Is this group the 4-permutation group?

Question

Recently I came along the set of $3\times 3$ matrices defined by taking any three vectors of (-1,-1,-1) or the three standard unit basis vectors as your columns. For example, one might take $$\left(\begin{array}{ccc} -1 &-1& -1\\ 1 &0 &0 \\ 0& 1 &0 \end{array}\right),$$ or the identity. Both are of the sort I am interested in. More specifically, I suspect that there is an isomorphism between this set of matrices and the 4-permutation group, $S_4$. However, I am not sure how to prove this. Help, of course, would be appreciated.

Answer 1

Hint. Your group naturally acts on the set $$\left\lbrace\left(\matrix{1\\0\\0}\right),\left(\matrix{0\\1\\0}\right),\left(\matrix{0\\0\\1}\right),\left(\matrix{-1\\-1\\-1}\right)\right\rbrace$$ by permutations, via the usual matrix-vector product.

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Example: your matrix $$\left(\begin{array}{ccc} -1 &1& 0\\ -1 &0 &1 \\ -1& 0 &0 \end{array}\right),$$ is the permutation $$\left(\matrix{1\\0\\0}\right) \to \left(\matrix{-1\\-1\\-1}\right) \to \left(\matrix{0\\0\\1}\right) \to \left(\matrix{0\\1\\0}\right)\to \left(\matrix{1\\0\\0}\right) $$

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Note also that the usual $3\times 3$ permutation matrices (whose columns are all standard unit basis vectors) correspond to the stabilizer of $\ \left(\matrix{-1\\-1\\-1}\right)$ and form a copy of $S_3$ inside $S_4$.