Main outstanding problem

Given that the prime values of the quadratic 27*x^2+27*x+7 are precisely those primes p = 1 (mod 3) for which ((p-1)/3)!^3 = 1 (mod p ), the nagging question is: what determines the value of ord[p]((p-1)/3)! ?

In short, what makes such a prime p be in the (earlier) sequence

3571, 4219, 13669, 25117, 55897, 89269, 102121, 170647, � (2)

rather than in the (earlier) sequence

7, 61, 331, 547, 1951, 2437, 7351, 8269, 9241, 10267, 23497, � (4)