I have two matrices in $\mathbb{R^3}$
$A=\begin{bmatrix}0&1&0\\ 1&0&0\\ 0&0&1 \end{bmatrix}$ $B=\begin{bmatrix}0&0&1\\ 1&0&0\\ 0&1&0 \end{bmatrix}$
I believe these matrices generate the group $SO_3(\mathbb{F2})$ which is isomorphic to $D3$, the dihedral group of order 3. As $A$ has order 2, $B$ has order 3 and $(AB)^2 = I_3$.
Have I got all that correct?
And then $A$ a change of basis matrix that maps the x axis $\to$ y axis. And $B$ is a change of basis matrix that maps x axis $\to$ y axis, y axis $\to$ z axis, z axis $\to$ x axis.
Have I got all that correct?