Gushel–Mukai varieties with many symmetries and an explicit irrational Gushel–Mukai threefold

We construct an explicit complex smooth Fano threefold with Picard number 1, index 1, and degree 10 (also known as a Gushel–Mukai threefold) and prove that it is not rational by showing that its intermediate Jacobian has a faithfull $\mathrm{PSL}(2,\mathbf{F}_{11}) $-action. Along the way, we construct Gushel–Mukai varieties of various dimensions with rather large (finite) automorphism groups.The starting point of all these constructions is an EPW sextic with a faithful $\mathrm{PSL}(2,\mathbf{F}_{11}) $-action discovered by Giovanni Mongardi in his thesis in 2013 and all this is joint work with him.