Use burnside's lemma to solve the follwing: You have a cake (evenly) split up in 6 parts and you want to divide 2 candles and chocolates over the cake to make a unique (thus all versions which are rotated or mirrored) cake (it is allowed to put multiple candles and/or chocolates on one piece of cake).
What I have so far is that this problem is isomorphic to $D_6$ and has the conjugation classes of $\{e\}, \{r,r^5\}, \{r^2,r^4\}, \{r^3\} , \{s,r^2s,r^4s\}, \{rs,r^3s,r^5s\}$ and according to me the $X^g$ which respectivily to the classes are $6^4, 0,0,3^2,5^2,3^2$. However filling this in in burnside's lemma gives $117,25$ which is not the awsner as it is not a natural number. Can somebody tell me what I am doing wrong here?
Your first count ($6^4$) is incorrect as it treats the two candles as different and the two pieces of chocolate as different. (You did not make this mistake in the other calculations.)
It should be $\left(\binom{6}{2}+6\right)^2 = 441$.