I have the disjoint cycle: $$(156)(2437).$$ Apparently the "method" would get us: $$(1,6)(1,5)(2,7)(2,3)(2,4).$$ Basically you take the first number, and put it as a transposition of the last number and go backwards till you use all of them. I don't understand where this comes form. I don't know why this method works. Does it even work for my case? Oh and the order is 12 with the permutation being odd.
I think I understand $$(1,5)(5,6)(2,4)(4,3)(3,7)$$ much better. Is this a correct way to do transpositions as well?