It is well-known there is an isomorphism between $GL(3,2)=PGL(3,2)$, the automorphism group of the Fano plane (i.e. the projective plane over the finite field with two elements), and $PSL(2,7)$, which is the automorphism group of the oriented projective line over the field with seven elements. (More details are on Wikipedia).
What I'd like to know is if there is a finite geometric reason that these two groups are isomorphic. For instance, some combinatorial mapping of these geometries that induces an isomorphism between their automorphism groups. I was talking to Richard Green today about exceptional stuff in low dimensions and he claimed that there wasn't really a nice way to see it, unlike, for instance, the construction of the exceptional (outer) automorphism of $S_6$ using synthemes and duads.