P 6/mmm
Number
191
Full symbol
P 6/m 2/m 2/m
Schoenflies
D 6h 1
Hall symbol
-P 6 2
Crystal system
hexagonal
Crystal class
6/mmm , order 24 — projected
Laue class
6/mmm , order 24
Patterson symmetry
P 6/mmm Patterson
map , which is computed from the intensities and therefore needs no phases. Friedel's law
makes it centrosymmetric whatever the crystal is, and a screw or a glide loses its
translation, because the map is a function of interatomic vectors rather than of positions.
Chirality
some operation turns an object inside out — a mirror, an inversion or an
inversion axis — so a structure built from one enantiomer cannot have this symmetry:
the group would demand the other hand alongside it.
Seen in the projection .
Lattice
P (primitive)
General position
24 equivalent
points — the coordinates , drawn
Setting
standard
Twinning by merohedry
This group’s point group already has the full symmetry of its lattice, so there is no operation left over to twin by: it cannot twin by merohedry. That is the ordinary case — a crystal twins by merohedry only when it is less symmetric than the lattice it sits on.
These are the twins by merohedry : every reflection of one orientation falls exactly on a reflection of the other, which is what makes them hard to spot and easy to refine wrongly. Two other kinds are not derived here and cannot be, from a space group alone. Reticular merohedry puts the two orientations on a common sublattice, so only some reflections overlap; finding those needs a search over superlattices and a decision about how much overlap counts. Pseudo-merohedry happens when a measured cell is accidentally close to a higher symmetry — that is a fact about one crystal and a tolerance, not about a space group.
Symmetry elements
2-fold rotation axis at x, y, 0 along [11̅0] inversion centre at 0, 0, 0 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 0, 0 2-fold rotation axis at 0, y, 0 along [010] inversion centre at 0, 1/2, 0 2-fold rotation axis at x, y, 0 along [120] inversion centre at 0, 0, 0 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 1, 0 2-fold rotation axis at x, y, 0 along [110] 2-fold rotation axis at x, y, 0 along [210] b 2₁ screw axis at x, y, 0 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [11̅0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [110], advancing 1/2, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [11̅0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, y, 0 along [120] 2₁ screw axis at x, y, 0 along [110], advancing 1/2, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 6̅ inversion axis at 1/3, 2/3, z along [001], inverting through 1/3, 2/3, 0 2₁ screw axis at x, y, 0 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention inversion centre at 1/2, 0, 0 2-fold rotation axis at x, 0, 0 along [100] 2-fold rotation axis at x, y, 0 along [210] 2₁ screw axis at 1/2, y, 0 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, y, 0 along [110] mirror plane at x, y, 0, perpendicular to [001] 2₁ screw axis at x, y, 0 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention inversion centre at 1/2, 1/2, 0 2-fold rotation axis at x, y, 0 along [11̅0] 2₁ screw axis at x, y, 0 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, 1/2, 0 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, 1/2, 0 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at 1/2, y, 0 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, y, 0 along [210] inversion centre at 1/2, 0, 0 2-fold rotation axis at x, 0, 0 along [100] 2₁ screw axis at x, y, 0 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 6̅ inversion axis at 2/3, 1/3, z along [001], inverting through 2/3, 1/3, 0 2₁ screw axis at x, y, 0 along [11̅0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [110], advancing 1/2, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [110], advancing 1/2, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, y, 0 along [120] 2₁ screw axis at x, y, 0 along [11̅0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, y, 1/2 along [11̅0] 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 0, 1 inversion centre at 0, 0, 1/2 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 0, 1/2 6-fold rotation axis at 0, 0, z along [001] 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 0, -1 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 0, -1/2 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 0, 0 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 0, 1/2 3-fold rotation axis at 0, 0, z along [001] 2-fold rotation axis at 0, 0, z along [001] 2₁ screw axis at x, y, 0 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention g glide plane at 0, y, z, perpendicular to [100], glide vector -1, 0, 0 mirror plane at 0, y, z, perpendicular to [100] g glide plane at 0, y, z, perpendicular to [100], glide vector 1, 0, 0 inversion centre at 0, 1/2, 1/2 2-fold rotation axis at 0, 1/2, z along [001] 2-fold rotation axis at 0, y, 1/2 along [010] g glide plane at 0, y, z, perpendicular to [100], glide vector -1, 0, 0 g glide plane at 0, y, z, perpendicular to [100], glide vector 1, 0, 0 2-fold rotation axis at x, y, 1/2 along [120] inversion centre at 0, 0, 1/2 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 1, 1/2 6-fold rotation axis at 0, 0, z along [001] 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 1, -1 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 1, -1/2 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 1, 0 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 1, 1/2 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 1, 1 3-fold rotation axis at 0, 0, z along [001] 2-fold rotation axis at 0, 0, z along [001] 2-fold rotation axis at x, y, 1/2 along [110] 2-fold rotation axis at x, y, 1/2 along [210] 2-fold rotation axis at x, y, 0 along [210] inversion centre at 0, 0, 0 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 0, 0 2-fold rotation axis at x, y, 0 along [110] 2-fold rotation axis at x, y, 0 along [120] 2₁ screw axis at x, y, 1/2 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention inversion centre at 0, 1/2, 0 2-fold rotation axis at 0, y, 0 along [010] 2₁ screw axis at x, y, 1/2 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, y, 0 along [11̅0] inversion centre at 0, 0, 0 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 1, 0 2₁ screw axis at x, y, 1/2 along [11̅0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 1/2 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 1/2 along [110], advancing 1/2, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention n glide plane at x, x, z, perpendicular to [110], glide vector -1/2, 1/2, 0 2₁ screw axis at x, y, 1/2 along [11̅0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention