I am trying to construct all the matrices for the point group D3d operations. I find that the matrices of E, i, S6, inverse of S6, C3, and inverse of C3 are easy, but I do not know how to construct the matrices of the three C2s and the three dihedral reflections. When I construct these matrices, they do not look like those of usual reflections or rotations. I cannot find any resources about finding these more complicated matrices. Any help would be appreciated.


One of the usual setup for the coordination system would be that $x$ axis coincident with a two-fold axis. If we use this setup, we can easily write out the matrix for rotation around this 2-fold axis: $$\begin{bmatrix} 1&0&0\\ 0&-1&0\\ 0&0&-1 \end{bmatrix}.$$

Then you can combine this with the transform matrix of the 3-fold axis to get the transform matrices of the other two 2-fold axes. That's just matrix multiplication.

In this setup, $yz$ is one of the reflection plane and the corresponding matrix is $$\begin{bmatrix} -1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix}.$$

Note that this is just the combination of the previous matrix with inversion. You can apply the 3-fold axis rotation to it (once again, matrix multiplication) to obtain the matrix with regard to other two reflection planes.


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