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Klein's simple group $H$ of order $168$ is the automorphism group of theplane quartic curve $C$, called Klein quartic. By Torelli Theorem, the fullautomorphism group $G$ of the Jacobian $J=J(C)$ is the group of order $336$,obtained by adding minus identity to $H$. The quotient variety $J/G$ can bealternatively represented as the quotient $\mathbb C^3/\tilde G$ of the complex$3$-space by the complex crystallographic group $\tilde G$, the extension of$G$ by the period lattice of the Klein quartic. Moreover, it turns out that$\tilde G$ is generated by affine complex reflections. According to aconjecture of Bernstein--Schwarzman, a quotient of $\mathbb C^n$ by anirreducible complex crystallographic group generated by reflections is aweighted projective space. The conjecture is known in dimension two and forcomplexifications of the real crystallographic groups generated by reflections.The case of $\tilde G$ is the first, and in a sense the smallest of the unknowncases. We compute the orbits and the stabilizers of the action of $G$ on $J$and deduce that $J/G=\mathbf C^3/\tilde G$ is a strongly simply connectedvariety with the same singularities as the weighted projective space $\mathbbP(1,2,4,7)$.

Action of the automorphism group on the Jacobian of Klein's quartic curve