The symmetric group S3

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 to the dihedral group $D_3$ 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, 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 are $\{e\}$, the three transpositions, and the two $3$-cycles. Its subgroups are $\{e\}$, three subgroups of order $2$, the alternating group $A_3$ of order $3$, and $S_3$. Only $\{e\}$, $A_3$ and $S_3$ are normal.