Problem:
Let $G$ be the set of 3-by-3 matrices such that they have exactly one nonzero entry in each row, and exactly one nonzero entry in each column, and the nonzero entries are only $\pm1$. Prove that $G \cong S_3 \rtimes H$ where $H$ is the group of 3-by-3 matrices that are diagonal with $\pm 1$ along the diagonal, and $S_3$ is the group of bijections of $\{1,2,3\}$
I can see that the semidirect product shuffles the columns of a matrix in $H$ to all the possible combinations, but I can't prove the necessary properties. They are,
$$H \triangleleft G, S_3 \leq G$$ $$H \cap S_3=\{e\}$$ $$H S_3 = G$$