Hello all I shall present my question and then some of the working out I have done.
Question
Let $\Gamma$ be the group of symmetries of a cube (including reflections) centred at the origin. I have managed to show the the order of $\Gamma $is $48$ and that it can be generated by the following matrices: (when the Lie Group is considered to be acting on $\mathbb{R}^3 $ by standard action)
$\kappa = \begin{pmatrix} -1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0&1 \end{pmatrix}$ $R_x= \begin{pmatrix} 1 & 0 & 0\\ 0 & 0 & -1 \\ 0 & 1&0 \end{pmatrix}$ $R_y= \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0 \\ -1 & 0&0 \end{pmatrix}$
A) Show that there are seven orbit representatives for the conjugacy classes
of isotropy subgroups.
B) Compute the seven conjugacy classes of isotropy subgroups.
My attempts
I find visualising things in 3d very difficult so I don't think I have done a great job but I will show where I managed to get.
Firstly the origin is quite easy, its isotropy subgroup is clearly the entire group.
Secondly any point on an axis is manageable as intuitively they will have $D_4$ symmetry.
I don't exactly know how to show isotropy subgroups are conjugate without direct calculation so any explanation on this would really help me, or any intuition behind why they are without directly calculating it.
Thanks a lot :)