Subgroup of Order $n^2-1$ in Symmetric Group $S_n$ when $n=5, 11, 71$ $G$ is a symmetric subgroup of symmetric group $S_n$ acting on $n$ objects where $n=5, 11, 71$ and order of $G$ is $n^2-1$.
Question: 


*

*Does $G$ (as defined above for $n=5, 11, 71$) exist?

*How can I compute such calculation ? Is there a online system/ website I can use? in this case, some introductory infromation will help.

This is only a partial attempt at tackling a more general underlying question.
Note that if $n \ge 5$ is odd, then $n^{2} - 1$ divides $n!$.
In fact $n^{2} = (n - 1) (n +1)$. Clearly $n-1$ divides $n!$. As to $n+1$, since $n$ is odd, $n+1$ is even, so
$$
n+1 = 2 \cdot \frac{n+1}{2}.
$$
Clearly also $2$ and $\frac{n+1}{2}$ divide $n!$,
Now if $n \ge 5$, we have that
$$
n - 1,  2, \frac{n+1}{2}
$$
are distinct.

Update 1
Direct computation with GAP show that $S_{7}$ has three conjugacy classes of subgroups of order $48$.

Update 2
Direct computation with GAP show that $S_{9}$ has seven conjugacy classes of subgroups of order $80$.
