3-cycle in symmetric group:S3

We consider the group $G$ 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 $\{1,2,3\}$.

We are interested in the conjugacy class of 3-cycles in this group, i.e., permutations that move all elements and have order three. The 3-cycles form a single conjugacy class and also form a single orbit under the action of the automorphism group of $S_{3}$.

The full list of elements in the conjugacy class is:

$\{(1,2,3),(1,3,2)\}$

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 $G$	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	${\frac {3!}{3}}=2$	$\operatorname {lcm} \{3\}=3$	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 $D_{2n}$, $n=3,2n=6$ ($n$ odd)	the generator of the cyclic piece of order $n$	2 (valid for $n\geq 3$)	$n=3$	element structure of dihedral groups, element structure of symmetric group:S3#Interpretation as dihedral group
As the special linear group $SL(2,q)$, $q=2$ ($q$ even)	the unique conjugacy class of semisimple elements not diagonalizable over the field itself	$q(q-1)=2(2-1)=2$	Must divide $q^{2}-1$ in general, in this case, it is $q^{2}-1=2^{2}-1=3$	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 $GA(1,q)$, $q=3$	the conjugacy class of all maps of the form $x\mapsto x+b,b\in \mathbb {F} _{3}^{\ast }$.	$q-1=2$	$q=3$	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