
Table of marks
Hi all,
I'm trying to figure out how the table of marks is calculated. For those of you unfarmiliar with it it is defined on the wikipedia page here (under the section marks), the important bit being:
For each pair of subgroups $\displaystyle H,K \le G$ define
$\displaystyle
m(K, H) = \left[G/K]^H\right = \# \left\{ gK \in G/K \mid HgK=gK \right\}.
$
Then finding the groups subgroups, and then putting them into conjugacy classes of subgroups, we get a string of representative subgroups $\displaystyle G_1, \dots, G_N$ where $\displaystyle G_1$ is the trivial group and $\displaystyle G_N$ is the whole group. Then we can make a N x N matrix where the i,j th entry is $\displaystyle m(G_i, G_j) $.
So for $\displaystyle S_3$:
$\displaystyle
\begin{array}{ccccc}
S_3 &1 &Z_2& Z_3& S_3\\
S_3/1 & 6 & . & . & . \\
S_3/Z_2 & 3 & 1 & . & . \\
S_3/Z_3 & 2 & 0 & 2 & . \\
S_3/S_3 & 1 & 1 & 1 & 1
\end{array}
$
I can calculate the top left entry obviously, and the bottom row, but I literally have no idea how to calculate the others. If someone could talk me through how to calculate another more difficult entry, that is a difficult $\displaystyle m(G_i, G_j) $, it would be of great help.
Many thanks in advance for any help, and have a merry christmas everyone!

Sorry wikipedia page here: Burnside ring  Wikipedia, the free encyclopedia , didnt seem to link correctly.