Rectangular stained glass window with different colors Suppose you have six squares of stained glass, all of different colors, and you would like to make a rectangular stained glass window in the shape of a 2 × 3 grid. How many different ways can you do this, taking symmetry into account? (Note that any pattern may be rotated 180°, flipped vertically, or flipped horizontally. You should count all the possible resulting patterns as the same window.) I've tried 6! / 3 but this is not correct. Any tips?

HINT: Representing the six colors by the numbers $1$ through $6$, we see that the following arrangements are equivalent:

$$\begin{bmatrix}1&2&3\\\4&5&6\end{bmatrix}\qquad\begin{bmatrix}3&2&1\\\6&5&4\end{bmatrix}\qquad\begin{bmatrix}4&5&6\\\1&2&3\end{bmatrix}\qquad\begin{bmatrix}6&5&4\\\3&2&1\end{bmatrix}$$

The first is the original; the second is flipped about the vertical axis; the third is flipped about the horizontal axis; and the last is rotated $180^\circ$ degrees (or flipped about each of the axes).

What _should_ you divide by?