The GAP Table of Marks Library TomLib. The concept of the table of marks of a finite group was first introduced by William Burnside in his famous book ”Theory of Groups of Finite order.” In fact many books refer to a table of marks as 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.

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