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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).
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.
Note that χ ρ is constant on conjugacy classes, that is, χ ρ (π) = χ ρ (σ −1 πσ) for all permutations σ. Over other fields the situation can become much more complicated. If the field K has characteristic equal to zero or greater than n then by Maschke's theorem the group algebra KS n is semisimple. In these cases the irreducible ...
If a finite group G has exactly two conjugacy classes of involutions with representatives t and z, then the Thompson order formula (Aschbacher 2000, 45.6) (Suzuki 1986, 5.1.7) states
The five conjugacy classes have representatives e, (12)(34), (12), (123), (1234) and, of these, the Möbius configuration corresponds to the conjugacy class e.
[2] Examples of ICC groups are the group of permutations of an infinite set that leave all but a finite subset of elements fixed, [1]: 908 and free groups on two generators. [1]: 908 In abelian groups, every conjugacy class consists of only one element, so ICC groups are, in a way, as far from being abelian as possible.