first, we use Cartesian coordinate to define the matrix representation of the Mirror, 180 degree Rotation and Parity.
i guess the reader can think the other matrix.
the group table is:
where i is identity matrix. we can see it is a group of order 8. there is a sub-group of . since the group table are diagonal symmetry, each element from its own class.
since a rotational group can be represented by 2X2 matrix instead of 3X3 by introducing complex number. those matrix is Pauli’s spinor.
the transform from 3X3 matrix to 2X2 matrix is that…… then we can map the mirror matrix and parity into 2X2 matrix.