Here is a question. As you see, the problem's established on finding an element of a certain order in $S_{10}$. I tried to do this question by using GAP. But, GAP couldn't handle the symmetric group $S_{10}$. What can we do in this situation? Is there a way for defining this large finite group for GAP? Thanks for your time.
I'd be really interested to see what you've tried. Anyhow, and in particular, for the symmetric group where conjugacy classes are well known, you can easily examine orders of their representatives:
gap> 15 in List(ConjugacyClasses(SymmetricGroup(10)),c->Order(Representative(c)));
true
Moreover, you can use the character table library without even constructing the group itself:
gap> 15 in OrdersClassRepresentatives(CharacterTable("S10"));
true