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Space Group Reflection Conditions

What a space group extinguishes: the integral, zonal and serial reflection conditions of any of the 230 groups, in any of the settings the International Tables list. Ask about one reflection and the page names the symmetry operation that removes it.

You supply
A space group number or symbol. Non-standard settings are included, so P21/n, Pbnm and I2/a answer for themselves rather than being redirected to a sibling with different conditions. A reflection is optional.
Reading it
Absences narrow the space group and rarely determine it: the 230 groups produce only 80 distinct patterns of absence, and 37 of them extinguish nothing at all. When your conditions fit more than one group, the page names the others.

Worked examples: P21/c – the commonest space group there is · P21/n – the same group, a different setting · P43212 – screw axes and nothing else · R3c – rhombohedral, in hexagonal axes · Fd3m – diamond, and (2 0 0) missing

Input

Space group

Any of 14, P2_1/c, P21/c or P 1 21/c 1. Subscripts may be written with an underscore or left out, and an inversion axis with a minus sign or without — Fd-3m and Fd3m both work.

Reflection (optional)
Position (optional)

Leave these empty for a general position. Put a point on a symmetry element — 0, 0, 0, or 1/4, 1/4, 1/4 — and the orbit collapses onto it.

Conditions belong to a setting, not to a space group number. Pnma, Pbnm and Pmcn are one space group with its axes labelled three ways, and the three tables differ: the mirror that empties hk0 in one of them empties 0kl in another. Every setting in the International Tables is here with its own operations, so P21/n and P21/a answer for themselves.

P42/ncm

Number 138
Full symbol P 42/n 21/c 2/m
Schoenflies D4h16
Crystal system tetragonal
Crystal class 4/mmm, order 16 — projected
Laue class 4/mmm, order 16
Lattice P (primitive)
General position 16 equivalent points — the coordinates, drawn
Setting standard

