I have an interesting problem in materials science that I think can be solved by obtaining a mapping, which is one-to-one, continuous and preserves the metric, from spherical 3-manifods (defined as $S^3/\Gamma$, where $\Gamma$ is a finite subgroup of $SO(4)$) to Euclidean space of some dimension $n$ $\left( \mathbb{R}^n \right)$. I believe this mapping is an isometric embedding.
I am primarily interested in obtaining a mapping to any $\mathbb{R}^n$ (not necessarily the smallest $n$ but something reasonable). For example, the group of proper rotations, $SO(3)$, I think, is a spherical 3-manifold ($S^3/(-I_{4\times4}$)), where $I_{4\times4}$ is the $4 \times 4$ identity matrix. This manifold can be isometrically embedded in $\mathbb{R}^9$ through the $3 \times 3$ matrix representation of that rotation. I am wondering if such embeddings are known for other spherical 3-manifolds. Any help is greatly appreciated.
Edit 1: I want to make this a little more specific if it helps. I am interested in the isometric embedding of $S^3/2O$, where $2O$ is the binary ocatahedral point group (https://en.wikipedia.org/wiki/Binary_octahedral_group). Alternatively, this space is equivalent to the quotient space $SO(3)/O$, where $O$ is the octahedral point group $(2,3,4)$.