Editing Tutte–Coxeter graph (section)

== The Tutte–Coxeter graph as a building ==

This graph is the [[Building (mathematics)#Spherical building|spherical building]] associated to the symplectic group <math>Sp_4(\mathbb{F}_{2})</math> (there is an exceptional isomorphism between this group and the symmetric group <math>S_6</math>).  More specifically, it is the incidence graph of a [[generalized quadrangle]].

Concretely, the Tutte-Coxeter graph can be defined from a 4-dimensional [[symplectic vector space]] <math>V</math> over <math>\mathbb{F}_{2}</math> as follows:
* vertices are either nonzero vectors, or isotropic 2-dimensional subspaces,
* there is an edge between a nonzero vector ''v'' and an isotropic 2-dimensional subspace <math>W\subset V</math> if and only if <math>v \in W</math>.