  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 faithful
$\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
Eisenbud-Popescu-Walter sextic with a faithful $\mathrm{PSL}(2,\mathbf{F}_{11})
$-action discovered by the second author in 2013.
