$p$-local finite groups where defined by Broto-Levi-Oliver as a generalization of finite groups studied at a prime $p$. Later on, the same authors, defined $p$-local compact groups as a generalization of compact Lie Groups at a prime $p$ and $p$-compact groups.

Examples of $p$-local finite groups which do not correspond to finite groups are known for every prime number $p$. These examples are called exotic $p$-local finite groups.

In the infinite case, very few cases have been studied. In this talk we will see a family of $p$-local compact groups which do not correspond to $p$-compact groups.