Juergen Ecker asked:

does anybody have an idea, what a presentation for the symmetric group
of order n could look like?

There are several presentations for the symmetric groups given in

H.S.M. Coxeter, W.O.J. Moser
Generators and relations for discrete groups.
Ergebnisse der Mathematik und ihrer Grenzgebiete,
Springer Verlag.

Select the one you like best.

Joachim Neubueser