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.
?