what is the smallest non-abelian finite group which has normal, non-abelian subgroups (plural)

Question

I am looking for smallest example of a group $G$ such that:

• $G$ is a finite, non-abelian group
• $G$ is not simple
• $G$ has non-trivial, proper, normal subgroups: $H_1, H_2, \dots $
• $H_1, H_2, \dots $ are not (all) abelian
• There is more than one normal subgroup (hence $H_1, H_2, \dots $)

In particular I would like to generate the multiplication table for such a group, if possible with permutations (all finite groups are subgroups of $S_n$, right!).

I do have access to gap, but I am just beginning with that

Since smallest possibility for non-abelian subgroup is order $6$, and I look for at least $2$ normal subgroups, then I believe $\text{Order}(G)$ must be at least $18$ (as if index of $H_1$ was $2$, then $H_1$ would be the unique normal subgroup). But none of the order $18$ finite groups fit all the above criteria.

?

Answer 1

$D_{12} \cong C_2 \times S_3$ is nonabelian of order $12$, has at least $2$ normal subgroups ($C_2 \times 1$ and $1 \times S_3$), and they are not all abelian.