Can you give an example of a non solvable group whose at least one Chief factor is a product of more than one simple group (i.e. it should not be a simple group).
Edit: Thanks @HallaSurvivor for suggestion. I am a Research scholar in mathematics. I have enough background in group theory.
The reason why I am asking this questions is that I have not study this concept of Chief series before. Almost all the example I have seen has the chief factors which are simple. But by definition it can be product of simple. So I just wanted to see such examples. Thank you!
You can construct such examples using wreath products.
The smallest example is $G=A_5 \wr C_2$, the wreath product of the simple group $A_5$ with the cyclic group of order $2$.
It has a normal subgroup $N \cong A_5 \times A_5$, which is also a chief factor, and $|G/N| = 2$. If $g \in G \setminus N$, and we denote the two direct factors of $N$ by $N_1$ and $N_2$, then $g^{-1}N_1g = N_2$ and $g^{-1}N_2g = N_1$.