What's the smallest normal subgroup of $D_{12}\times D_{12}$ that contains $(s,s)$ where $s$ is reflection?

Question

What's the smallest normal subgroup of $D_{12}\times D_{12}$ that contains $(s,s)$ where $s$ is reflection?

My guess is $\langle s,r^2\rangle \times \langle s,r^2\rangle$ but I'm not too sure how to show it is the smallest. Is this correct?

Answer 1

Since we know all normal subgroups of $D_n$, see

Normal subgroups of dihedral groups

it is clear that $\langle s,r^2\rangle$ is the smallest normal subgroup of $D_n$ containing $s$. Now Goursat's Lemma describes normal subgroups in the direct product. This is also explained an MSE, e.g.,

Subgroups of a direct product