Transposition 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 transpositions in this group, i.e., permutations that flip two elements and fix the third element. The transpositions 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:

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

Arithmetic functions

Function	Value	Explanation
order of the whole group	6
size of conjugacy class	3
size of automorphism class	3
number of conjugacy classes in automorphism class	1
order of elements in conjugacy class	2

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 2 + 1	$\frac{3!}{(2)(1)}=3$	$\mathrm{l}\mathrm{c}\mathrm{m}\{2,1\}=2$	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 conjugacy class of all "reflections" i.e. things outside the cyclic subgroup of index two	$n=3$	2	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 unipotent elements (note that the conjugacy class is unique when $q$ is even)	$q^2-1=2^2-1=3$	characteristic prime, i.e., prime whose power is $q$. In this case, it is 2	element structure of special linear group of degree two over a finite field, element structure of symmetric group:S3#Interpretation as special linear group of degree two
As the general affine group $GA(1,q)$, $q=3$	the conjugacy class of all maps of the form $x\mapsto 2x+b,b\in {\mathbb{F}}_3$.	$q=3$	order of 2 in ${\mathbb{F}}_3^{\ast }$, which equals 2	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
As the general semilinear group $\mathrm{\Gamma }L(1,p^2)$, $p=2,p^2=4$	the conjugacy class of all maps whose field automorphism part is the non-identity map. Note that for generic $p$, there are $p-1$ such conjugacy classes, but since $p=2$, the conjugacy class is unique.	$p+1=2+1=3$	Since this is the conjugacy class containing the mapping that is simply the field automorphism, it has order 2.	element structure of general semilinear group of degree one over a finite field, element structure of symmetric group:S3#Interpretation as general semilinear group of degree one