Here is the “natural embedding” of the 10 vertices of the Petersen Graph in 5 dimensions.  
The rows are the position of the vertices in R5.  
The symmetries of the Petersen graphs are permutations of the vertices, that act as 10x10 permutation matrices on the left, and they are isomorphic to group of 5x5 Orthogonal matrices that act by multiplication on the right.

The next figure shows the projection of the the 5-dimensional embedding into R3.  The vertices are at the 6 faces of the cube, or the 4 vertices of a tetrahedron.