Conjugacy Class in a matrix group with columns of standard basis I want to find the conjugacy classes of this group:
$$G=\{(a_1,a_2,a_3):a_i=\pm e_j\}\subset GL_3(\mathbb C)$$
I know that $S_3$ is one of the subgroups of this group, so I can check which elements of $S_3$ continues to be conjugate in our group,
I know that in $Z(G)$ there is two elements $I,-I$.
I know that I have to check the diagonal matrices on which ones are conjugate, but I don't know how to finish and find the others conjugacy class of this group, and I don't Know how to find the number of the elements of the conjugacy class.
I'll be thankful for help
 A: This group is a wreath product of a cyclic group of order $2$ (i.e. $\pm 1$) with a symmetric group of order $3$. Thus $S_3$ is also a factor group and one can use the classes of $S_3$ as first approximation (i.e. classes of $G$ that will map into the class of $S_3$), which also gives how coordinates are permuted.

*

*The class of the identity in $S_3$ gives matrices that are diagonal with 0,1,2,3 entries that are $-1$, thus
4 classes, including the two in the centre and two of size 3 outside the centre.


*Next come matrices that map to elements of order $2$. Their shape must be $\left(\begin{array}{ccc}0&*&0\\*&0&0\\0&0&*\\
\end{array}\right)$ (or conjugate). You get 4 classes distingiuished by determinant $\pm1$ and element order 2 or 4. All have 6 elements.


*Finally, you get two classes that map to elements of order $3$. Their shape must be $\left(\begin{array}{ccc}0&*&0\\0&0&*\\*&0&0\\
\end{array}\right)$ (or conjugate). One has elements of order 3 and determinant 1, the other order $6$ and determinant $-1$. Both have 8 elements.