n glide plane at x, x, z, perpendicular to [110], glide vector -1/2, 1/2, 0 a glide plane at 1/4, y, z, perpendicular to [100], glide vector -1/2, 0, 0 a glide plane at 1/4, y, z, perpendicular to [100], glide vector 1/2, 0, 0 2-fold rotation axis at x, y, 1/2 along [120] a a glide plane at 1/4, y, z, perpendicular to [100], glide vector -1/2, 0, 0 a glide plane at 1/4, y, z, perpendicular to [100], glide vector 1/2, 0, 0 n glide plane at x, x, z, perpendicular to [11̅0], glide vector 1/2, 1/2, 0 2₁ screw axis at x, y, 1/2 along [110], advancing 1/2, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention n glide plane at x, x, z, perpendicular to [11̅0], glide vector 1/2, 1/2, 0 2₁ screw axis at x, y, 1/2 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 6̅ inversion axis at 1/3, 2/3, z along [001], inverting through 1/3, 2/3, 1 6̅ inversion axis at 1/3, 2/3, z along [001], inverting through 1/3, 2/3, 1/2 6̅ inversion axis at 1/3, 2/3, z along [001], inverting through 1/3, 2/3, -1 6̅ inversion axis at 1/3, 2/3, z along [001], inverting through 1/3, 2/3, -1/2 6̅ inversion axis at 1/3, 2/3, z along [001], inverting through 1/3, 2/3, 0 6̅ inversion axis at 1/3, 2/3, z along [001], inverting through 1/3, 2/3, 1/2 3-fold rotation axis at 1/3, 2/3, z along [001] b glide plane at x, x, z, perpendicular to [210], glide vector -1, -1/2, 0 b glide plane at x, x, z, perpendicular to [210], glide vector -1, 1/2, 0 b glide plane at x, x, z, perpendicular to [210], glide vector 0, -1/2, 0 b glide plane at x, x, z, perpendicular to [210], glide vector 0, 1/2, 0 b glide plane at x, x, z, perpendicular to [210], glide vector 1, -1/2, 0 b glide plane at x, x, z, perpendicular to [210], glide vector -1, -1/2, 0 b glide plane at x, x, z, perpendicular to [210], glide vector -1, 1/2, 0 b glide plane at x, x, z, perpendicular to [210], glide vector 0, -1/2, 0 b glide plane at x, x, z, perpendicular to [210], glide vector 0, 1/2, 0 b glide plane at x, x, z, perpendicular to [210], glide vector 1, -1/2, 0 2₁ screw axis at x, y, 1/2 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 1/2 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention g glide plane at x, 0, z, perpendicular to [010], glide vector 0, -1, 0 mirror plane at x, 0, z, perpendicular to [010] g glide plane at x, 0, z, perpendicular to [010], glide vector 0, 1, 0 2-fold rotation axis at 1/2, 0, z along [001] 2-fold rotation axis at x, 0, 1/2 along [100] inversion centre at 1/2, 0, 1/2 g glide plane at x, 0, z, perpendicular to [010], glide vector 0, -1, 0 g glide plane at x, 0, z, perpendicular to [010], glide vector 0, 1, 0 b glide plane at x, 1/4, z, perpendicular to [010], glide vector 0, -1/2, 0 b glide plane at x, 1/4, z, perpendicular to [010], glide vector 0, 1/2, 0 2-fold rotation axis at x, y, 1/2 along [210] b glide plane at x, 1/4, z, perpendicular to [010], glide vector 0, -1/2, 0 b glide plane at x, 1/4, z, perpendicular to [010], glide vector 0, 1/2, 0 a glide plane at x, x, z, perpendicular to [120], glide vector -1/2, -1, 0 a glide plane at x, x, z, perpendicular to [120], glide vector 1/2, -1, 0 a glide plane at x, x, z, perpendicular to [120], glide vector -1/2, 0, 0 a glide plane at x, x, z, perpendicular to [120], glide vector 1/2, 0, 0 a glide plane at x, x, z, perpendicular to [120], glide vector -1/2, 1, 0 a glide plane at x, x, z, perpendicular to [120], glide vector -1/2, -1, 0 a glide plane at x, x, z, perpendicular to [120], glide vector 1/2, -1, 0 a glide plane at x, x, z, perpendicular to [120], glide vector -1/2, 0, 0 a glide plane at x, x, z, perpendicular to [120], glide vector 1/2, 0, 0 a glide plane at x, x, z, perpendicular to [120], glide vector -1/2, 1, 0 g glide plane at 1/2, y, z, perpendicular to [100], glide vector -1, 0, 0 mirror plane at 1/2, y, z, perpendicular to [100] g glide plane at 1/2, y, z, perpendicular to [100], glide vector 1, 0, 0 inversion centre at 1/2, 1/2, 1/2 mirror plane at x, x, z, perpendicular to [110] mirror plane at x, x, z, perpendicular to [11̅0] 2₁ screw axis at x, 1/2, 1/2 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at 1/2, y, 1/2 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 1/2 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention mirror plane at x, y, 1/2, perpendicular to [001] g glide plane at x, 1/2, z, perpendicular to [010], glide vector 0, -1, 0 mirror plane at x, 1/2, z, perpendicular to [010] g glide plane at x, 1/2, z, perpendicular to [010], glide vector 0, 1, 0 2-fold rotation axis at 1/2, 1/2, z along [001] 2-fold rotation axis at x, y, 1/2 along [11̅0] 2₁ screw axis at x, y, 1/2 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, y, 1/2 along [110] g glide plane at 1/2, y, z, perpendicular to [100], glide vector -1, 0, 0 g glide plane at 1/2, y, z, perpendicular to [100], glide vector 1, 0, 0 g glide plane at x, 1/2, z, perpendicular to [010], glide vector 0, -1, 0 g glide plane at x, 1/2, z, perpendicular to [010], glide vector 0, 1, 0 2₁ screw axis at x, 1/2, 1/2 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at 1/2, y, 1/2 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention b glide plane at x, 3/4, z, perpendicular to [010], glide vector 0, -1/2, 0 b glide plane at x, 3/4, z, perpendicular to [010], glide vector 0, 1/2, 0 2-fold rotation axis at x, y, 1/2 along [210] b glide plane at x, 3/4, z, perpendicular to [010], glide vector 0, -1/2, 0 b glide plane at x, 3/4, z, perpendicular to [010], glide vector 0, 1/2, 0 inversion centre at 1/2, 0, 1/2 g glide plane at x, 0, z, perpendicular to [010], glide vector 0, -1, 0 mirror plane at x, 0, z, perpendicular to [010] 2-fold rotation axis at 1/2, 0, z along [001] 2-fold rotation axis at x, 0, 1/2 along [100] g glide plane at x, 0, z, perpendicular to [010], glide vector 0, -1, 0 2₁ screw axis at x, y, 1/2 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 6̅ inversion axis at 2/3, 1/3, z along [001], inverting through 2/3, 1/3, 1 6̅ inversion axis at 2/3, 1/3, z along [001], inverting through 2/3, 1/3, 1/2 6̅ inversion axis at 2/3, 1/3, z along [001], inverting through 2/3, 1/3, -1 6̅ inversion axis at 2/3, 1/3, z along [001], inverting through 2/3, 1/3, -1/2 6̅ inversion axis at 2/3, 1/3, z along [001], inverting through 2/3, 1/3, 0 6̅ inversion axis at 2/3, 1/3, z along [001], inverting through 2/3, 1/3, 1/2 3-fold rotation axis at 2/3, 1/3, z along [001] 2₁ screw axis at x, y, 1/2 along [11̅0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 1/2 along [110], advancing 1/2, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 1/2 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention n glide plane at x, x, z, perpendicular to [11̅0], glide