A worked example: the hexagon

Cosets, stabilisers and kernels on the smallest group where they all differ — every colouring of a D₆ figure, drawn.

The coloured crystals are built from one recipe: a colouring is not something you invent, it is something you choose a subgroup for. That recipe is easier to see on a finite group than on a wallpaper group, and the hexagon is the smallest case where every distinction the catalogue makes is already visible. Everything on this page is computed from the group rather than tabulated — the subgroup lattice, the cosets, the pictures.

D₆ here means the hexagon's twelve symmetries — six rotations 1, r, r², r³, r⁴, r⁵ by multiples of 60°, and six reflections s, rs, …, r⁵s. (This is the crystallographer's D₆, the one behind the site's 632 and *632, not the algebraist's group of order 6.) The one relation you need is that reflecting reverses a turn, s r s = r⁻¹.

1. Copies are labelled by group elements

Draw one asymmetric motif in one of the twelve slivers of the hexagon. The group produces eleven more, and — because the motif has no symmetry of its own — exactly one copy per group element. So a copy is a group element. Here each of the twelve has its own colour:

Note that six copies read as left-handed and six as right-handed. That is the reflections doing their work, and it is why the motif has to be chiral: a symmetric blob would make the picture unreadable.

2. Cosets

Fix a subgroup H ≤ G. The left coset of an element g is

gH = { gh : h ∈ H }.

Three facts do all the work. Every element lies in exactly one coset, so the cosets partition the group. Every coset has |H| elements, so there are |G|/|H| of them. And gH = g′H exactly when g⁻¹g′ ∈ H.

Take H = {1, s}, a single reflection. Then |H| = 2, so there are six cosets:

{1, s}   {r, rs}   {r², r²s}   {r³, r³s}   {r⁴, r⁴s}   {r⁵, r⁵s}

Six cosets, six colours, and the two copies sharing a colour are the two halves of a hexagon vertex's neighbourhood — a left-handed one and its mirror image. That is not a coincidence: it is the whole idea.

3. The colouring

Paint the copy at g with the colour gH. It is well defined by the third fact above — two elements share a colour precisely when they differ by something in H — and it automatically respects the symmetry, since acting by x sends the colour gH to the colour xgH, the same way for every copy at once. That is the definition the catalogue uses, with E(2) × S_k in place of D₆ × S_k.

4. Left cosets or right cosets?

Both are used in the literature, and the choice is not free once the picture is drawn. Copies were labelled by sending the base copy to g·x₀, so the symmetry acts on copies by left multiplication. Ask which partitions of G that action preserves: if the block containing 1 is B, then aB is a block for every a, which forces B to be a subgroup and the blocks to be its left cosets. So

colours = orbits of right multiplication by H;   symmetry = left multiplication by G.

The construction works precisely because those two actions commute. That is the whole content of "the colouring respects the symmetry", and it is what makes the stabiliser come out as H rather than something else.

Take the other convention literally and colour the copy at g by the right coset Hg. For H = {1, s} the two partitions are

gH:  {1,s} {r,rs} {r²,r²s} {r³,r³s} {r⁴,r⁴s} {r⁵,r⁵s}
Hg:  {1,s} {r,r⁵s} {rs,r⁵} {r²,r⁴s} {r²s,r⁴} {r³,r³s}

— genuinely different, and only four of the twelve symmetries permute the right-hand blocks at all. Those four are the normaliser of H, since a·Hg = aHg is again a right coset exactly when aH = Ha. So mixing the conventions does not give a worse colouring; it gives something that is not a colouring.

None of this changes the count. The map gH ↦ Hg⁻¹ is a bijection between the two coset spaces, subgroups and their conjugacy classes are left–right symmetric, and normality is too — so a catalogue built the other way round has the same 16 subgroups, the same 10 classes and the same palette sizes. Every card below shows both, and they coincide exactly on the normal rows.

5. Every colouring of the hexagon

There are … subgroups in … conjugacy classes, so there are that many colourings up to a turn of the hexagon. Conjugate subgroups give the same picture rotated, which is why the classes are what get counted — the same reason the coloured crystals are counted up to an affine map of the plane.

Grouped by palette size, as catalogue D is. The number of colours is the index of H, so the headings below are also a walk down the subgroup lattice: index 1 at the top, index 12 — the free orbit, one colour per symmetry — at the bottom.

6. What to read off them

Two colourings can have the same number of colours and be structurally different. Both six-colour entries above have six colours and twelve copies. But H = {1, s} is not normal: its kernel is trivial, the colour group is all of D₆, and twelve symmetries act on six colours — each colour keeping a two-element stabiliser. Whereas H = {1, r³} is normal: stabiliser and kernel coincide, the colour group is S₃, and six act on six. The first is not regular, the second is. That distinction is exactly why the pair notation Γ/K cannot be a complete name for a coloured crystal, and why the catalogue prints H as well as the kernel.

Some colourings could be clocks and some could not. A colour group that is cyclic can be read as "advance the phase by 1/N", which is what turns a colouring into an animation — and into a crystal one dimension up. Of the classes above only those with cyclic colour group qualify; the ones with colour group S₃ or D₆ cannot be run as a clock at all, because phases form a circle and the group of phase-advances is cyclic. That is the same obstruction that leaves 185 of the 269 coloured crystals with no film to their name.

The three two-colourings are the magnetic case. Colour-preserving against colour-swapping is Shubnikov's prime, spin up against spin down. On the hexagon there are three of them, one for each subgroup of index 2: C₆ — rotations one colour, reflections the other, which with a chiral motif is left-handed against right-handed — and the two S₃s, which alternate around the hexagon in the two inequivalent ways.

Now read catalogue D again: every entry there is this construction with D₆ replaced by one of the 17 wallpaper groups, and the plate below each label is the same picture on an infinite figure.