I am studying group theory using Armstrong's Groups and Symmetry, one of the biggest problem is that there is no solution manual available. Thus I will rely on you guys!
Find all the rotational symmetries of a cube
I'm not sure if I've done it right. I think there are in total 18 rotational symmetries. There 9 rotational symmetries if rotated by an axis connecting the centre of opposite faces. There are 8 rotational symmetries if connecting opposite vertices as an axis. With the identity, there are a total of 18 rotational symmetries. Am I right?
You counted the rotations about the long diagonal wrongly, because there are 4 such diagonals. There are also more. You could rotate through an axis that goes through the midpoints of opposite edges. That's all you missed out. You may also be interested in reading about the orbit-counting theorem.