Reidemeister-Schreier rewriting process. How to find the appropriate transversal element? It's not clear for me how to find the appropriate transversal element in the next example below for an example for $x,y,z$ elements. Please would someone show this process completely? ![enter image description here](

The group $S_3$ is defined as $\langle a,b \mid a^2 = b^3 = 1, ba = ab^2\rangle$, where $a = (12), b = (123)$.

The elements $ba$ and $ab ^ 2$ are both equal $(23)$. Since there is a homomorphism from $G$ to $G/H = S_3$, the elements $ba$ and $ab ^ 2$ are in the same coset. Therefore, the Schreier representative $ba$ is equal to $ab ^ 2$.