Symmetry elements

2-fold rotation axis at x, y, 0 along [11̄0]inversion centre at 0, 0, 0inversion centre at 0, 1/2, 02₁ screw axis at x, y, 0 along [11̄0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at x, y, 0 along [11̄0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a conventioninversion centre at 1/2, 0, 0inversion centre at 0, 0, 02₁ 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 convention2₁ 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 convention2-fold rotation axis at x, y, 0 along [110]b2₁ screw axis at 0, y, 1/4 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at 0, y, 1/4 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a conventioninversion centre at 1/2, 1/2, 0n glide plane at x, y, 0, perpendicular to [001], glide vector 1/2, 1/2, 02₁ 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 convention2-fold rotation axis at x, y, 0 along [11̄0]n glide plane at x, y, 0, perpendicular to [001], glide vector 1/2, 1/2, 02₁ 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 convention2-fold rotation axis at x, y, 0 along [110]2-fold rotation axis at x, y, 1/2 along [11̄0]inversion centre at 0, 0, 1/22₁ screw axis at x, 0, 1/4 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at x, 0, 1/4 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ 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 conventioninversion centre at 0, 0, 02₁ 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 conventioninversion centre at 1/2, 0, 0a2₁ screw axis at x, y, 0 along [11̄0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention4̄ inversion axis at 1/4, 3/4, z along [001], inverting through 1/4, 3/4, 1/42₁ screw axis at x, y, 0 along [11̄0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a conventioninversion centre at 0, 1/2, 1/22₁ screw axis at x, 1/2, 1/4 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at 1/2, y, 1/4 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at x, 1/2, 1/4 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at 1/2, y, 1/4 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a conventioninversion centre at 0, 1/2, 02₁ screw axis at x, y, 1/2 along [11̄0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention4₂ screw axis at 1/4, 1/4, z along [001], advancing 0, 0, 1/2 per turn -- its own mirror image, so the winding drawn is a conventionmirror plane at x, x, z, perpendicular to [110]2₁ screw axis at x, y, 1/2 along [11̄0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention4₂ screw axis at 1/4, 1/4, z along [001], advancing 0, 0, 1/2 per turn -- its own mirror image, so the winding drawn is a convention2-fold rotation axis at 1/4, 1/4, z along [001]4̄ inversion axis at 3/4, 1/4, z along [001], inverting through 3/4, 1/4, 1/4inversion centre at 1/2, 0, 1/2c glide plane at 1/4, y, z, perpendicular to [100], glide vector 0, 0, 1/2c glide plane at 1/4, y, z, perpendicular to [100], glide vector 0, 0, 1/22₁ 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 conventioninversion centre at 0, 0, 1/22₁ screw axis at x, 0, 1/4 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at x, 0, 1/4 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention2-fold rotation axis at x, y, 0 along [11̄0]inversion centre at 0, 0, 0c glide plane at x, 1/4, z, perpendicular to [010], glide vector 0, 0, 1/2c glide plane at x, 1/4, z, perpendicular to [010], glide vector 0, 0, 1/22₁ 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 conventionn glide plane at x, x, z, perpendicular to [11̄0], glide vector 1/2, 1/2, 04̄ inversion axis at 1/4, 3/4, z along [001], inverting through 1/4, 3/4, 5/42-fold rotation axis at x, y, 1/2 along [110]4̄ inversion axis at 1/4, 3/4, z along [001], inverting through 1/4, 3/4, -3/44̄ inversion axis at 1/4, 3/4, z along [001], inverting through 1/4, 3/4, -1/44̄ inversion axis at 1/4, 3/4, z along [001], inverting through 1/4, 3/4, 1/44̄ inversion axis at 1/4, 3/4, z along [001], inverting through 1/4, 3/4, 3/42-fold rotation axis at 1/4, 3/4, z along [001]n glide plane at x, x, z, perpendicular to [11̄0], glide vector 1/2, 1/2, 02₁ screw axis at 0, y, 3/4 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at 0, y, 3/4 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a conventionn glide plane at x, y, 1/2, perpendicular to [001], glide vector 1/2, 1/2, 0mirror plane at x, x, z, perpendicular to [11̄0]n glide plane at x, x, z, perpendicular to [110], glide vector -1/2, 1/2, 02₁ 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 conventioninversion centre at 1/2, 1/2, 1/22-fold rotation axis at x, y, 1/2 along [11̄0]n glide plane at x, y, 1/2, perpendicular to [001], glide vector 1/2, 1/2, 0n glide plane at x, x, z, perpendicular to [110], glide vector -1/2, 1/2, 02₁ 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 convention2₁ screw axis at 0, y, 1/4 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at 0, y, 1/4 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a conventionn glide plane at x, x, z, perpendicular to [11̄0], glide vector 1/2, 1/2, 04̄ inversion axis at 3/4, 1/4, z along [001], inverting through 3/4, 1/4, 5/42-fold rotation axis at x, y, 1/2 along [110]4̄ inversion axis at 3/4, 1/4, z along [001], inverting through 3/4, 1/4, -3/44̄ inversion axis at 3/4, 1/4, z along [001], inverting through 3/4, 1/4, -1/44̄ inversion axis at 3/4, 1/4, z along [001], inverting through 3/4, 1/4, 1/44̄ inversion axis at 3/4, 1/4, z along [001], inverting through 3/4, 1/4, 3/42-fold rotation axis at 3/4, 1/4, z along [001]n glide plane at x, x, z, perpendicular to [11̄0], glide vector 1/2, 1/2, 0c glide plane at x, 3/4, z, perpendicular to [010], glide vector 0, 0, 1/2c glide plane at x, 3/4, z, perpendicular to [010], glide vector 0, 0, 1/2inversion centre at 0, 0, 02-fold rotation axis at x, y, 0 along [11̄0]2₁ screw axis at x, 0, 3/4 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at x, 0, 3/4 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ 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 conventioninversion centre at 0, 0, 1/22₁ 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 conventionc glide plane at 3/4, y, z, perpendicular to [100], glide vector 0, 0, 1/2c glide plane at 3/4, y, z, perpendicular to [100], glide vector 0, 0, 1/2inversion centre at 1/2, 0, 1/22₁ screw axis at x, y, 1/2 along [11̄0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention4₂ screw axis at 3/4, 3/4, z along [001], advancing 0, 0, 1/2 per turn -- its own mirror image, so the winding drawn is a convention4̄ inversion axis at 1/4, 3/4, z along [001], inverting through 1/4, 3/4, 3/42₁ screw axis at x, y, 1/2 along [11̄0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a conventionmirror plane at x, x, z, perpendicular to [110]4₂ screw axis at 3/4, 3/4, z along [001], advancing 0, 0, 1/2 per turn -- its own mirror image, so the winding drawn is a convention2-fold