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The Del Pezzo surface $Y$ of degree 5 is the blow up of the plane in 4general points, embedded in $\mathbb{P}^5$ by the system of cubics passingthrough these points. It is the simplest example of the Buchsbaum-Eisenbudtheorem on arithmetically-Gorenstein subvarieties of codimension 3 beingPfaffian. Its automorphism group is the symmetric group $\mathfrak S_5$. Wegive canonical explicit $\mathfrak S_5$-invariant Pfaffian equations through a$6 \times 6$ antisymmetric matrix. We give concrete geometric descriptions ofthe irreducible representations of $\mathfrak S_5$. Finally, we give $\mathfrakS_5$-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 ofdegree $4$.

$\mathfrak S_5$-equivariant syzygies for the Del Pezzo Surface of Degree 5