Full description

The concept of a table of marks was introduced by W. Burnside in his
1955 book Theory of Groups of Finite Order. Therefore a table of
marks is sometimes called a Burnside matrix.
The table of marks of a finite group G is a matrix whose rows and
columns are labelled by the conjugacy classes of subgroups of G and
where for two subgroups H and K the (H, K)-entry is the number of
fixed points of K in the transitive action of G on the cosets of H
in G. So the table of marks characterizes the set of all permutation
representations of G. Moreover, the table of marks gives a compact
description of the subgroup lattice of G, since from the numbers of
fixed points the numbers of conjugates of a subgroup K contained in
a subgroup H can be derived.
For small groups the table of marks of G can be constructed directly
in GAP by first computing the entire subgroup lattice of G. However,
for larger groups this method is unfeasible. The GAP Table of Marks
library provides access to several hundred tables of marks and their
maximal subgroups.