I am looking for an example of a locally compact group $G$ that is type $I$ and has a unitary representation $(\pi, H_\pi)$ of $G$ such that $\pi(G)\cong B(H)$, as C*-algebras, where $H$ is an infinite dimensional (separable) Hilbert space. By $\pi(G)$, I really mean the (extension to $C^*_{\rm{max}}(G)$) of the integrated representation of $\pi$.
I know that I have to look for non-second countable examples, otherwise $\pi(G)$ would be separable as a C*-algebra (while $B(H)$ is not). I also have to avoid virtually abelian groups and compact groups because of theorem such as Peter-Weyl.
Thank you for your help !