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 $n \ge 3$)	$n = 3$	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