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!



Poincaré dodecahedron space
You can read the book by Seifert and Threlfall "A textbook of topology", pages 223-225: They 
start by writing down a presentation of $\pi\_1$ of the Poincare homology sphere (by reading off generators and relators from the identification of faces of the spherical dodecahedron). Then they describe how to transform this presentation to the one of the icosahedral group. No computers are required, just some patience with their somewhat archaic style. 


