A tesseract has $384$ symmetries. How do I get the actual $384$ matrices corresponding to them?
My attempt: The Tesseract has $8$ hyper-faces (which are regular 3-d cubes). One can rotate it so that each of them gets a turn being the "bottom" face. Then, multiply these rotation matrices by the symmetries of a 3-d cube (which one can get from the answer here). These matrices somehow need to be embedded into the $4 \times 4$ identity metrix. I'm struggling with the details and how I'd confirm I got the right ones.
I would tackle it like this:
Assume you have coordinate directions $x,y,z,w$. The positive $x$ direction can map to the positive or negative direction of any of the $4$ dimensions. So you have $8$ choices there. That gives you a $+1$ or $-1$ in one cell of the matrix, mapping the first input dimension to one of the output dimensions.
Next you get to map the second input dimension to one of the remaining $3$ dimensions, again with a choice of sign. That's another entry in your matrix.
Likewise for third and fourth. You get a total of $8×6×4×2=384$ combinations.
Focusing a bit more on the matrices, you can start with all zeros. Then you put a $+1$ or $-1$ in any column of the first row. Then likewise for the second row but only where the first row is zero. Then likewise for third row where both first and second row have zero. And finally a $+1$ or $-1$ in the fourth row in the sole column where all three rows above have zeros.
Your approach with focusing on one cube-shaped cell likely will make things more complicated, because it breaks the symmetry, making one dimension handled differently form the others.