rotation axis at 3/4, 3/4, z along [001]cinversion centre at 0, 1/2, 02₁ screw axis at x, 1/2, 3/4 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at 1/2, y, 3/4 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at x, 1/2, 3/4 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at 1/2, y, 3/4 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a conventioninversion centre at 0, 1/2, 1/22₁ screw axis at x, y, 0 along [11̄0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at x, y, 0 along [11̄0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention4̄ inversion axis at 3/4, 1/4, z along [001], inverting through 3/4, 1/4, 3/4inversion centre at 1/2, 0, 02₁ 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 conventioninversion centre at 0, 0, 02₁ screw axis at x, 0, 3/4 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at x, 0, 3/4 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention2-fold rotation axis at x, y, 1/2 along [11̄0]inversion centre at 0, 0, 1/22₁ 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 convention2-fold rotation axis at x, y, 0 along [110]n glide plane at x, y, 0, perpendicular to [001], glide vector 1/2, 1/2, 02-fold rotation axis at x, y, 0 along [11̄0]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 conventioninversion centre at 1/2, 1/2, 0n glide plane at x, y, 0, perpendicular to [001], glide vector 1/2, 1/2, 02₁ 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 convention2₁ screw axis at 0, y, 3/4 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at 0, y, 3/4 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2-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 conventioninversion centre at 0, 0, 02₁ 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 conventioninversion centre at 1/2, 0, 02₁ screw axis at x, y, 0 along [11̄0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at x, y, 0 along [11̄0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a conventioninversion centre at 0, 1/2, 02-fold rotation axis at x, y, 0 along [11̄0]inversion centre at 0, 0, 04₂ screw axis at 1/4, 1/4, z along [001], advancing 0, 0, 1/2 per turn -- its own mirror image, so the winding drawn is a convention4₂ screw axis at 3/4, 3/4, z along [001], advancing 0, 0, 1/2 per turn -- its own mirror image, so the winding drawn is a convention4̄ inversion axis at 1/4, 3/4, z along [001], inverting through 1/4, 3/4, -3/44̄ inversion axis at 1/4, 3/4, z along [001], inverting through 1/4, 3/4, -1/44̄ inversion axis at 1/4, 3/4, z along [001], inverting through 1/4, 3/4, 1/44̄ inversion axis at 1/4, 3/4, z along [001], inverting through 1/4, 3/4, 3/44̄ inversion axis at 1/4, 3/4, z along [001], inverting through 1/4, 3/4, 5/44̄ inversion axis at 3/4, 1/4, z along [001], inverting through 3/4, 1/4, -3/44̄ inversion axis at 3/4, 1/4, z along [001], inverting through 3/4, 1/4, -1/44̄ inversion axis at 3/4, 1/4, z along [001], inverting through 3/4, 1/4, 1/44̄ inversion axis at 3/4, 1/4, z along [001], inverting through 3/4, 1/4, 3/44̄ inversion axis at 3/4, 1/4, z along [001], inverting through 3/4, 1/4, 5/42-fold rotation axis at 1/4, 1/4, z along [001]2-fold rotation axis at 1/4, 3/4, z along [001]2-fold rotation axis at 3/4, 1/4, z along [001]2-fold rotation axis at 3/4, 3/4, z along [001]2₁ screw axis at x, 0, 1/4 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at x, 0, 3/4 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at x, 1/2, 1/4 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at x, 1/2, 3/4 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at x, 0, 1/4 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at x, 0, 3/4 along [100], advancing 1/2, 0, 0 per turn -- its own mirror image, so the winding drawn is a convention2-fold rotation axis at x, y, 0 along [11̄0]2-fold rotation axis at x, y, 1/2 along [11̄0]2-fold rotation axis at x, y, 0 along [11̄0]2₁ screw axis at x, y, 0 along [11̄0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at x, y, 1/2 along [11̄0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at x, y, 0 along [11̄0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2-fold rotation axis at x, y, 0 along [11̄0]2-fold rotation axis at x, y, 1/2 along [11̄0]2-fold rotation axis at x, y, 0 along [11̄0]2₁ screw axis at x, y, 0 along [11̄0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at x, y, 1/2 along [11̄0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at x, y, 0 along [11̄0], advancing 1/2, -1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2-fold rotation axis at x, y, 0 along [11̄0]2-fold rotation axis at x, y, 1/2 along [11̄0]2-fold rotation axis at x, y, 0 along [11̄0]2₁ screw axis at 0, y, 1/4 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at 0, y, 3/4 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at 1/2, y, 1/4 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at 1/2, y, 3/4 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at 0, y, 1/4 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ screw axis at 0, y, 3/4 along [010], advancing 0, 1/2, 0 per turn -- its own mirror image, so the winding drawn is a convention2₁ 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 convention2₁ 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 convention2₁ 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 convention2-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 convention2₁ 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 convention2₁ 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 convention2-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 convention2₁ 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 convention2₁ 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 conventioninversion centre at 0, 0, 0inversion centre at 0, 0, 1/2inversion centre at 0, 0, 0inversion centre at 0, 1/2, 0inversion centre at 0, 1/2, 1/2inversion centre at 0, 1/2, 0inversion centre at 0, 0, 0inversion centre at 0, 0, 1/2inversion centre at 0, 0, 0inversion centre at 1/2, 0, 0inversion centre at 1/2, 0, 1/2inversion centre at 1/2, 0, 0inversion centre at 1/2, 1/2, 0inversion centre at 1/2, 1/2, 1/2inversion centre at 1/2, 1/2, 0inversion centre at 1/2, 0, 0inversion centre at 1/2, 0, 1/2inversion centre at 1/2, 0, 0inversion centre at 0, 0, 0inversion centre at 0, 0, 1/2inversion centre at 0, 0, 0inversion centre at 0, 1/2, 0inversion centre at 0, 1/2, 1/2inversion centre at 0, 1/2, 0inversion centre at 0, 0, 0inversion centre at 0, 0, 1/2inversion centre at 0, 0, 04̄ inversion axis at 1/4, 3/4, z along [001], inverting through 1/4, 3/4, 1/44̄ inversion axis at 1/4, 3/4, z along [001], inverting through 1/4, 3/4, 3/44̄ inversion axis at 3/4, 1/4, z along [001], inverting through 3/4, 1/4, 1/44̄ inversion axis at 3/4, 1/4, z along [001], inverting through 3/4, 1/4, 3/4
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Look along

