Let G be a finite group with n conjugacy classes of subgroups C_1, ldots, C_n and representatives H_i in C_i, i = 1, ldots, n. The table of marks of G is defined to be the n times n matrix M = (m_{ij}) where m_{ij} is the number of fixed points of the subgroup H_j in the action of G on the cosets of H_i in G.
Since H_j can only have fixed points if it is contained in a one point stablizer the matrix M is lower triangular if the classes C_i are sorted according to the following condition; if H_i is contained in a conjugate of H_j then i leq j.
Moreover, the diagonal entries m_{ii} are nonzero since m_{ii} equals the index of H_i in its normalizer in G. Hence M is invertible. Since any transitive action of G is equivalent to an action on the cosets of a subgroup of G, one sees that the table of marks completely characterizes permutation representations of G.
The entries m_{ij} have further meanings. If H_1 is the trivial subgroup of G then each mark m_{i1} in the first column of M is equal to the index of H_i in G since the trivial subgroup fixes all cosets of H_i. If H_n = G then each m_{nj} in the last row of M is equal to 1 since there is only one coset of G in G. In general, m_{ij} equals the number of conjugates of H_i which contain H_j, multiplied by the index of H_i in its normalizer in G. Moreover, the number c_{ij} of conjugates of H_j which are contained in H_i can be derived from the marks m_{ij} via the formula
[ c_ij = fracm_ij m_j1m_i1 m_jj. ]
Both the marks m_{ij} and the numbers of subgroups c_{ij} are needed for the functions described in this chapter.
GAP 3.4.4