Combinatorics containig cube

I assumed we're coloring a physical cube, like a block of wood, meaning we are free to rotate it but reflection symmetries are not possible.

Now, if we hold the cube in place, there are $6! = 720$ ways to color the sides. But since we can rotate it, we have to divide by the size of the rotational symmetry group of the cube to get the correct count. There are 24 rotation symmetries of the cube. You can convince yourself of this as follows: You can move a given face $F$ to any of the 6 faces of the cube via rotation. This also determines where the face opposite $F$ goes. Once you've done this, there are 4 ways to rotate the faces adjacent to $F$ . So, there are 24 total rotations. Therefore, there are $\frac{720}{24} = 30$ distinct colorings of the cube