Looking for confirmation that the following method constructs a geometric object whose symmetry is described by a finite group $G$.
Let $G$ be a finite group which is a subgroup of the symmetric group $S_n$. $S_n$ can be represented as a regular $n-1$ simplex. This simplex can be cut into $n!$ identical pieces such that every piece contains the simplexes’ center, such as cutting a line segment in half, cutting an equilateral triangle into six identical pieces, a tetrahedron cut into 24 pieces, etc.
Each piece is given a color (or more intuitively a “painting”). When a group action is performed on the simplex, the pieces of the simplex are relocated to different pieces. If one piece of the simplex can be relocated to another piece of the simplex using a group action defined by one of the elements of $G$, then those two pieces are given the same color (or “painting”). Would the resulting colored simplex possess a symmetry described by $G$?
For example, the subgroups of $S_3$ are the trivial group, $C_2$, $C_3$, and $S_3$. Using the partitioned triangle below, when each piece has a unique color, the triangle’s symmetry is described by the trivial group. When pieces 1 and 6 are colored red, pieces 2 and 5 are colored green, and pieces 3 and 4 are colored blue, the symmetry is described by $C_2$. When pieces 1, 3, and 5 are painted white and pieces 2, 4, and 6 are painted black, the symmetry is described by $C_3$. And when each piece is given the same color, the symmetry is described by $S_3$. Would this method work for any finite group, ignoring how impractical constructing such an object may be?
The motive is, given a geometric representation of a finite group $G$ with a normal subgroup $N$ and quotient group $G/N$, do the normal subgroup and quotient group “manifest themselves” in the representation geometrically? This will be it’s own question.
