Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$ with $1$ and $2$ appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rotated to obtain a dice that has the same numbers and colors on the corresponding faces as the other one. Find the number of distinct dice that can be designed.
In the above question, I fixed $2$ faces above and below containing number $1$ and $2$ and via circular permutation method can say $(4-1)!$ Permutations of side $4$ faces and since $2$ opposite faces are of same color and we have 3 sets of opposite faces and $2$ different colors so $2×2×2$ ie $8$ diff. Color combinations. And if we combine numerical permutations with colored ones we get $8×3!$, i.e. $48$.
BUT my doubt is when we fixed the above two faces and said the circular permutations $6$ can't we switch the faces and rotate the dice two get a similar dice.