$PSL(2,7)$ acts on the projective plane over $\mathbb{F}\_2$ (the Fano plane) through its identification with $GL(3,2)$. It also acts on the projective plane over $\mathbb{C}$ through either of its pair of 3-dimensional complex representations. Does the Fano plane embed in $\mathbb{P}\_\mathbb{C}^2$ so that the action restricts? To make the question precise:



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> Does there exist a 7-point orbit in $\mathbb{P}\_\mathbb{C}^2$ so that the permutation representation of $PSL(2,7)$ obtained from the action on this orbit is conjugate to the permutation representation of $PSL(2,7)$ on the 7 points of the Fano plane?
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Does the Fano plane "embed" in the complex projective plane?
(As suggested, comment turned into answer) : The answer is "no". ${\rm PSL}(2,7)$ has two non-conjugate subgroups of index $7$, each isomorphic to the symmetric group $S\_{4}$. Either of the three dimensional irreducible representations of ${\rm PSL}(2,7)$ restrict irreducibly to each subgroup of ${\rm PSL}(2,7)$ isomorphic to ${\rm S}\_{4}$ (a direct and easy calculation), so no such subgroup stabilizes any one-dimensional subspace.


