I figured out that there are 24 rotational symmetries, shown below.
Rotation fixing:
faces = 9
diagonals = 8
edges = 6
identity = 1
total = 24

Now, I don't know why the symmetry group of the 3-cube has 48 elements; I know it has to do something with reflection but am unable to picture this. Also, my teacher attached the following hint that I have hard time understanding: "Consider the action of the symmetry group on the set of four diagonals." How is this relevant?