Does the octic,

$\tag{1} x^8+3x^7-15x^6-29x^5+79x^4+61x^3+29x+16 = nx^2$

for any constant *n* have Galois group of order 1344? Its discriminant *D* is a perfect square,

$D = (1728n^4-341901n^3-11560361n^2+3383044089n+28121497213)^2$

Surely (1) is not an isolated result. How easy is it to find another family with the same Galois group and same form,

$\tag{2} \text{octic poly in}\ x = nx^2$

Perusing Kluener's "*A Database For Number Fields*" this seems to be the only one. (Though I was able to find a 2-parameter family of a different form.)