  A fake projective plane is a complex surface with the same Betti numbers as
$\mathbb{C} P^2$ but not biholomorphic to it. We study the fake projective
plane $\mathbb{P}_{\operatorname{fake}}^2 = (a = 7, p = 2, \emptyset, D_3 2_7)$
in the Cartwright-Steger classification. In this paper, we exploit the large
symmetries given by $\operatorname{Aut}(\mathbb{P}_{\operatorname{fake}}^2) =
C_7 \rtimes C_3$ to construct an embedding of this surface into $\mathbb{C}
P^5$ as a system of $56$ sextics with coefficients in $\mathbb{Q}(\sqrt{-7})$.
For each torsion line bundle $T \in
\operatorname{Pic}(\mathbb{P}_{\operatorname{fake}}^2)$, we also compute and
study the linear systems $|nH + T|$ with small $n$, where $H$ is an ample
generator of the N\'eron-Severi group.
