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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?

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up vote 1 down vote accepted

Yes, all is correct (except that I had to figure out your matrix $B$ based on the information afterwards).

Note also that $D_3\cong S_3$, and these matrices just generate the symmetric group of the standard basis $i,j,k$.

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