Proving S4 is generated by a 2-cycle and a 3-cycle

[lus1450]

Homework Statement

Show $S_4$ (symmetric group on $4$ letters) can be generated by two elements $x$ and $y$ such that $x^2 = y^3 = (xy)^4$

Homework Equations

The Attempt at a Solution

I'm guessing I can use $(12)$ and $(143)$. I got this since I know $S_n = \langle (12), (13), (14) \rangle$ by theorem in my text, and $(13)(14) = (143)$. I know by another theorem that $S_n = \langle (12),(1234) \rangle \text{ and } (1234) = (143)(143)(12)$, so I believe that means I am allowed to say they both generate the same group. Furthermore, $(12)(143) = (1432)$, which is a $4$-cycle that will be the identity when raised to the fourth. I know here it's not written rigorously, but that is my mindset. Is this correct? Thanks in advance

 

[brmath]

It looks good to me.