Pentago is a board game with a field with 36 points on an 6x6 board. The board is divided into 4 smaller squares which are 3x3 points big. Each of these smaller squares can be rotated. On each field, either a white marble, black marble or nothing can be placed
The question is, how many possible board positions exist, that are not equivalent to something you could get by turning the squares or looking at the game at another angle?
I tried to tackle this problem with group theory by applying the Burnside Lemma.
Here are the permutations and the corresponding amount of cycles i have came up with:
identity - 36 cycles
rotating whole board - 9 cycles
rotating small square 1 - 30 cycles
rotating small square 2 - 30 cycles
rotating small square 3 - 30 cycles
rotating small square 4 - 30 cycles
mirroring at diagonal axis 1 - 21 cycles
mirroring at diagonal axis 2 - 21 cycles
mirroring at horizontal axis 1 - 18 cycles
mirroring at horizontal axis 2 - 18 cycles
Each field can either be empty, white or black (3 states), so by plugging in these numbers in the equation we get
$\frac{1}{10}\left(3^{36}+3^{9}+4 \times 3^{30}+2 \times 3^{21}+2 \times 3^{18}\right) = \frac{75459110760472392}{5}$
There must be a mistake somewhere, because this is not an integer. Does anyone have a clue, what could be the mistake?