Non-PORC behaviour in groups of order p7

Abstract We investigate a family of 3-generator groups G ( p , x , y ) indexed by a prime p > 3 and integers x , y . The groups all have order p 7 and class 3. If x and y are coprime to p , then the order of the automorphism group of G ( p , x , y ) is one of four polynomials in p , where the choice of polynomial depends on the number of roots in GF ( p ) of the polynomial g ( t ) , where g ( t ) = t 3 − x t − y . If x and y are integers such that the Galois group of g ( t ) over the rationals is S 3 , then the number of roots of g ( t ) over GF ( p ) is not a PORC function. So for most pairs of integers x , y the order of the automorphism group of G ( p , x , y ) is not a PORC function. Nevertheless, the frequency with which the different orders of automorphism group arise over all x , y is describable in terms of PORC functions.