I am looking for examples of totally disconnected, locally compact groups, which are not sigma-compact. For a start any such an example would do, so that I can a feeling for those groups and how to find them. (In fact I was not very successful so far in finding appropriate examples. Discrete uncountable groups do not count...)

In particular I am looking for such groups $G$, which are algebraically a (semi)direct product of subgroups $N$ and $H$, with the topology being different to the product topology.

Note that sigma-compactness is an obstruction to such an example of a group topology, which is the reason for this particular restriction.