(a) In cubic symmetry, there are six C₄ rotations (90° rotations about the x, y, and z axes), eight C₃ rotations (120° rotations about the [111] directions, or cube corners), and three C₂ rotations (180° rotations about the x, y, and z axes), among other symmetry operations. Write down 3 × 3 matrix representations of all of these operations, using the x, y, and z unit vectors as the basis functions of the matrix representations.

(b) Does this set of matrices, along with the identity matrix, form a group? Is multiplication of these matrices commutative in any or all cases?

(c) Can you come up with a 2 × 2 representation which has the same multiplication properties? It does not matter what your matrices are, as long as they have the same multiplication properties. 

How about a 1 × 1 representation? What is the simplest possible matrix representation that has the same multiplication properties?