A special Debarre-Voisin fourfold

Let $\mathbf G$ be the finite simple group $\mathrm{PSL}(2,\mathbf F_{11})$. It has an irreducible representation $V_{10}$ of dimension 10. In this note, we study a special trivector $\sigma\in \bigwedge^3V_{10}^\vee$ which is $\mathbf G$-invariant. Following the construction of Debarre-Voisin, we obtain a smooth hyperk\"ahler fourfold $X_6^\sigma\subset\mathrm{Gr}(6,V_{10})$ with many symmetries. We will also look at the associated Peskine variety $X_1^\sigma\subset \mathbf P(V_{10})$, which is highly symmetric as well and admits 55 isolated singular points. It will help us to understand better the geometry of the special Debarre-Voisin fourfold $X_6^\sigma$.