[GAP Forum] size of permutation groups

[Dima Pasechnik]

Alan, 
By far the fastest practical way to check if you got S_n is to
find few (pseudo)random elements, and compute their orders.
When  doing this, it's important not to trigger a computation of 
the strong stab. chain. 
Here is a snippet from my old GAP files :

# q is a group
gprimes:=function(q)
local ra, i;
ra:=[];
for i in [1..500] do
ra:=Union(FactorsInt(Order(PseudoRandom(q))),ra);
od;
return ra;
end;

Roughly speaking, the primes occurring will provide a fingerprint
matching S_n. I can't recall who invented this simple trick.

HTH,
Dmitrii