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Conjugacy classes may be referred to by describing them, or more briefly by abbreviations such as "6A", meaning "a certain conjugacy class with elements of order 6", and "6B" would be a different conjugacy class with elements of order 6; the conjugacy class 1A is the conjugacy class of the identity which has order 1.
The outer automorphism also exchanges permutations of type (12)(345) with (123456) (class 2 1 3 1 with class 6 1). For each of the other cycle types in S 6, the outer automorphism fixes the class of permutations of the cycle type. On A 6, it interchanges the 3-cycles (like (123)) with elements of class 3 2 (like (123)(456)).
The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a concise form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on any representative of the respective conjugacy class of G (because characters are class functions).
Thus the conjugacy class within the Euclidean group E(n) of inversion in a point is the set of inversions in all points. Since a combination of two inversions is a translation, the conjugate closure of a singleton containing inversion in a point is the set of all translations and the inversions in all points.
Characters are class functions, that is, they each take a constant value on a given conjugacy class. More precisely, the set of irreducible characters of a given group G into a field F form a basis of the F-vector space of all class functions G → F. Isomorphic representations have the same characters.
For groups realized over algebraically closed fields, there is a single conjugacy class of Borel subgroups. Borel subgroups are one of the two key ingredients in understanding the structure of simple (more generally, reductive ) algebraic groups, in Jacques Tits ' theory of groups with a ( B , N ) pair .
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The entries in the same row are in the same conjugacy class. Every entry appears once in each column, as seen in the file below. Every entry appears once in each column, as seen in the file below. The positions of permutations with inversion sets symmetric to each other have positions in the table that are symmetric to each other.