Point at an element to read what it is.

  • 12 2-fold axes
  • 14 21 screw axes
  • 4 4 inversion axes
  • 2 42 screw axes
  • 4 c glides
  • 8 inversion centres
  • 3 mirror planes
  • 5 n glides

Point 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 4/mmm — the 16 operations of P42/ncm 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.

ab[231], upper hemisphere, same hand as the start (from x,y,z) [231̄], lower hemisphere, opposite hand — an inverted image (from x,y,-z)[2̄3̄1̄], lower hemisphere, opposite hand — an inverted image (from -x,-y,-z) [2̄3̄1], upper hemisphere, same hand as the start (from -x,-y,z)[3̄21], upper hemisphere, same hand as the start (from -y,x,z) [3̄21̄], lower hemisphere, opposite hand — an inverted image (from -y,x,-z)[32̄1̄], lower hemisphere, opposite hand — an inverted image (from y,-x,-z) [32̄1], upper hemisphere, same hand as the start (from y,-x,z)[23̄1̄], lower hemisphere, same hand as the start (from x,-y,-z) [23̄1], upper hemisphere, opposite hand — an inverted image (from x,-y,z)[2̄31], upper hemisphere, opposite hand — an inverted image (from -x,y,z) [2̄31̄], lower hemisphere, same hand as the start (from -x,y,-z)[3̄2̄1̄], lower hemisphere, same hand as the start (from -y,-x,-z) [3̄2̄1], upper hemisphere, opposite hand — an inverted image (from -y,-x,z)[321], upper hemisphere, opposite hand — an inverted image (from y,x,z) [321̄], lower hemisphere, same hand as the start (from y,x,-z)abmirror plane perpendicular to [001]mirror plane perpendicular to [010]mirror plane perpendicular to [11̄0]mirror plane perpendicular to [100]mirror plane perpendicular to [110]4-fold rotation axis along [001], with a 4̄ inversion axis on the same line4-fold rotation axis along [001], with a 4̄ inversion axis on the same line2-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 [11̄0]2-fold rotation axis along [11̄0]2-fold rotation axis along [11̄0]2-fold rotation axis along [11̄0]2-fold rotation axis along [11̄0]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]inversion centre at the origin

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, the rim is ninety degrees away from it, and the angle between any two directions survives the projection. 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.

