The Sylow p -subgroups of the symmetric group of degree p are the wreath product of two cyclic groups of order p. For instance, when p = 3, a Sylow 3-subgroup of Sym(9) is generated by a = (1 4 7)(2 5 8)(3 6 9) and the elements x = (1 2 3), y = (4 5 6), z = (7 8 9), and every element of the Sylow 3-subgroup has the form a x y z for ⁠ 0 ≤ ≤ i , j , k , l ≤ ≤ 2 ⁠.