Show the two different representation of the fundamental groups of the Klein bottle $G=<a,b:aba^{-1} b>$ and $H=<c,d:c^2 d^{-2}>$.
As we calculate in different and gett one result as $\pi\_1(K)=G$, and other as $\pi\_1(K)=H$, and I need to show $G \cong H$.

I try to do this by constructing a function $f:G\to H$ and try to define $f(a)$ is in terms of $c$ and $d$, and $f(b)$ is also defined in terms of $c$ and $d$, such that $f(e\_G)=f(aba^{-1} b)=f(a)f(b)f(a^{-1})f(b)=c^2d^{-2}=e\_H$.

I am not sure how to define such $f$ for $a$ and $b$, please help me. Thanks a lot!