Proving that the stabilizer of $\{2,4,6\}$ in $D_{6}$ is isomorphic to $S_{3}.$

Question

For me $D_{6}$ is the dihedral group of $2n$ elements as on Wikipedia.

the stabilizer of $\{2,4,6\}$ in $D_{6}$ is defined here Understanding how reflection acts on rotation in $D_{6}.$ which is the following set $\{id. , \rho^{2}, \rho^{4}$, reflection through the axis of symmetry passing through the vertex 2 and between vertices 4 & 6, reflection through the axis of symmetry passing through the vertex 4 and between vertices 2 & 6, reflection through the axis of symmetry passing through the vertex 6 and between vertices 2 & 4 } where $\rho$ means rotation.

My trial to create a homomorphism is this:\

$() \rightarrow id., (12) \rightarrow \rho^{2}, (13) \rightarrow \rho^{4}, (23) \rightarrow $ reflection through the axis of symmetry passing through the vertex 2 and between vertices 4 & 6, $(1 2 3) \rightarrow $ reflection through the axis of symmetry passing through the vertex 4 and between vertices 2 & 6 and $(132) \rightarrow $ reflection through the axis of symmetry passing through the vertex 6 and between vertices 2 & 4 .

Is this a correct isomorphism? Is there a smarter way of saying that they are isomorphic?

Answer 1

You do not need to construct an explicit isomorphism.

1. Compute the number of elements in the stabilizer. It must be $6$.

2. Prove that the stabilizer is not commutative.

3. Use the fact that there is exactly one non-commutative group of order $6$, the group $S_3$.