$\DeclareMathOperator\SO{SO}$Consider the $7$-dimensional irreducible representation of $\SO(3)$. I am seeking a description of the orbit space 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 irreps of $\SO(3)$. Any help or reference would be appreciated.

$\DeclareMathOperator\SO{SO}$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.

Consider the $7$-dimensional irreducible representation $SO(3)$. I am seeking a description of the orbit space 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 irreps of $SO(3)$. Any help or reference would be appreciated.

$\DeclareMathOperator\SO{SO}$Consider the $7$-dimensional irreducible representation of $\SO(3)$. I am seeking a description of the orbit space 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 irreps of $\SO(3)$. Any help or reference would be appreciated.