Cycle decomposition of Permutations into disjoint cycles is unique (up to reordering of cycles).
A proof can be found here.
The condition of disjoint is crucial. For example, the permutation (1 3 2) can be factored into (2 3)(1 2), where the two cycles are not disjoint. (1 3 2)=(1 2)(1 3) is also another decomposition, the two cycles are also not disjoint.
Wolframalpha can calculate permutations, a useful tool to replace manual calculations. Take note though that Wolframalpha’s convention is multiplying permutations from left to right, while most books follow the convention of multiplying right to left.