Orbit space of $\mathrm{SO}(3)$ irreducible representations $\DeclareMathOperator\SO{SO}$Consider the $7$-dimensional $\mathbb R^7$ real irreducible orthogonal representation of $\SO(3)$. I am seeking a description of the orbit space (when the action is restricted to the sphere) that is as complete as possible. What is it isometric to? What are the orbit types? etc...
It would be even better if there is a nice description of orbits spaces for all real irreps of $\SO(3)$. Any help or reference would be appreciated.
 A: I don't know where the orbit types in this case were first explicitly classified, but it is done in my paper Second order families of special Lagrangian 3-folds, Perspectives in Riemannian geometry, 63–98, CRM Proc. Lecture Notes 40, Amer. Math. Soc., Providence, RI, 2006. MR2237106.  See Proposition 1 of Section 3.  I include the zero, but it's easy to remove that and see what all the nonzero orbit types are.
Because the ring of $\mathrm{SO}(3)$-invariant polynomial invariants on the irreducible representation of dimension $7$ is generated in degrees $2$, $4$, $6$, $10$, and $15$ (with the square of the $15$-degree polynomial expressible as a polynomial in the lower degree ones), you actually have to go quite a distance before you can distinguish all of the orbits.  The possible stabilizer types are $\mathrm{SO}(2)$, $A_4$, $S_3$, $\mathbb{Z}_3$, $\mathbb{Z}_2$, and $\{1\}$.
However, it's not hard to distinguish the $\mathbb{Z}_2$-quotient that you get by dividing by the action of $\mathrm{O}(3)$.
One way to describe the $\mathrm{O}(3)$-orbits on $S^6\subset\mathbb{R}^7$ is as un-ordered triples of points on the $2$-sphere modulo the rotations.