vector 1/2, 1/2, 0 2₁ screw axis at x, y, 1/2 along [110], advancing 1/2, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention n glide plane at x, x, z, perpendicular to [11̅0], glide vector 1/2, 1/2, 0 a glide plane at 3/4, y, z, perpendicular to [100], glide vector -1/2, 0, 0 a glide plane at 3/4, y, z, perpendicular to [100], glide vector 1/2, 0, 0 2-fold rotation axis at x, y, 1/2 along [120] a glide plane at 3/4, y, z, perpendicular to [100], glide vector -1/2, 0, 0 a glide plane at 3/4, y, z, perpendicular to [100], glide vector 1/2, 0, 0 n glide plane at x, x, z, perpendicular to [110], glide vector -1/2, 1/2, 0 2₁ screw axis at x, y, 1/2 along [11̅0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention n glide plane at x, x, z, perpendicular to [110], glide vector -1/2, 1/2, 0 g glide plane at x, x, z, perpendicular to [120], glide vector -1, -1, 0 g glide plane at x, x, z, perpendicular to [120], glide vector 0, -1, 0 g glide plane at x, x, z, perpendicular to [120], glide vector -1, 0, 0 mirror plane at x, x, z, perpendicular to [120] g glide plane at x, x, z, perpendicular to [120], glide vector 1, 0, 0 g glide plane at x, x, z, perpendicular to [120], glide vector -1, -1, 0 g glide plane at x, x, z, perpendicular to [120], glide vector 0, -1, 0 g glide plane at x, x, z, perpendicular to [120], glide vector -1, 0, 0 g glide plane at x, x, z, perpendicular to [120], glide vector 1, 0, 0 2₁ screw axis at x, y, 1/2 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, y, 0 along [11̅0] inversion centre at 0, 0, 0 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 0, 1 2₁ screw axis at x, y, 1/2 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at 0, y, 0 along [010] inversion centre at 0, 1/2, 0 g glide plane at x, x, z, perpendicular to [210], glide vector -1, -1, 0 g glide plane at x, x, z, perpendicular to [210], glide vector -1, 0, 0 g glide plane at x, x, z, perpendicular to [210], glide vector 0, -1, 0 mirror plane at x, x, z, perpendicular to [210] g glide plane at x, x, z, perpendicular to [210], glide vector 0, 1, 0 g glide plane at x, x, z, perpendicular to [210], glide vector -1, -1, 0 g glide plane at x, x, z, perpendicular to [210], glide vector -1, 0, 0 g glide plane at x, x, z, perpendicular to [210], glide vector 0, -1, 0 g glide plane at x, x, z, perpendicular to [210], glide vector 0, 1, 0 2-fold rotation axis at x, y, 0 along [120] 2-fold rotation axis at x, y, 0 along [110] inversion centre at 0, 0, 0 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 1, 1 2-fold rotation axis at x, y, 0 along [210] 2-fold rotation axis at x, y, 1/2 along [210] 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 0, 1 6-fold rotation axis at 0, 0, z along [001] 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 0, -1 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 0, -1/2 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 0, 0 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 0, 1/2 3-fold rotation axis at 0, 0, z along [001] 2-fold rotation axis at 0, 0, z along [001] 2-fold rotation axis at x, y, 1/2 along [110] inversion centre at 0, 0, 1/2 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 0, 1/2 2-fold rotation axis at x, y, 1/2 along [120] 2₁ screw axis at x, y, 0 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention g glide plane at 0, y, z, perpendicular to [100], glide vector -1, 0, 0 mirror plane at 0, y, z, perpendicular to [100] 2-fold rotation axis at 0, y, 1/2 along [010] inversion centre at 0, 1/2, 1/2 2-fold rotation axis at 0, 1/2, z along [001] g glide plane at 0, y, z, perpendicular to [100], glide vector -1, 0, 0 2₁ screw axis at x, y, 0 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 1, 1 6-fold rotation axis at 0, 0, z along [001] 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 1, -1 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 1, -1/2 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 1, 0 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 1, 1/2 3-fold rotation axis at 0, 0, z along [001] 2-fold rotation axis at 0, 0, z along [001] 2-fold rotation axis at x, y, 1/2 along [11̅0] inversion centre at 0, 0, 1/2 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 1, 1/2 2₁ screw axis at x, y, 0 along [11̅0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [110], advancing 1/2, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention c 2₁ screw axis at x, y, 0 along [11̅0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, y, 0 along [120] 2₁ screw axis at x, y, 0 along [110], advancing 1/2, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 6̅ inversion axis at 1/3, 2/3, z along [001], inverting through 1/3, 2/3, 1 2₁ screw axis at x, y, 0 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention inversion centre at 1/2, 0, 0 2-fold rotation axis at x, 0, 0 along [100] 2-fold rotation axis at x, y, 0 along [210] 2₁ screw axis at 1/2, y, 0 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, y, 0 along [110] inversion centre at 1/2, 1/2, 0 mirror plane at x, y, 0, perpendicular to [001] 2₁ screw axis at x, 1/2, 0 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, y, 0 along [11̅0] 2₁ screw axis at x, y, 0 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, 1/2, 0 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at 1/2, y, 0 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, y, 0 along [210] 2-fold rotation axis at x, 0, 0 along [100] inversion centre at 1/2, 0, 0 2₁ screw axis at x, y, 0 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 6̅ inversion axis at 2/3, 1/3, z along [001], inverting through 2/3, 1/3, 1 2₁ screw axis at x, y, 0 along [11̅0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [110], advancing 1/2, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [110], advancing 1/2, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, y, 0 along [120] 2₁ screw axis at x, y, 0 along [11̅0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, y, 0 along [210] 2-fold rotation axis at x, y, 0 along [110] inversion centre at 0, 0, 0 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 0, 1 2-fold rotation axis at x, y, 0 along [120] 2-fold rotation axis at 0, y, 0 along [010] inversion centre at 0, 1/2, 0 2-fold rotation axis at x, y, 0 along [11̅0] inversion centre at 0, 0, 0 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 1, 1 6-fold rotation axis at 0, 0, z along [001] 6-fold rotation axis at 0, 0, z along [001] 6-fold rotation axis at 0, 0, z along [001] 6-fold rotation axis at 0, 0, z along [001] 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 1, -1 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 1, -1/2 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 