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](https://en.wikipedia.org/wiki/PSL(2,7)#Actions_on_projective_spaces)).


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](https://math.colorado.edu/~rmg/) 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.



The exceptional isomorphism between PGL(3,2) and PSL(2,7): geometric origin?
V. Dotsenko's construction, on math.stackexchange:


<https://math.stackexchange.com/questions/1401/why-psl-3-mathbb-f-2-cong-psl-2-mathbb-f-7/1450#1450>


may fit your requirement "combinatorial mapping of these geometries that induces an isomorphism".


