A $3 \times 3 \times 3$ magic cube is a three-dimensional array of the consecutive integers $1$ through $27$, with the special property that the sum along any row, any column, any pillar, or any of the four space diagonals is equal to the same number. (Rotations of a certain solution are considered the same solution and therefore not counted.)
How many different $3 \times 3 \times 3$ magic cubes are there?
I got that the sum of each of these should be equal to $\frac19\left(\frac{27\cdot28}{2}\right)=42$, but I have no idea how to proceed after that.
Any help would be appreciated!
Up to rotation and reflection, there are four.