= Crystallographic orbit
A \b[crystallographic orbit] is the set of points generated from one point $\mathbf r$ by all operations of a specified <space group> $G$:
$$
G\mathbf r=\{g\mathbf r:g\in G\}.
$$
Because the <space group> includes lattice translations, the full orbit is infinite. The familiar finite list of equivalent fractional positions lists representatives within one conventional <unit cell>.<Crystallographic orbit in the IUCr Online Dictionary of Crystallography>[Definition]{ref}
For an original example in $P\bar1$, take $\mathbf r=(0.2,0.3,0.4)$. Inversion gives $(-0.2,-0.3,-0.4)$, represented in the same cell as $(0.8,0.7,0.6)$. Lattice translations repeat both sites throughout space. At the origin, inversion produces no additional site, so there is only one representative per primitive cell. The smaller <crystallographic site multiplicity> comes from the larger <site-symmetry group>.
If $\mathbf s=g\mathbf r$, their <site-symmetry groups> obey
$$
G_{\mathbf s}=gG_{\mathbf r}g^{-1}.
$$
Indeed, an operation fixing $\mathbf r$ becomes an operation fixing $g\mathbf r$ after conjugation. This connects spatial equivalence to <conjugate subgroups>.
Several distinct <crystallographic orbits> can belong to the same <Wyckoff position>. For example, replacing the generic coordinates above by $(0.1,0.2,0.3)$ usually produces a different pair, while both pairs have trivial <site-symmetry groups>. A <Wyckoff position> classifies the conjugacy type; an orbit supplies one particular symmetry-generated set of positions.<Crystallographic orbit in the IUCr Online Dictionary of Crystallography>[Crystallographic orbits and site-symmetry groups; Crystallographic orbits and Wyckoff positions]{ref}
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