Group generated by $x,y$. With relations $x^3=y^2=(xy)^2=1$. 
Let $G$ be a group generated by $x,y$ with the relations $x^3=y^2=(xy)^2=1$. Then show that the order of $G$ is 6.

My attempt: So writing down the elements of $G$ we have $\{1,x,x^2,y,\}$. Other elements include $\{xy, xy^2, x^2y\}$ it seems I am counting more than $6$. Are some of these equal? how do I prove that?

One group presentation for the dihedral group $D_n$ is $\langle x,y|x^2=1,y^n=1,(xy)^2=1\rangle $. Hence the group is indeed isomorphic to $D_3$. Here $x$ with $x^2=1$ corresponds to a reflection, and $y$ with $y^3=1$ to a rotation of $60$ degrees. Finally we have $xyx^{-1}=y^{-1}=y^2$, which is how rotation and reflection interact. So all elements are given by $\{1,y,y^2,x,xy,xy^2\}$.
Sorry, I have interchanged $x$ and $y$ here.
