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Does {7,4|3} have a realization on the Klein quartic?

It was pointed out to me recently that the polyhedron [{7,4|3}](https://www.abstract-polytopes.com/atlas/336/208/14.html) has the same automorphism group as the [Klein quartic](https://en.wikipedia.org/wiki/Klein_quartic).

Specifically {7,4|3} is:

$\langle \rho_0,\rho_1,\rho_2 \mid \rho_0^2, \rho_1^2, \rho_2^2, (\rho_0\rho_2)^2, (\rho_0\rho_1)^7, (\rho_1\rho_2)^4, (\rho_0\rho_1\rho_2\rho_1)^3\rangle$

Naturally this leads to the question: Does {7,4|3} have a realization on the Klein quartic?  If so where are its vertices?

It *seems* obvious that it should exist. We already know there exists a faithful map from this automorphism group to concrete symmetries, so we just take the image of the generators under this map and place the vertex at the intersection of spaces fixed by $\rho_1$ and $\rho_2$.

However, there's no guarantee as far as I can tell that these spaces will have a non-empty intersection. And while the existence of the map is trivial, actually deriving it beyond what I'm capable of. I just don't know the tools I would need to solve this other than by guess and check. That only works if there is a solution, and I'm beginning to suspect there is not.

Does {7,4|3} have a realization on the Klein quartic?  If so where are its vertices?

<sub>This question has been reposted from [math stackexchange](https://math.stackexchange.com/questions/4884427/does-7-43-have-a-realization-on-the-klein-quartic)</sub>