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: 


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*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.

$S_n$ admits a subgroup (not necessarily symmetric) of order $n^2-1$ if and only if neither of $n\pm 1$ is prime. In particular, it admits such a subgroup if $n$ is odd and at least $5$.
