An example of a group with $1+np$ Sylow $p$-subgroups I want to find an example of a group $G$ with $1+p$ Sylow $p$-subgroups, where $p$ is a fixed prime. My problem is I don't know a lot of known Sylow subgroups, and the answer of this depends on $p$.
Then I'd like to know if this can be done easily for $1+np$.
Any suggestions?
Thanks.

For $p=2$, the Klein's group $\{1,3,5,7\}$ (mod 8 multiplication) has
3 subgroups (they are the only proper subgroups).
For $p=3$, and $p=5$, the examples are $S_4$ and $S_5$ respectively.
