The Foster graph is a distance-transitive symmetric bipartite cubic graph on 90 vertices (in fact, the only such graph).
I have seen this graph described as the incidence structure of the unique flag-transitive triple cover of the generalized quadrangle associated with $Sp_4(2)$, but this is both not constructive (as I don't know how to build said triple cover without appeal to the Foster graph) and rather heavy on abstractions (surely I shouldn't have to define a bunch of incidence geometry axioms and symplectic groups to talk about this object!).
Are there any "nice" constructions that give rise to this graph or to its associated incidence structure? Something in the vein of "count the following structures on the icosahedron as the points, and these other structures as your lines, where incidence is defined by blah" - ideally it would make it obvious that the resulting object inherits some kinds of symmetries from the base objects it's constructed out of.