# How to correctly enumerate all the schemes of this hexahedral coloring problem?

Choose several colors from the given six different colors to dye six faces of a cube, and dye each two faces with common edges into different colors. How many different dyeing schemes are there?

Note: if we dye two identical cubes, we can make the six corresponding faces of the two cubes dyed the same by proper flipping, then we say that the two cubes have the same dyeing scheme.

``Show[Graphics3D[   Rotate[Cuboid[{0, 0, 0}, {1, 2, 1}], 0 Degree, {0, 0, 1}],    Axes -> True], i = 1;   Graphics3D[   Table[Text[Style[ToString[i++], 15], {x, y, z}], {x, 0, 1}, {y, 0,      2, 2}, {z, 0, 1}]]] `` ``g0 = Graph[(Sort /@        Flatten[Map[Thread[#[] \[UndirectedEdge] #[]] &,         {{1, {2, 3, 5}},          {2, {1, 4, 6}},          {3, {1, 4, 7}},          {4, {2, 3, 8}},          {5, {1, 6, 7}},          {6, {2, 5, 8}},          {7, {3, 5, 8}},          {8, {4, 6, 7}}}]]) // DeleteDuplicates,     VertexLabels -> "Name"]; ChromaticPolynomial[g0, 6]  poly = CycleIndexPolynomial[DihedralGroup,    Array[Subscript[a, ##] &, 6]] ``

The result of the above code is `198030`, but the reference answer is `230`. How to correctly list all the solutions to this problem?