Non-PORC behaviour in groups of order p7

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)) isone 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)-xt-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. Crown Copyright (C) 2016 Published by Elsevier Inc. All rights reserved.