as in the title given the application $F(x_1,x_2,x_3)=(-\frac{1}{2}x_1-\frac{\sqrt3}{2}x_2+\frac{1}{2};-\frac{\sqrt3}{2}x_1+\frac{1}{2}x_2+\frac{\sqrt3}{6};-x_3)$
1)find all isometries of $\mathbb R^3$ that commute with F
2)find an isometry G of $\mathbb R^3$ that commute with F and write its expression.
my considerations:
this kind of application reminds me a "-" rotation around z axes of $\pi/6$ , so a rotation of $-\pi/6$ is it right ? but i don't have ideas on finding isometries, I thought I could find another application G (not an isometry) just creating $3*3$ matrix and imposing the standard condition $G*F=F*G$