1, 0 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 1, 1/2 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 1, 1 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 0, -1 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 0, -1/2 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 0, 0 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 0, 1/2 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 0, 1 6̅ inversion axis at 1/3, 2/3, z along [001], inverting through 1/3, 2/3, -1 6̅ inversion axis at 1/3, 2/3, z along [001], inverting through 1/3, 2/3, -1/2 6̅ inversion axis at 1/3, 2/3, z along [001], inverting through 1/3, 2/3, 0 6̅ inversion axis at 1/3, 2/3, z along [001], inverting through 1/3, 2/3, 1/2 6̅ inversion axis at 1/3, 2/3, z along [001], inverting through 1/3, 2/3, 1 6̅ inversion axis at 2/3, 1/3, z along [001], inverting through 2/3, 1/3, -1 6̅ inversion axis at 2/3, 1/3, z along [001], inverting through 2/3, 1/3, -1/2 6̅ inversion axis at 2/3, 1/3, z along [001], inverting through 2/3, 1/3, 0 6̅ inversion axis at 2/3, 1/3, z along [001], inverting through 2/3, 1/3, 1/2 6̅ inversion axis at 2/3, 1/3, z along [001], inverting through 2/3, 1/3, 1 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 1, -1 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 1, -1/2 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 1, 0 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 1, 1/2 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 1, 1 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 0, -1 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 0, -1/2 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 0, 0 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 0, 1/2 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 0, 1 3-fold rotation axis at 0, 0, z along [001] 3-fold rotation axis at 0, 0, z along [001] 3-fold rotation axis at 2/3, 1/3, z along [001] 3-fold rotation axis at 1/3, 2/3, z along [001] 3-fold rotation axis at 0, 0, z along [001] 3-fold rotation axis at 0, 0, z along [001] 2-fold rotation axis at 0, 0, z along [001] 2-fold rotation axis at 0, 1/2, z along [001] 2-fold rotation axis at 0, 0, z along [001] 2-fold rotation axis at 1/2, 0, z along [001] 2-fold rotation axis at 1/2, 1/2, z along [001] 2-fold rotation axis at 1/2, 0, z along [001] 2-fold rotation axis at 0, 0, z along [001] 2-fold rotation axis at 0, 1/2, z along [001] 2-fold rotation axis at 0, 0, z along [001] 2-fold rotation axis at x, y, 0 along [11̅0] 2-fold rotation axis at x, y, 1/2 along [11̅0] 2-fold rotation axis at x, y, 0 along [11̅0] 2₁ screw axis at x, y, 0 along [11̅0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 1/2 along [11̅0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [11̅0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, y, 0 along [11̅0] 2-fold rotation axis at x, y, 1/2 along [11̅0] 2-fold rotation axis at x, y, 0 along [11̅0] 2₁ screw axis at x, y, 0 along [11̅0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 1/2 along [11̅0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [11̅0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, y, 0 along [11̅0] 2-fold rotation axis at x, y, 1/2 along [11̅0] 2-fold rotation axis at x, y, 0 along [11̅0] 2-fold rotation axis at 0, y, 0 along [010] 2-fold rotation axis at 0, y, 1/2 along [010] 2-fold rotation axis at 0, y, 0 along [010] 2₁ screw axis at 1/2, y, 0 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at 1/2, y, 1/2 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at 1/2, y, 0 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at 0, y, 0 along [010] 2-fold rotation axis at 0, y, 1/2 along [010] 2-fold rotation axis at 0, y, 0 along [010] 2-fold rotation axis at x, y, 0 along [120] 2-fold rotation axis at x, y, 1/2 along [120] 2-fold rotation axis at x, y, 0 along [120] 2-fold rotation axis at x, y, 0 along [120] 2-fold rotation axis at x, y, 1/2 along [120] 2-fold rotation axis at x, y, 0 along [120] 2₁ screw axis at x, y, 0 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 1/2 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, y, 0 along [120] 2-fold rotation axis at x, y, 1/2 along [120] 2-fold rotation axis at x, y, 0 along [120] 2₁ screw axis at x, y, 0 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 1/2 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, y, 0 along [120] 2-fold rotation axis at x, y, 1/2 along [120] 2-fold rotation axis at x, y, 0 along [120] 2₁ screw axis at x, y, 0 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 1/2 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [120], advancing 1/2, 1, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, y, 0 along [110] 2-fold rotation axis at x, y, 1/2 along [110] 2-fold rotation axis at x, y, 0 along [110] 2₁ screw axis at x, y, 0 along [110], advancing 1/2, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 1/2 along [110], advancing 1/2, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [110], advancing 1/2, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, y, 0 along [110] 2-fold rotation axis at x, y, 1/2 along [110] 2-fold rotation axis at x, y, 0 along [110] 2₁ screw axis at x, y, 0 along [110], advancing 1/2, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 1/2 along [110], advancing 1/2, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [110], advancing 1/2, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, y, 0 along [110] 2-fold rotation axis at x, y, 1/2 along [110] 2-fold rotation axis at x, y, 0 along [110] 2-fold rotation axis at x, y, 0 along [210] 2-fold rotation axis at x, y, 1/2 along [210] 2-fold rotation axis at x, y, 0 along [210] 2-fold rotation axis at x, y, 0 along [210] 2-fold rotation axis at x, y, 1/2 along [210] 2-fold rotation axis at x, y, 0 along [210] 2-fold rotation axis at x, y, 0 along [210] 2-fold rotation axis at x, y, 1/2 along [210] 2-fold rotation axis at x, y, 0 along [210] 2-fold rotation axis at x, y, 0 along [210] 2-fold rotation axis at x, y, 1/2 along [210] 2-fold rotation axis at x, y, 0 along [210] 2₁ screw axis at x, y, 0 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 1/2 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 1/2 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 1/2 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, y, 0 along [210], advancing 1, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, 0, 0 along [100] 2-fold rotation axis at x, 0, 1/2 along [100] 2-fold rotation axis at x, 0, 0 along [100] 2₁ screw axis at x, 1/2, 0 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, 1/2, 1/2 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention 2₁ screw axis