Does the Fano plane mnemonic for octonion multiplication have any deeper meaning?

http://en.wikipedia.org/wiki/File:FanoPlane.svg

The symmetry group of the Fano plane is PSL(2,7), the second-smallest nonabelian simple group. It is also the smallest Hurwitz group, and the group of automorphisms of the Klein quartic.

http://en.wikipedia.org/wiki/PSL(2,7)

I guess I'm wondering if Hurwitz' classification of normed division algebras

http://en.wikipedia.org/wiki/Hurwitz's_theorem_(composition_algebras)

and Hurwitz' theorem on automorphisms of Riemann surfaces

http://en.wikipedia.org/wiki/Hurwitz's_automorphisms_theorem

are directly related in some way.