Use the Todd-Coxeter Algorithm to analyse the group generated by two elements x,y, with the following relations determine the order of the group and identify the group if you can.
$$x^3=y^3=1, xyx = yxy$$
Given function is,
$xyx = yxy$
Multiply $x^2$ to both sides we get, $x^3yx = x^2yxy$
Here we know that $x^3 = 1$ so,
$yx = yxyx^2$
on applying Right cancellation law $y = yxyx$
On applying Left cancellation Law $xyx = 1$
Similarly
$yxy = 1$
so, $xyx = yxy = 1$
Hence the order of the Group is 1 and the Types of Group is Cyclic.
Best Answer
Note that I did what you were looking for in light of Derek's neat hint
You see that $[G:\langle y\rangle]=8$ so the possibilities for $|G|$ could be $24$.