Abstract algebra: finding order or group

Question

Find the order of a group defined by: $$G=\{u,v\,|\,u^4=v^3=1,\,u\cdot v=u^2\cdot v^2\}.$$

When I first looked at the problem it looked like dihedral group but it's not. I am unable to find order of group.

Answer 1

Since $uv=u^2v^2\Longleftrightarrow uv=uuvv\Longleftrightarrow 1=vu$, $G$ is generated both by $u$ and bt $v$ (since each one is the inverse of the other one). But then $u^4=u^3=1$. So $u=1$. Therefore, $|G|=1$.