In the symmetric group the elements are given as $\sigma_{1}, \sigma_{2} ,\dots$ and $\rho, \rho^{2},\dots$.
In my notes I am only given what each one is equal to in a certain symmetry group.
Is this standard notation to represent permutations that are ordered in a certain way? If so what does each represent with their given index/power.
Geometrically $\rho$ is a rotation, and $\sigma$ is a reflection (from German Spiegelung).
The permutations in $S_3$ correspond to 2 rotations over $120^\circ$ respectively $240^\circ$, 3 reflections, and identity. They are the symmetries of an equilateral triangle.
When we pick $\rho$ to represent either of the rotations (123) or (321), and $\sigma$ to represent one of the reflections (12), (23), or (31), all other permutations can be written as the combination of these 2.