This question is linked to a previous one that I asked:
Attempting to find a specific similarity (equivalence) matrix
I have a group of 24 elements, with two generators. I need to find an equivalence matrix $Q$ such that $A_i=Q^{-1}B_iQ$ where $B_i$ are permutation matrices and $A_i$ are block diagonal matrices, all with the same block diagonal form. The same $Q$ for all $A_i$ and $B_i$ needs to be found.
This is related to the symmetry group $O$ (rotations of a cube).
I read: http://www.mathworks.co.uk/matlabcentral/newsreader/view_thread/295993 and through suggested methods can find $Q$ for a specific pair $A_i,B_i$ but not a specific $Q$ that works for all pairs. I also find that each $Q$ is complex but it needs to be real.
I have a feeling that the form of the $Q$ that I'm looking for has no zeros and is fully populated with $1$ and $-1$, maybe multiplied by a constant, say $1/24$.
Help is much appreciated!
Edit: To clarify, for each element $g_1,...,g_{24}$ in my group $G$, I have a corresponding matrix representation $B_1,...,B_{24}$ with their corresponding block diagonal form $A_1,...,A_{24}$. I know all $A_i$ and $B_i$ (I have calculated these using the two generators).
Edit 2: The block diagonal $A_i$ take the form shown in the image.
The blocks with the same colour represent equivalent blocks (see comments from Marc below).