at x, 1/2, 0 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention 2-fold rotation axis at x, 0, 0 along [100] 2-fold rotation axis at x, 0, 1/2 along [100] 2-fold rotation axis at x, 0, 0 along [100] inversion centre at 0, 0, 0 inversion centre at 0, 0, 1/2 inversion centre at 0, 0, 0 inversion centre at 0, 1/2, 0 inversion centre at 0, 1/2, 1/2 inversion centre at 0, 1/2, 0 inversion centre at 0, 0, 0 inversion centre at 0, 0, 1/2 inversion centre at 0, 0, 0 inversion centre at 1/2, 0, 0 inversion centre at 1/2, 0, 1/2 inversion centre at 1/2, 0, 0 inversion centre at 1/2, 1/2, 0 inversion centre at 1/2, 1/2, 1/2 inversion centre at 1/2, 1/2, 0 inversion centre at 1/2, 0, 0 inversion centre at 1/2, 0, 1/2 inversion centre at 1/2, 0, 0 inversion centre at 0, 0, 0 inversion centre at 0, 0, 1/2 inversion centre at 0, 0, 0 inversion centre at 0, 1/2, 0 inversion centre at 0, 1/2, 1/2 inversion centre at 0, 1/2, 0 inversion centre at 0, 0, 0 inversion centre at 0, 0, 1/2 inversion centre at 0, 0, 0 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 1, 0 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 1, 1/2 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 1, 1 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 0, 0 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 0, 1/2 6̅ inversion axis at 0, 0, z along [001], inverting through 0, 0, 1 6̅ inversion axis at 1/3, 2/3, z along [001], inverting through 1/3, 2/3, 0 6̅ inversion axis at 1/3, 2/3, z along [001], inverting through 1/3, 2/3, 1/2 6̅ inversion axis at 1/3, 2/3, z along [001], inverting through 1/3, 2/3, 1 6̅ inversion axis at 2/3, 1/3, z along [001], inverting through 2/3, 1/3, 0 6̅ inversion axis at 2/3, 1/3, z along [001], inverting through 2/3, 1/3, 1/2 6̅ inversion axis at 2/3, 1/3, z along [001], inverting through 2/3, 1/3, 1 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 1, 0 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 1, 1/2 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 1, 1 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 0, 0 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 0, 1/2 6̅ inversion axis at 0, 0, z along [001], inverting through 1, 0, 1 Screw axes as a helix — shows the handedness half-arrows — the Tables' symbol Enlarge
Look along a −a b −b c −c [111]
Point at an element to read what it is.
22 2-fold axes20 21 screw axes3 3-fold axes1 6-fold axis6 6 inversion axes9 a glides9 b glides16 g glides8 inversion centres10 mirror planes4 n glidesPoint at any element for its type, its direction and where it sits. Axes are
drawn thicker the higher their order; a screw axis or a glide plane is dashed, because its
operation carries a translation the pure one does not. Switch a class off to see through a
dense group.
The cell drawn here is a representative one: its shape obeys every constraint this group's symmetry imposes and nothing else, because a space group fixes the equalities among the cell constants and never their values. Read the right angles, the equal axes and the 120° where they appear; do not read the axial ratios.
The crystal class, projected
Every space group leaves a crystal class behind when its translations are
taken away, and this is 6/mmm — the
24 operations of
P 6/mmm seen as directions rather than as
places. On the left, one general direction and everywhere the class sends it; on the right,
the same class written as its axes and mirrors.
a b [231], upper hemisphere, same hand as the start (from x,y,z)
[231̅], lower hemisphere, opposite hand — an inverted image (from x,y,-z) [2̅3̅1̅], lower hemisphere, opposite hand — an inverted image (from -x,-y,-z)
[2̅3̅1], upper hemisphere, same hand as the start (from -x,-y,z) [1̅21], upper hemisphere, same hand as the start (from x-y,x,z)
[1̅21̅], lower hemisphere, opposite hand — an inverted image (from x-y,x,-z) [12̅1̅], lower hemisphere, opposite hand — an inverted image (from -x+y,-x,-z)
[12̅1], upper hemisphere, same hand as the start (from -x+y,-x,z) [3̅1̅1], upper hemisphere, same hand as the start (from -y,x-y,z)
[3̅1̅1̅], lower hemisphere, opposite hand — an inverted image (from -y,x-y,-z) [311̅], lower hemisphere, opposite hand — an inverted image (from y,-x+y,-z)
[311], upper hemisphere, same hand as the start (from y,-x+y,z) [3̅2̅1̅], lower hemisphere, same hand as the start (from -y,-x,-z)
[3̅2̅1], upper hemisphere, opposite hand — an inverted image (from -y,-x,z) [321], upper hemisphere, opposite hand — an inverted image (from y,x,z)
[321̅], lower hemisphere, same hand as the start (from y,x,-z) [2̅11̅], lower hemisphere, same hand as the start (from -x,-x+y,-z)
[2̅11], upper hemisphere, opposite hand — an inverted image (from -x,-x+y,z) [21̅1], upper hemisphere, opposite hand — an inverted image (from x,x-y,z)
[21̅1̅], lower hemisphere, same hand as the start (from x,x-y,-z) [131̅], lower hemisphere, same hand as the start (from -x+y,y,-z)
[131], upper hemisphere, opposite hand — an inverted image (from -x+y,y,z) [1̅3̅1], upper hemisphere, opposite hand — an inverted image (from x-y,-y,z)
[1̅3̅1̅], lower hemisphere, same hand as the start (from x-y,-y,-z) a b mirror plane perpendicular to [001] mirror plane perpendicular to [010] mirror plane perpendicular to [11̅0] mirror plane perpendicular to [100] mirror plane perpendicular to [110] mirror plane perpendicular to [120] mirror plane perpendicular to [210] 6-fold rotation axis along [001], with a 6̅ inversion axis on the same line 6-fold rotation axis along [001], with a 6̅ inversion axis on the same line 2-fold rotation axis along [010] 2-fold rotation axis along [010] 2-fold rotation axis along [010] 2-fold rotation axis along [010] 2-fold rotation axis along [010] 2-fold rotation axis along [11̅0] 2-fold rotation axis along [11̅0] 2-fold rotation axis along [11̅0] 2-fold rotation axis along [11̅0] 2-fold rotation axis along [11̅0] 2-fold rotation axis along [100] 2-fold rotation axis along [100] 2-fold rotation axis along [100] 2-fold rotation axis along [100] 2-fold rotation axis along [100] 2-fold rotation axis along [110] 2-fold rotation axis along [110] 2-fold rotation axis along [110] 2-fold rotation axis along [110] 2-fold rotation axis along [110] 2-fold rotation axis along [120] 2-fold rotation axis along [120] 2-fold rotation axis along [120] 2-fold rotation axis along [120] 2-fold rotation axis along [120] 2-fold rotation axis along [210] 2-fold rotation axis along [210] 2-fold rotation axis along [210] 2-fold rotation axis along [210] 2-fold rotation axis along [210] inversion centre at the origin Point at any pole or symbol to have it name itself. Click it to keep it on screen; click it again, click empty space, or press Escape to let go.
