We are making bracelets with 6 stones in a ring, with three different colors of stone... We are making bracelets with 6 stones in a ring, with three different colors of stone. A bracelet must contain at least one stone of each color. Two bracelets are considered to be identical if one is simply a rotation or a flip of the other. How many different bracelets are possible?
What I have so far:
number of possible bracelets = $3^6 - 3(2^6) + 3(1^6) = 540$
There are 12 possible rotations and flips, 6 of each.
|Inv(e)| = 540 
|Inv($r$)| = |Inv($r^5$)| = 0
|Inv($r^2$)| = |Inv($r^4$)| = 0
|Inv($r^3$)| = 6
|Inv($f$)| = |Inv($f^1$)| = |Inv($f^2$)| = |Inv($f^3$)| = |Inv($f^4$)| = |Inv($f^5$)| = 6
|Inv(any combination of rotations and flips)| = 0
Doesn't the number of unique bracelets = $\sum|Inv()|\over12$? When I do this, I get 48.5. What am I doing wrong?

Most of the numbers are right, but remember that there are two kinds of flips: three of them fix no vertices, while three of them fix two vertices.
For the latter kind of flip, there are $36$ invariant necklaces, giving us the correct count of $(540 + 0 + 0 + 6 + 0 + 0 + 6 + 6 + 6 + 36 + 36 + 36)/12 = 56$.
