  We prove that the Klein cubic threefold $F$ is the only smooth cubic
threefold which has an automorphism of order 11. We compute the period lattice
of the intermediate Jacobian of $F$ and study its Fano surface $S$. We compute
also the set of fibrations of $S$ onto a curve of positive genus and the
intersection between the fibres of these fibrations. These fibres generate an
index 2 sub-group of the N\'eron-Severi group and we obtain a set of generators
of this group. The N\'eron-Severi group of $S$ has rank $25=h^{1,1}$ and
discriminant $11^{10}$.
