Template:Particular automorphism class
We consider the group defined as symmetric group:S3, i.e., the symmetric group of degree three, which for convenience we take to be the symmetric group acting on the set .
We are interested in the conjugacy class of 3cycles in this group, i.e., permutations that move all elements and have order three. The 3cycles form a single conjugacy class and also form a single orbit under the action of the automorphism group of .
The full list of elements in the conjugacy class is:
Arithmetic functions
Function 
Value 
Explanation

order of the whole group 
6 

size of conjugacy class 
2 

size of automorphism class 
2 

number of conjugacy classes in automorphism class 
1 

order of elements in conjugacy class 
3 

Description in alternative interpretations of the whole group
Interpretation of 
Description of conjugacy class 
Verification of size computation 
Verification of element order computation 
More information

As the symmetric group of degree three 
corresponds to the partition 3 


element structure of symmetric groups, element structure of symmetric group:S3#Interpretation as symmetric group, conjugacy class size formula in symmetric group

As the dihedral group , ( odd) 
the generator of the cyclic piece of order 
2 (valid for ) 

element structure of dihedral groups, element structure of symmetric group:S3#Interpretation as dihedral group

As the special linear group , ( even) 
the unique conjugacy class of semisimple elements not diagonalizable over the field itself 

Must divide in general, in this case, it is 
element structure of special linear group of degree two over a finite field, element structure of symmetric group:S3#Interpretation as general linear group of degree two

As the general affine group , 
the conjugacy class of all maps of the form . 


element structure of general affine group of degree one over a finite field, element structure of symmetric group:S3#Interpretation as general affine group of degree one
