A partial order on the 240 packings of PG(3,2)

It has long been known that the most symmetrical solutions of Kirkman’s Schoolgirl Problem can be constructed from the $240$ packings of the projective space $PG(3,2)$, but it seems to have escaped notice that these packings have the structure of a partially ordered set. In this paper, we construct a shellable Bruhat-like graded partial order on the packings of $PG(3,2)$ that refines the partial order on the product of four chains $[8]\times [5]\times [3]\times [2]$ and defines a Lehmer code on the packings. The partial order exists because the packings of $PG(3,2)$ form a quasiparabolic set (in the sense of Rains–Vazirani) that is in bijective correspondence with a certain collection of maximal orthogonal subsets of the $E_8$ root system. The $E_8$ construction also induces transitive actions of the Weyl groups of type $D_n$ on the packings for $5\le n\le 8$, and these actions are faithful for $n<8$. It is possible to define both the signed permutation action and the partial order using the combinatorics of labelled Fano planes.