How can I prove that the elements
$$x^j\cdot y^l\cdot x\cdot y^k$$
with $x=(12)$ , $y=(1234)$ , $0\le j \le 1$ , $0\le l\le 2$ , $0\le k\le 3$ form the symmetric group $S_4$ ?
Is should be enough to show that the elements are pairwise different, but how can I do this by hand ?