Determining the minimum dimension required for embedding a finite group Consider the groups $S_3$ and $S_4$ which are the symmetric groups on 3 and 4 elements respectively. We note that $S_3$ can be realized geometrically as the set of all rotations and reflections of a triangle. And $S_4$ similarly is the set of all rotations and reflections of a tetrahedron.
But Consider a stranger group such as $(\Bbb{Z}/2\Bbb{Z})^2$ this is described as the set of rotations and reflections of a cross-like-structure that leave it unchanged. 
https://en.wikipedia.org/wiki/Klein_four-group
So here's my question. For a given finite group $G$. How do we find the minimum euclidean dimension such that there exists a polyhedron whose set of rotations and reflections that leave it unchanged correspond to the group?
 A: Okay well partial answer: the answer should is the minimum dimension of a faithful real representation, certainly it can't be better than this.  Morally, you just choose a generic point in the space and take a convex hull of that point along with the other $|G|-1$ elements of its $G$ orbit.
Unfortunately this may have extra symmetries, take for example the action of a cyclic group of order 3 on $\mathbb{R}^2$ acting by rotations.  Any point you choose for this construction (other than the origin) will give you an equilateral triangle which has extra symmetries.
This isn't too bad though we can fix this by symmetrically truncating the polytope we get this way by hyperplanes passing through $\epsilon $-neighborhoods of our original vertices which are not perpendicular to the line from the origin to that vertex ($\epsilon$ sufficiently small so none of the truncations interact).
So now this is a purely representation theoretic question, about bounding the dimension of the smallest faithful real representation.  I'm tempted to guess $log_2(|G|)$ is the worst case attained by $(\mathbb{Z}/2\mathbb{Z})^n$, but I'm not sure off the top of my head.
A: There are groups for which no such polyhedron exists, and hence there is no minimum dimension.
For example, consider a group $G$ of order $|G|=2^{\mathfrak{c}}$. If $G$ is the group of symmetries of a polyhedron in $n$-dimensional Euclidean space for some $n\in\mathbb{N}$, then $G$ is a subgroup of $\operatorname{Sym}(\Bbb{R}^n)$, the group of symmetries of $\Bbb{R}^n$. But $|\operatorname{Sym}(\Bbb{R}^n)|=\mathfrak{c}$, so it cannot contain $G$ as a subset, let alone a subgroup, a contradiction.
