Physics relevant examples of non-simple groups constructed from a semidirect product?

Question

In the first part of this article, the difference between simple and non-simple groups is explained. It says that non-simple groups can for example be constructed by taking the direct product of $G \times H$ of two groups $G$ and $H$. If I understand this right, the standard model group $SU(3)_c \times SU(2)_L \times U(1)$ is a physics example of this type of non-simple group, where the components are normal subgroups (?).

Another possibility to construct a non-simple group is by making use of the simidirect product $G\rtimes_{\phi} H$ which means that the product of elements of the two subgroups $G$ and $H$ is twisted by some homomorphism $g_1\phi_{h_2}(g_2)$.

What are some interesting physics examples and applications of this second type of non-simple groups?

Comments

I think this belongs to Q&A section, although a little explanation of the motivation will make it a nicer question.