Symmetry group of the dodecahedron and its subsets > Let $G$ be the symmetry group of the dodecahedron. Indicate subsets of the dodecahedron on which $G$ acts by all possible permutations. I know that $G \simeq A_5 \times \mathbb{Z}_2$. Its order is $120=2^3\cdot3\cdot5$, so there may be $2,3,4$ or $5$ subsets on which $G$ acts by all permutations. I don't see any possibilites other than $5$ Kepler's cubes which are inscribed in the dodecahedron. How can this problem be solved rigorously?

The group $A_5$ is simple; the only normal subgroups of $G = A_5 \times C_2$ are thus $A_5$, $C_2$, $\\{1\\}$, and $A_5\times C_2$; the corresponding quotients are $C_2$, $A_5$, $A_5\times C_2$ and $\\{1\\}$. You're asking about surjective homomorphisms $G\to S_n$, i.e. for which $n$ we can realize $S_n$ as a quotient of $G$. We have $\\{1\\}=S_1$ and $C_2=S_2$; the other two groups are not isomorphic to any $S_n$ (note in particular that the action on the Kepler cubes is not by all permutations). Only $n = 1$ and $n = 2$ are possible.

If the question is "find all the subsets $E$ of the dodecahedron such that $G$ acts on $E$ by all permutations" then the only possibility is $E=$ {the center} (there is no orbit with just 2 elements). If rather you're looking for $S_1,\dots,S_n$ such that $G$ permutes them, then we can also have $n=2$: set $S_1$ be the $A_5$-orbit of $P$ and $S_2$ the orbit of $-P$, where $P$ is a generic point (so that $-P\
otin S_1$).