$x-\,$axis. This *partial* homogeneity is due to the fact that the orthogonal transformations involved in the corresponding covering groups have the $x-\,$axis as their axis of rotation.

[\*[3]{}[|c]{}|]{} & &\
Class & $y$ & $z$\
& &\
& &\
${\mathcal{G}_2}$ & $-\frac{1}{2} \, d_1 \cos\theta$ & $\pm \frac{1}{2} 
\sqrt{d_1^2 \sin^2 \theta - |a|^2}$\
& &\
& &\
${\mathcal{G}_3}$ & $\pm \frac{\sqrt{3}}{6} \sqrt{d_1^2 \sin^2 \theta - 
|a|^2} - \frac{1}{2} \, d_1\cos\theta$ & $\mp \frac{1}{2} 
\sqrt{d_1^2 \sin^2 \theta - |a|^2} - \frac{\sqrt{3}}{6} \, 
d_1\cos\theta$\
& &\
& &\
${\mathcal{G}_4}$ & $\pm \frac{1}{2} \sqrt{d_1^2 \sin^2 \theta - |a|^2} - 
\frac{1}{2} \, d_1\cos\theta$ & $\mp \frac{1}{2} \sqrt{d_1^2 
\sin^2 \theta - |a|^2} - \frac{1}{2} \, d_1\cos\theta$\
& &\
& &\
${\mathcal{G}_5}$ & $\pm \frac{\sqrt{3}}{2} \sqrt{d_1^2 \sin^2 \theta - 
|a|^2} - \frac{1}{2} \, d_1\cos\theta$ & $\mp \frac{1}{2} 
\sqrt{d_1^2 \sin^2 \theta - |a|^2} - \frac{\sqrt{3}}{2} \, 
d_1\cos\theta$\
& &\

We now fit the multiple images $\{C_0,C_1,C_2\}$ within manifolds of class ${\mathcal{G}_6}$. The generators for the covering group of a manifold of this class are $\alpha=(A_1,a)$, $\beta=(A_2,b+c)$ and $\mu=(A_2,b-c)$. The vectors $\{a,b,c\}$ are mutually orthogonal but their lengths are free parameters. For manifolds of class ${\mathcal{G}_6}$ we have two possible configurations, both of them with $C_0$ located at 