Permute—Wolfram Language Documentation [proxy]

Permute[expr,perm]

permutes the positions of the elements of expr according to the permutation perm.

Permute[expr,gr]

returns the list of permuted forms of expr under the elements of the permutation group gr.

Details

Permute works with any nonatomic expressions, operating on the first level of expressions.
Permute reorders the elements of an expression but never changes its length.
The permutation perm can be given in disjoint cyclic form or as a permutation list.
When perm is given in cyclic form Cycles[{cyc₁,cyc₂,…}], a cycle {p₁,p₂,…} moves the elements of expr in a cyclic manner so that expr[[p_i]] is moved to position p_i+1.
When perm is given as a permutation list, the result is equivalent to the use of Permute[expr,PermutationCycles[perm]].
A permutation group gr can be specified by generators, with head PermutationGroup, or in named form, with head SymmetricGroup, AlternatingGroup, ….

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Cyclic permutation of three elements in a list:

Take the lowercase alphabet:

Exchange the first and last character:

Permute several characters:

Permute an expression under all elements of a group:

Permute the parts of an expression:

Permute the parts of an expression under all elements of a group:

Give a permutation in list form. The length of the expression does not change:

The eight possible rotations and reflections of a square:

Permute never changes the number of parts of an expression. It simply reorders them:

However, Part can change the number of parts:

When applied to the identity permutation list, Permute is the inverse of PermutationReplace:

Permute can also be used as an alternative to PermutationList:

Another way of inverting the action of Permute is using FindPermutation:

When all parts of the expression are different, the permutation can be uniquely recovered:

Permute is a right action with respect to the product of permutations: