneously without changing any corner:

https://wolfram.com/xid/0b0kvcg79u6-cof889

https://wolfram.com/xid/0b0kvcg79u6-6hj7cu

Edges and corners cannot be mixed (as the action of the group on the cube is not transitive), but any two corners or any two edges can be swapped:

https://wolfram.com/xid/0b0kvcg79u6-zgc9t5

Wolfram Research (2010), PermutationGroup, Wolfram Language function, https://reference.wolfram.com/language/ref/PermutationGroup.html.
Text
Wolfram Research (2010), PermutationGroup, Wolfram Language function, https://reference.wolfram.com/language/ref/PermutationGroup.html.
Wolfram Research (2010), PermutationGroup, Wolfram Language function, https://reference.wolfram.com/language/ref/PermutationGroup.html.
CMS
Wolfram Language. 2010. "PermutationGroup." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PermutationGroup.html.
Wolfram Language. 2010. "PermutationGroup." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PermutationGroup.html.
APA
Wolfram Language. (2010). PermutationGroup. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PermutationGroup.html
Wolfram Language. (2010). PermutationGroup. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PermutationGroup.html
BibTeX
@misc{reference.wolfram_2025_permutationgroup, author="Wolfram Research", title="{PermutationGroup}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/PermutationGroup.html}", note=[Accessed: 08-July-2025
]}
BibLaTeX
@online{reference.wolfram_2025_permutationgroup, organization={Wolfram Research}, title={PermutationGroup}, year={2010}, url={https://reference.wolfram.com/language/ref/PermutationGroup.html}, note=[Accessed: 08-July-2025
]}