There are two factors of 5 in the product, not counting n.  This implies that n is at least 10 since the factorial needs to be at least 10 for two factor of 5.  But n cannot be 10 since that would make the product have three factors of 5.  So n is at least 11.

All the prime factors in the product are at most 7, not counting n.  Then n cannot be 13 or higher, as this would add at least two higher primes to the factorial where the product can only add primes from n, which is only one unless n>=143, which is far too large for the puzzle.  So n is at most 12.

The intersection of the constraints is n=11 or n=12.  Both 11! and 12! have 11 as a factor.  In order for 11 to appear in the product the factor of 11 must come from n.  Then n must be 11.

Pull out the calculator and verify: 11! = 39916800 and 6*7*24*60*60*11 = 3991680.