The concept of a table of marks was introduced by W. Burnside in his book ``Theory of Groups of Finite Order'', see Bur55. 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 A and B the (A, B)--entry is the number of fixed points of B in the transitive action of G on the cosets of A 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 B contained in a subgroup A can be derived.
A table of marks of a given group G can be constructed from the subgroup lattice of G (see Constructing Tables of Marks). For several groups, the table of marks can be restored from the GAP library of tables of marks (see The Library of Tables of Marks).
Given the table of marks of G, one can display it (see Printing Tables of Marks) and derive information about G and its Burnside ring from it (see Attributes of Tables of Marks, Properties of Tables of Marks, Other Operations for Tables of Marks). Moreover, tables of marks in GAP provide an easy access to the classes of subgroups of their underlying groups (see Accessing Subgroups via Tables of Marks).
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