$S_{10}$, the symmetric group and GAP

Question

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.

What do you mean by "GAP couldn't handle the symmetric group $S_{10}$"? Please tell what exactly have you tried and what happened. — Olexandr Konovalov, Jul 28 '13 at 20:25

score 2 · Accepted Answer · edited Jul 08 '20 at 09:15

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

$S_{10}$, the symmetric group and GAP

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