  We define a notion of vexillar design for the flag variety in the spirit of
the spherical designs introduced by Delsarte, Goethals and Seidel. For a finite
subgroup of the orthogonal group, we explain how conditions on the group have
the orbits of any flag under the group action be a design and point out why the
minima of a lattice in the sense of the general Hermite constant forming a
4-design implies being extreme. The reasoning proves useful to show the
extremality of many new expected examples ($E_8$, $\La_{24}$, Barnes-Wall
lattices, Thompson-Smith lattice for instance) that were out of reach until
now.
