cycles commute: for example, in S6 there is the equality . Every element of Sn can be written as a product of disjoint cycles; this representation is unique up to the order of the factors, and the freedom present in representing each individual cycle by choosing its starting point. Cycles admit the following conjugation property with any permutation , this property is often used to obtain its generators and relations. Special elements Certain elements of the symmetric group of {1, 2, ..., n} are of particular interest (these can be generalized to the symmetric group of any finite totally ordered set, but not to that of an unordered set). The is the one given by: This is the unique maximal element with respect to the Bruhat order and the longest element in the symmetric group with respect to generating set consisting of the adjacent transpositions , . This is an involution, and consists of (non-adjacent) transpositions so it thus has sign: which is 4-periodic in n. In S2n, the perfect shuffle is the permutation that splits the set into 2 piles and interleaves them. Its sign is also Note that the reverse on n elements and perfect shuffle on 2n elements have the same sign; these are important to the classification of Clifford algebras, which are 8-periodic. Conjugacy classes The conjugacy classes of Sn correspond to the cycle types of permutations; that is, two elements of Sn are conjugate in Sn if and only if they consist of the same number of disjoint cycles of the same lengths. For instance, in S5, (1 2 3)(4 5) and (1 4 3)(2 5) are conjugate; (1 2 3)(4 5) and (1 2)(4 5) are not. A conjugating element of Sn can be constructed in "two line notation" by placing the "cycle notations" of the two conjugate permutations on top of one another. Continuing the previous example, which can be written as the product of cycles as (2 4). This permutation then relates (1 2 3)(4 5) and (1 4 3)(2 5) via conjugation, that is, It is clear that such a permutation is not unique. Conjugacy classes of correspond to integer partitions of : to the partition with and , is associated the set of permutations with cycles of lengths . Then is a conjugacy class of , whose elements are said to be of cycle-type . Low degree groups The low-degree