---
tag: 09H9Y0Q11W
kind: example
name: The symmetric group S3
msc: [20B30]
instance_of: [0515A6D2Y1, 0JZ27DMJ5D]
uses: [0RJMNEN3BJ, 09GK3DE0WV, 0D1VWQGZZ7]
sources:
  - url: https://en.wikipedia.org/wiki/Dihedral_group_of_order_6
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Dihedral_group_of_order_6
    locator: Permutations of a set of three objects
  - url: https://en.wikipedia.org/wiki/Dihedral_group_of_order_6
    locator: Conjugacy classes
same_as:
  wikidata: Q2204630
  wikipedia: Dihedral group of order 6
---

## Example

The symmetric group $S_3$ has the six elements $e$, $(1\ 2)$, $(1\ 3)$, $(2\ 3)$, $(1\ 2\ 3)$ and $(1\ 3\ 2)$.
It is [[isomorphic|08MG7H8JW5]] to the [[dihedral group|0JZ27DMJ5D]] $D_3$ of the symmetries of an equilateral triangle, when you number the vertices $1$, $2$, $3$.
The $3$-cycles are the rotations, and the transpositions are the reflections.
It is not [[abelian|082GHNPJ53]], because $(1\ 2)(2\ 3) = (1\ 2\ 3)$ but $(2\ 3)(1\ 2) = (1\ 3\ 2)$.
Thus $S_3$ is the smallest group that is not abelian.
Its [[conjugacy classes|0DXD4QSZ8A]] are $\{e\}$, the three transpositions, and the two $3$-cycles.
Its subgroups are $\{e\}$, three subgroups of order $2$, the [[alternating group|0Z5905Q73D]] $A_3$ of order $3$, and $S_3$.
Only $\{e\}$, $A_3$ and $S_3$ are [[normal|0CB0DX5BVV]].