I would like to calculate all possible combinations of symmetries, that a $10\times 10$ board of Dots and Boxes can have. For this purpose we only care about what lines have been drawn, but not which player owns which box. For example, a $3\times 3$ board might look as follows:
o-o o
| |
o-o-o
| |
o o-o
I've looked up the following notation: $\rho_0, \rho_1, \rho_2, \rho_3$ for $0$, $90$, $180$ and $270$ counterclockwise rotation; $\mu_1, \mu_2$ for mirroring about vertical and horizontal axis; $\delta_1$, $\delta_2$ for mirroring about top-left-bottom-right and top-right-bottom-left diagonals respectively.
The example above would therefore have $(\rho_0,\rho_2,\delta_1,\delta_2)$ symmetries.
I wrote a script to generate all $3\times 3$ boards and calculate which combinations of symmetries they have, but I'm not sure whether the result generalizes to $10\times 10$ case, as for example, $2\times 2$ boards have less possible combinations than $3\times 3$. Is there an analytical way to do it?
All possible combinations of symmetries for $3 \times 3$ boards are:
$(\rho_0)$
$(\rho_0,\rho_2)$
$(\rho_0,\mu_1)$
$(\rho_0,\mu_2)$
$(\rho_0,\delta_1)$
$(\rho_0,\delta_2)$
$(\rho_0,\rho_1,\rho_2,\rho_3)$
$(\rho_0,\rho_2,\mu_1,\mu_2)$
$(\rho_0,\rho_2,\delta_1,\delta_2)$
$(\rho_0,\rho_1,\rho_2,\rho_3,\mu_1,\mu_2,\delta_1,\delta_2)$