## Symmetry group of 3x3 array isomorphic to S3xS3xS2

Hello everyone,

I'm having some trouble constructing an isomorphism between groups. I have a set of parameters $\left{j_{kl} | 1 \leq k,l \leq 3 \right}$ and a coefficient depending on these parameters. It is generally written as the 3x3 array
$\left{ \begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix} \right}$
(usually surrounded by curly brackets, but the tex code doesn't seem to work).

The coefficient (a 9j symbol, Wigner 9j-Symbol -- from Wolfram MathWorld) is (up to a sign, which is irrelevant) unchanged upon any permutation of the rows or columns and under transposition of the array. Thus one can define symmetry operations as such permutations that leave the symbol invariant - a simple example would be the interchanging of rows 1 and 2.

It's not hard to show that these symmetry operations form a group, but I now wish to show that this group is isomorphic to the group $S_3 \times S_3 \times S_2$. The statement seems nearly trivial : both of the $S_3$'s that appear are linked to the permutations of the three rows and columns, and the group $S_2$ corresponds to the transposition operation. However, I can't seem to find a complete, satisfying argument or proof.

What I have found is that labelling the rows $r1,r2,r3$ shows that the row operations on the symbol form a group isomorphic to $S_3$. The same holds for the column operations by labeling the columns $c1,c2,c3$. Furthermore, any row permutation acts as an identity operator on $c1,c2,c3$ and any column permutation does not affect $r1,r2,r3$. Thus, the subgroup of row and column operations form a group isomorphic to $S_3 \times S_3$.
I don't see, however, how I could nicely introduce transpositions and show that the total group is isomorphic to $S_3 \times S_3 \times S_2$.

Can anyone help finding a nice argument? It does not matter if it is entirely different from the semi-proof above.