Both discs are the same sphere seen from the same place: every direction is joined to the far pole and marked where that line crosses the equator, so the centre of the disc is straight up c and the rim is ninety degrees away from it. The projection is conformal : two curves cross on the disc at the angle they cross at on the sphere. That is not the same as being able to read the angle between two poles off the disc — separation is not to scale, and measuring it needs a Wulff net or the direction vectors themselves. a points down the page and b across it.
A filled dot is a direction in the upper half of the sphere and an open circle one in the lower; where the group has a mirror across the page the two coincide, and the mark is a dot inside a circle. A comma means the operation that produced it turned the object inside out.
24 poles, drawn as 12 marks. 6/mmm has a mirror across the page, so each pole in the upper half of the sphere lands exactly on top of one in the lower half. Every one of those pairs is drawn as the Tables draw it — a dot inside a circle — and pointing at it names both.
12 of the 24 poles carry a
comma. Those images are of the opposite hand, produced by an operation that turns
the object inside out — a mirror, an inversion, or an inversion axis. A structure
built from a single enantiomer cannot have this symmetry, because the group would demand
the other one alongside it.
The International Tables turn the page for monoclinic groups and draw the unique axis in the plane of the paper. This projects along c whatever the group is, so that all three diagrams on this page share one orientation — in a group with the unique axis b , the two-fold therefore lies across the disc instead of standing at its centre.
General position
The 24 symmetry
operations of
P 6/mmm , written as coordinate triplets. Each one
says where the group sends a point at x , y , z .
x ,y ,z -x ,-y ,-z x -y ,x ,z -x +y ,-x ,-z -y ,x -y ,z y ,-x +y ,-z -x ,-y ,z x ,y ,-z -x +y ,-x ,z x -y ,x ,-z y ,-x +y ,z -y ,x -y ,-z -y ,-x ,-z y ,x ,z -x ,-x +y ,-z x ,x -y ,z -x +y ,y ,-z x -y ,-y ,z y ,x ,-z -y ,-x ,z x ,x -y ,-z -x ,-x +y ,z x -y ,-y ,-z -x +y ,y ,z
Wyckoff positions
A point of the cell is held in place by whichever operations send it to itself, and points
held the same way form one Wyckoff position .
P 6/mmm has
18, listed from the general position — the one no operation
fixes — down to the most symmetric site it has.
Multiplicity
Site symmetry
Order
Coordinates
Draw it
24
1
1
x , y , z the
24 points
12
m 2
2x , x , z the
12 points
12
m 2
x , x , z the
12 points
12
m 2
x , y , 0the
12 points
12
m 2
x , y , 1/2the
12 points
6
mm 24
0, 1/2, z the
6 points
6
mm 24
0, y , 0the
6 points
6
mm 24
0, y , 1/2the
6 points
6
mm 24
x , -x , 0the
6 points
6
mm 24
x , -x , 1/2the
6 points
4
3m
6
1/3, 2/3, z the
4 points
3
mmm 8
0, 1/2, 0the
3 points
3
mmm 8
0, 1/2, 1/2the
3 points
2
6mm
12
0, 0, z the
2 points
2
6 m 212
1/3, 2/3, 0the
2 points
2
6 m 212
1/3, 2/3, 1/2the
2 points
1
6/mmm
24
0, 0, 0the
1 point
1
6/mmm
24
0, 0, 1/2the
1 point
Multiplicity times the order of the site symmetry is
24, the number of operations, in every row
— a point with more symmetry holding it has fewer copies, and the two numbers are
computed separately here so that they have to agree.
The International Tables give each position a letter and print
its site symmetry oriented against the directions of the space-group symbol
— 4a , and m .mm rather than mmm . Neither is here. A letter is
an assignment the Tables make, not a consequence of the symmetry, and the dots are a notation
with no way of checking it short of copying Tables pages; the symmetry itself, the
multiplicity and the coordinates all follow from the operations and are derived.
One member of each set is printed. Every position is several subspaces of the
cell that the group carries onto one another, so the Tables may name a different one —
the same position either way. Follow a row to put that point in the form and draw its orbit.
