We embed the moduli space $Q$ of 5 points on the projective line $S_5$-equivariantly into $\mathbb{P} (V)$, where $V$ is the 6-dimensional irreducible module of the symmetric group $S_5$. This module splits with respect to the icosahedral group $A_5$ into the two standard 3-dimensional representations. The resulting linear projections of $Q$ relate the action of $A_5$ on $Q$ to those on the regular icosahedron.

MSC Classifications:

14L24, 20B25 show english descriptions Geometric invariant theory [See also 13A50]
Finite automorphism groups of algebraic, geometric, or combinatorial structures [See also 05Bxx, 12F10, 20G40, 20H30, 51-XX] 14L24 - Geometric invariant theory [See also 13A50]
20B25 - Finite automorphism groups of algebraic, geometric, or combinatorial structures [See also 05Bxx, 12F10, 20G40, 20H30, 51-XX]

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