Example of a finite group

Question

Hi everyone,

I am looking for an example of finite group $G$ such that (a) the number of elements of order $2$ group $G$ is $p(p+1)/2$ or $p(p-1)/2$ ($p$ is a prime divisor of order $G$) (b) the number of Sylow $p-$subgroups of $G$ is $1$ and order of a Sylow $p-$subgroup $G$ is $p^2$ (c) for every prime divisor $r$ (not $p$, $G$ has not any element of order $p^2$) of order $G$, group $G$ has a element of order $rp$.

Any comment and/or advice would be greatly appreciated.

Answer 1

The group $G = C_2 \times C_2 \times C_3 \times C_3$ has this property (for $p=3$).