Where those points are
b 0.08, 0.21, 0.31 — mirror image 0.08, 0.87, 0.31 — as given 0.13, 0.21, 0.31 — as given 0.13, 0.92, 0.31 — mirror image 0.21, 0.08, 0.31 — as given 0.21, 0.13, 0.31 — mirror image a 0.08, 0.21, 0.69 — as given 0.08, 0.87, 0.69 — mirror image 0.13, 0.21, 0.69 — mirror image 0.79, 0.87, 0.31 — mirror image 0.79, 0.92, 0.31 — as given 0.13, 0.92, 0.69 — as given 0.87, 0.08, 0.31 — mirror image 0.21, 0.08, 0.69 — mirror image 0.21, 0.13, 0.69 — as given 0.87, 0.79, 0.31 — as given 0.92, 0.13, 0.31 — as given 0.92, 0.79, 0.31 — mirror image c 0.79, 0.87, 0.69 — as given 0.79, 0.92, 0.69 — mirror image 0.87, 0.08, 0.69 — as given 0.87, 0.79, 0.69 — mirror image 0.92, 0.13, 0.69 — mirror image 0.92, 0.79, 0.69 — as given 0.13, 0.21, 0.31 — as given 0.87, 0.79, 0.69 — mirror image 0.92, 0.13, 0.31 — as given 0.08, 0.87, 0.69 — mirror image 0.79, 0.92, 0.31 — as given 0.21, 0.08, 0.69 — mirror image 0.87, 0.79, 0.31 — as given 0.13, 0.21, 0.69 — mirror image 0.08, 0.87, 0.31 — as given 0.92, 0.13, 0.69 — mirror image 0.21, 0.08, 0.31 — as given 0.79, 0.92, 0.69 — mirror image 0.79, 0.87, 0.69 — as given 0.21, 0.13, 0.31 — mirror image 0.87, 0.08, 0.69 — as given 0.13, 0.92, 0.31 — mirror image 0.08, 0.21, 0.69 — as given 0.92, 0.79, 0.31 — mirror image 0.21, 0.13, 0.69 — as given 0.79, 0.87, 0.31 — mirror image 0.13, 0.92, 0.69 — as given 0.87, 0.08, 0.31 — mirror image 0.92, 0.79, 0.69 — as given 0.08, 0.21, 0.31 — mirror image
drag to rotate · scroll to zoom · click a point to keep it
Look along a −a b −b c −c [111]
Point at a position to read its coordinates, or click one to keep it.
0.13, 0.21, 0.31 is a general position.P 6/mmm other than the identity leaves it where it is, so the group sends it to 24 distinct points — one per operation, which is what makes the multiplicity equal to the order of the group.
That is the position x , y , z
That is the general position the drawing opens on. Type a point into the
form to move it — the picture and the multiplicity follow.
A filled circle is the point as you gave it; a hollow one is a mirror image of it, produced by an operation of the second kind — an inversion, a mirror, a glide or a rotoinversion. The International Tables mark the same distinction with a comma inside the circle. It matters for anything chiral: those copies are not superimposable on the one you typed.
The 24 points
# x y z From 1 0.13 0.21 0.31 x ,y ,z 2 0.87 0.79 0.69 -x ,-y ,-z 3 0.92 0.13 0.31 x -y ,x ,z 4 0.08 0.87 0.69 -x +y ,-x ,-z 5 0.79 0.92 0.31 -y ,x -y ,z 6 0.21 0.08 0.69 y ,-x +y ,-z 7 0.87 0.79 0.31 -x ,-y ,z 8 0.13 0.21 0.69 x ,y ,-z 9 0.08 0.87 0.31 -x +y ,-x ,z 10 0.92 0.13 0.69 x -y ,x ,-z 11 0.21 0.08 0.31 y ,-x +y ,z 12 0.79 0.92 0.69 -y ,x -y ,-z 13 0.79 0.87 0.69 -y ,-x ,-z 14 0.21 0.13 0.31 y ,x ,z 15 0.87 0.08 0.69 -x ,-x +y ,-z 16 0.13 0.92 0.31 x ,x -y ,z 17 0.08 0.21 0.69 -x +y ,y ,-z 18 0.92 0.79 0.31 x -y ,-y ,z 19 0.21 0.13 0.69 y ,x ,-z 20 0.79 0.87 0.31 -y ,-x ,z 21 0.13 0.92 0.69 x ,x -y ,-z 22 0.87 0.08 0.31 -x ,-x +y ,z 23 0.92 0.79 0.69 x -y ,-y ,-z 24 0.08 0.21 0.31 -x +y ,y ,z
Reflection conditions
None. This space group extinguishes no reflection systematically: it has
a primitive lattice, no glide plane and no screw axis, so nothing in its symmetry forces a
structure factor to zero. Thirty-seven of the 230 are like this, and for those, absences tell
you nothing at all.
Is a reflection there?
Put three Miller indices into the form and this says whether
P 6/mmm allows that reflection, and which
symmetry operation removes it if it does not.
The conditions, drawn
Each picture is one plane of the reciprocal lattice, drawn as a net of the reflections that
plane contains. The shape of the net is schematic — a space group is symmetry and not
metric, so there are no cell constants here to draw — but the equalities among its lengths
and angles are the group's own, which is why a monoclinic
h 0l net is oblique where its hk 0 net is not.
Zonehk 0h 0l 0kl
-5 5 0 - allowed. -4 5 0 - allowed. -3 5 0 - allowed. -2 5 0 - allowed. -1 5 0 - allowed. 0 5 0 - allowed. 1 5 0 - allowed. 2 5 0 - allowed. 3 5 0 - allowed. 4 5 0 - allowed. 5 5 0 - allowed. -5 4 0 - allowed. -4 4 0 - allowed. -3 4 0 - allowed. -2 4 0 - allowed. -1 4 0 - allowed. 0 4 0 - allowed. 1 4 0 - allowed. 2 4 0 - allowed. 3 4 0 - allowed. 4 4 0 - allowed. 5 4 0 - allowed. -5 3 0 - allowed. -4 3 0 - allowed. -3 3 0 - allowed. -2 3 0 - allowed. -1 3 0 - allowed. 0 3 0 - allowed. 1 3 0 - allowed. 2 3 0 - allowed. 3 3 0 - allowed. 4 3 0 - allowed. 5 3 0 - allowed. -5 2 0 - allowed. -4 2 0 - allowed. -3 2 0 - allowed. -2 2 0 - allowed. -1 2 0 - allowed. 0 2 0 - allowed. 1 2 0 - allowed. 2 2 0 - allowed. 3 2 0 - allowed. 4 2 0 - allowed. 5 2 0 - allowed. -5 1 0 - allowed. -4 1 0 - allowed. -3 1 0 - allowed. -2 1 0 - allowed. -1 1 0 - allowed. 0 1 0 - allowed. 