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Symmetries of cross-ratios and the equation for Möbius structures

We consider orthogonal representations $η_n:S_n \curvearrowright \mathbb{R}^N$ of the symmetry groups $S_n$, $n\ge 4$, with $N=n!/8$ motivated by symmetries of cross-ratios. For $n=5$ we find the decomposition of $η_5$ into irreducible components and show that one of the components gives the solution to the equations, which describe Möbius structures in the class of sub-Möbius structures. In this sense, the condition defining Möbius structures is hidden already in symmetries of cross-ratios.