  The Del Pezzo surface $Y$ of degree 5 is the blow up of the plane in 4
general points, embedded in $\mathbb{P}^5$ by the system of cubics passing
through these points. It is the simplest example of the Buchsbaum-Eisenbud
theorem on arithmetically-Gorenstein subvarieties of codimension 3 being
Pfaffian. Its automorphism group is the symmetric group $\mathfrak S_5$. We
give canonical explicit $\mathfrak S_5$-invariant Pfaffian equations through a
$6 \times 6$ antisymmetric matrix. We give concrete geometric descriptions of
the irreducible representations of $\mathfrak S_5$. Finally, we give $\mathfrak
S_5$-invariant equations for the embedding of $Y$ inside $(\mathbb{P}^1)^5$,
and show that they have the same Hilbert resolution as for the Del Pezzo of
degree $4$.
