The Poincaré homology sphere $X$ is constructed by identifying opposite faces of a dodecahedron by a (clockwise) twist of 36 degree.

Many books say its fundamental group $\pi_1(X)$ is the binary icosahedron group, my question is how to prove this. In Ratcliffe's hyperbolic manifold book, there is a theorem of Poincaré about computing $\pi_1$ of such spaces, following this method one can check $\pi_1(X)$ has a presentation of $6$ generators, $a, b, c, d, e, f$ and relations $a=bd=ce=df=eb=fc$ and $f=be$, how to show this is isomorphic to the 120-order icosahedron group? Is there any easy way to see this? Thanks!