[Math] Using Todd-Coxeter algorithm to identify the group

Question

From Artin's Algebra (6.9.2):

Use the Todd-Coxeter Algorithm to identify the group generated by two elements $x,y$, with the following relations: $x^2=y^2=1,xyx=yxy$.

To do so, I first need to choose a subgroup for which $G$ acts on its cosets. Then Todd-Coxeter algorithm will give the permutation representation $\rho$ of $G$. How do I ensure that the representation does not contain redundant relations?

For example, if I end up with $\rho(x)=\rho(y)=1$, then it is not the required identification. I think I need to make the representation faithful. So, which subgroup should I choose? Does any subgroup work?

Edit: Using the subgroup generated by $\{y\}$, I found $x=(1\ 2),y=(2\ 3)$. But still, I chose the subgroup at random. Had I choose $G$ as the subgroup, I'd end up with the trivial representation.

Answer 1

I did the T.C. version for finding the index of $H=\langle x\rangle$ of order $2$:

We see that $[G:H]=3$ SO $|G|=6$. Now:

$$1xy=1y=2\neq 3=2x=1yx$$

It means that $G\cong S_3$.