Assume G is a group, x,y is in G; x and y are not identity, but $x^3=1$ and $y^2=1$ and $(xy)^2=1$. Find the order of G and the group table

Question

So I am stuck with this problem and I can't seem to find the relationship with the x, y and identity in dealing with size of group and how they connect with $(xy)^2=1$. Can someone help me with this?

Answer 1

Since $(xy)^2=1$, which means $xy=(xy)^{-1}=y^{-1}x^{-1}=yx^2$. Thus, every element of $G$ can be reduced to a form $y^ix^j$, where $i=0,1$ and $j=0,1,2$ (since $y^2=1=x^3$). Thus there are at most six elements in $G$. You can check in fact it is equal to $6$.