How many different colourings are there of the cube using 3 different colours? A cube is called coloured if each of its faces is coloured by one of 3 given colours. Two colourings are considered to be the same if there is a rotation of the cube carrying one colouring to the other. How would you prove there are exactly 57 different colourings of the cube? 
I think the approach is to use Burnsides orbit counting theorem?

Yes your approach is correct, and do you see that you should use a group of order 24? Which one? See also here for the full answer. But first try it yourself!
