  Let $S$ be the (minimal) Enriques surface obtained from the symmetric quartic
surface $(\sum_{i<j}x_ix_j)^2=kx_1x_2x_3x_4$ in $\mathbb{P}^3$ with $k\neq
0,4,36$, by taking quotient of the Cremona action $(x_i) \mapsto (1/x_i)$. The
automorphism group of $S$ is a semi-direct product of a free product
$\mathcal{F}$ of four involutions and the symmetric group $\mathfrak{S}_4$. Up
to action of $\mathcal{F}$, there are exactly $29$ elliptic pencils on $S$.
