  A 3 dimensional analogue of Sakai's theory concerning the relation between
rational surfaces and discrete Painlev\'e equations is studied. For a family of
rational varieties obtained by blow-ups at 8 points in general position in
${\mathbb P}^3$, we define its symmetry group using the inner product that is
associated with the intersection numbers and show that the group is isomorphic
to the Weyl group of type $E_7^{(1)}$. By normalizing the configuration space
by means of elliptic curves, the action of the Weyl group and the dynamical
system associated with a translation are explicitly described. As a result, it
is found that the action of the Weyl group on ${\mathbb P}^3$ preserves a one
parameter family of quadratic surfaces and that it can therefore be reduced to
the action on ${\mathbb P}^1\times {\mathbb P}^1$.
