The Poincare 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 Poincare 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!