Point at any pole or any symbol to have it name itself.

16 poles, drawn as 8 marks. 4/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.

8 of the 16 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 16 symmetry operations of P42/ncm, written as coordinate triplets. Each one says where the group sends a point at x, y, z.

  1. x,y,z
  2. -x,-y,-z
  3. -y+1/2,x,z+1/2
  4. y+1/2,-x,-z+1/2
  5. -x+1/2,-y+1/2,z
  6. x+1/2,y+1/2,-z
  7. y,-x+1/2,z+1/2
  8. -y,x+1/2,-z+1/2
  9. x+1/2,-y,-z+1/2
  10. -x+1/2,y,z+1/2
  11. -y,-x,-z
  12. y,x,z
  13. -x,y+1/2,-z+1/2
  14. x,-y+1/2,z+1/2
  15. y+1/2,x+1/2,-z
  16. -y+1/2,-x+1/2,z

Where those points are

b0.13, 0.21, 0.31 — as given0.21, 0.13, 0.31 — mirror image0.29, 0.37, 0.31 — mirror imagea0.37, 0.29, 0.31 — as given0.63, 0.79, 0.19 — as given0.79, 0.63, 0.19 — mirror image0.71, 0.87, 0.19 — mirror image0.87, 0.71, 0.19 — as given0.13, 0.29, 0.81 — mirror image0.29, 0.13, 0.81 — as given0.21, 0.37, 0.81 — as given0.37, 0.21, 0.81 — mirror imagec0.63, 0.71, 0.69 — mirror image0.71, 0.63, 0.69 — as given0.79, 0.87, 0.69 — as given0.87, 0.79, 0.69 — mirror image
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Look along

0.13, 0.21, 0.31 is a general position. No operation of P42/ncm leaves it where it is, so the group sends it to 16 distinct points — one per operation, which is what makes the multiplicity equal to the order of the group.

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 16 points

#xy zFrom
1 0.13 0.21 0.31 x,y,z
2 0.87 0.79 0.69 -x,-y,-z
3 0.29 0.13 0.81 -y+1/2,x,z+1/2
4 0.71 0.87 0.19 y+1/2,-x,-z+1/2
5 0.37 0.29 0.31 -x+1/2,-y+1/2,z
6 0.63 0.71 0.69 x+1/2,y+1/2,-z
7 0.21 0.37 0.81 y,-x+1/2,z+1/2
8 0.79 0.63 0.19 -y,x+1/2,-z+1/2
9 0.63 0.79 0.19 x+1/2,-y,-z+1/2
10 0.37 0.21 0.81 -x+1/2,y,z+1/2
11 0.79 0.87 0.69 -y,-x,-z
12 0.21 0.13 0.31 y,x,z
13 0.87 0.71 0.19 -x,y+1/2,-z+1/2
14 0.13 0.29 0.81 x,-y+1/2,z+1/2
15 0.71 0.63 0.69 y+1/2,x+1/2,-z
16 0.29 0.37 0.31 -y+1/2,-x+1/2,z

Reflection conditions

ClassConditionFrom
0kl l = 2n zonal
h0l l = 2n zonal
hk0 h + k = 2n zonal
h00 h = 2n serial
0k0 k = 2n serial
00l l = 2n serial

Reading down the table: integral conditions come from the lattice centring and apply to every hkl; zonal conditions come from glide planes and empty one plane of reciprocal space; serial conditions come from screw axes and empty one row. The same three kinds are named under the stick pattern on the HKL page, which applies the integral ones only.

A class in this table stands for its own zone and every zone the point group carries it onto. In P63/mmc the row for hhl also governs (−2, 1, l) and (1, −2, l); in a cubic group the row for hhl also governs hkh and hkk. Seventeen of the settings here extinguish reflections that no literal reading of a class label covers, which is why the check below asks the symmetry operations rather than this table.

Is a reflection there?

Put three Miller indices into the form and this says whether P42/ncm allows that reflection, and which symmetry operation removes it if it does not.

What the absences do not tell you

These same conditions are produced by one other space group. 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. Only one is listed here.