1 1 0 - allowed. 2 1 0 - allowed. 3 1 0 - allowed. 4 1 0 - allowed. 5 1 0 - allowed. -5 0 0 - allowed. -4 0 0 - allowed. -3 0 0 - allowed. -2 0 0 - allowed. -1 0 0 - allowed. 0 0 0 - the origin of reciprocal space, where the direct beam goes. Not a reflection. 1 0 0 - allowed. 2 0 0 - allowed. 3 0 0 - allowed. 4 0 0 - allowed. 5 0 0 - allowed. -5 -1 0 - allowed. -4 -1 0 - allowed. -3 -1 0 - allowed. -2 -1 0 - allowed. -1 -1 0 - allowed. 0 -1 0 - allowed. 1 -1 0 - allowed. 2 -1 0 - allowed. 3 -1 0 - allowed. 4 -1 0 - allowed. 5 -1 0 - allowed. -5 -2 0 - allowed. -4 -2 0 - allowed. -3 -2 0 - allowed. -2 -2 0 - allowed. -1 -2 0 - allowed. 0 -2 0 - allowed. 1 -2 0 - allowed. 2 -2 0 - allowed. 3 -2 0 - allowed. 4 -2 0 - allowed. 5 -2 0 - allowed. -5 -3 0 - allowed. -4 -3 0 - allowed. -3 -3 0 - allowed. -2 -3 0 - allowed. -1 -3 0 - allowed. 0 -3 0 - allowed. 1 -3 0 - allowed. 2 -3 0 - allowed. 3 -3 0 - allowed. 4 -3 0 - allowed. 5 -3 0 - allowed. -5 -4 0 - allowed. -4 -4 0 - allowed. -3 -4 0 - allowed. -2 -4 0 - allowed. -1 -4 0 - allowed. 0 -4 0 - allowed. 1 -4 0 - allowed. 2 -4 0 - allowed. 3 -4 0 - allowed. 4 -4 0 - allowed. 5 -4 0 - allowed. -5 -5 0 - allowed. -4 -5 0 - allowed. -3 -5 0 - allowed. -2 -5 0 - allowed. -1 -5 0 - allowed. 0 -5 0 - allowed. 1 -5 0 - allowed. 2 -5 0 - allowed. 3 -5 0 - allowed. 4 -5 0 - allowed. 5 -5 0 - allowed. a *b *hk 0-5 5 1 - allowed. -4 5 1 - allowed. -3 5 1 - allowed. -2 5 1 - allowed. -1 5 1 - allowed. 0 5 1 - allowed. 1 5 1 - allowed. 2 5 1 - allowed. 3 5 1 - allowed. 4 5 1 - allowed. 5 5 1 - allowed. -5 4 1 - allowed. -4 4 1 - allowed. -3 4 1 - allowed. -2 4 1 - allowed. -1 4 1 - allowed. 0 4 1 - allowed. 1 4 1 - allowed. 2 4 1 - allowed. 3 4 1 - allowed. 4 4 1 - allowed. 5 4 1 - allowed. -5 3 1 - allowed. -4 3 1 - allowed. -3 3 1 - allowed. -2 3 1 - allowed. -1 3 1 - allowed. 0 3 1 - allowed. 1 3 1 - allowed. 2 3 1 - allowed. 3 3 1 - allowed. 4 3 1 - allowed. 5 3 1 - allowed. -5 2 1 - allowed. -4 2 1 - allowed. -3 2 1 - allowed. -2 2 1 - allowed. -1 2 1 - allowed. 0 2 1 - allowed. 1 2 1 - allowed. 2 2 1 - allowed. 3 2 1 - allowed. 4 2 1 - allowed. 5 2 1 - allowed. -5 1 1 - allowed. -4 1 1 - allowed. -3 1 1 - allowed. -2 1 1 - allowed. -1 1 1 - allowed. 0 1 1 - allowed. 1 1 1 - allowed. 2 1 1 - allowed. 3 1 1 - allowed. 4 1 1 - allowed. 5 1 1 - allowed. -5 0 1 - allowed. -4 0 1 - allowed. -3 0 1 - allowed. -2 0 1 - allowed. -1 0 1 - allowed. 0 0 1 - allowed. 1 0 1 - allowed. 2 0 1 - allowed. 3 0 1 - allowed. 4 0 1 - allowed. 5 0 1 - allowed. -5 -1 1 - allowed. -4 -1 1 - allowed. -3 -1 1 - allowed. -2 -1 1 - allowed. -1 -1 1 - allowed. 0 -1 1 - allowed. 1 -1 1 - allowed. 2 -1 1 - allowed. 3 -1 1 - allowed. 4 -1 1 - allowed. 5 -1 1 - allowed. -5 -2 1 - allowed. -4 -2 1 - allowed. -3 -2 1 - allowed. -2 -2 1 - allowed. -1 -2 1 - allowed. 0 -2 1 - allowed. 1 -2 1 - allowed. 2 -2 1 - allowed. 3 -2 1 - allowed. 4 -2 1 - allowed. 5 -2 1 - allowed. -5 -3 1 - allowed. -4 -3 1 - allowed. -3 -3 1 - allowed. -2 -3 1 - allowed. -1 -3 1 - allowed. 0 -3 1 - allowed. 1 -3 1 - allowed. 2 -3 1 - allowed. 3 -3 1 - allowed. 4 -3 1 - allowed. 5 -3 1 - allowed. -5 -4 1 - allowed. -4 -4 1 - allowed. -3 -4 1 - allowed. -2 -4 1 - allowed. -1 -4 1 - allowed. 0 -4 1 - allowed. 1 -4 1 - allowed. 2 -4 1 - allowed. 3 -4 1 - allowed. 4 -4 1 - allowed. 5 -4 1 - allowed. -5 -5 1 - allowed. -4 -5 1 - allowed. -3 -5 1 - allowed. -2 -5 1 - allowed. -1 -5 1 - allowed. 0 -5 1 - allowed. 1 -5 1 - allowed. 2 -5 1 - allowed. 3 -5 1 - allowed. 4 -5 1 - allowed. 5 -5 1 - allowed. a *b *hk 1
Neither layer loses a reflection to the symmetry. Whatever conditions this group has, none of them reaches this zone.
Whatever the structure is, the pattern in the zero layer cannot break the symmetry the Laue group carries into it, which here is 6mm . The upper layer keeps the same 6mm . The reverse does not follow: a group with no conditions at all draws a perfectly symmetrical net, so a symmetrical picture is no evidence of a Laue class.
Drawn from the symmetry operations, not from the table above: the class labels
there stand for whole symmetry orbits, so a picture built by applying them literally would show
some genuinely absent reflections as present. The two are checked against each other rather than
derived from each other.
What the absences do not tell you
These same conditions are produced by
36 other space groups.
Systematic absences cannot separate them, and no amount of care with the data will change
that: the choice between them is made on intensity statistics, on the structure solving, or
on chemistry.
Moving the origin multiplies every structure factor by a phase and changes no intensity, so the two origin choices the tables give for 24 space groups have identical conditions — which is why this table does not ask which one you are looking at, and the coordinates above it do.