Package:pari
Version: 2.6.1

Tested with development git-fed73f7.

This also happens in pari-stable, version 2.5.5

Tested with development git-6d50e19.

Here is an example gp session demonstrating the problem:
? p=x^11 + 627*x^4 - 584
%8 = x^11 + 627*x^4 - 584
? factormod(p,1069)
%9 =
[Mod(1, 1069)*x^4 + Mod(650, 1069)*x^3 + Mod(713, 1069)*x^2 + Mod(98, 1069)*x + Mod(433, 1069) 1]

[Mod(1, 1069)*x^7 + Mod(419, 1069)*x^6 + Mod(601, 1069)*x^5 + Mod(10, 1069)*x^4 + Mod(266, 1069)*x^3 + Mod(831, 1069)*x^2 + Mod(10
11, 1069)*x + Mod(169, 1069) 1]

? polgalois(p)
%10 = [110, -1, 4, "F_110(11)=11:10"]
?

The factormod() command shows that the Galois group of p contains a 7-cycle. But this is not consistent with the order of the Galois group being 110 as reported by polgalois.