What does it mean to say "the 222 space group"?
This is a crystallographer’s shorthand for a certain type of space group.

You may have heard of the 14 Bravais lattices, e.g. cubic, monoclinic, orthorhombic. These lattices all assume perfect sphere’s at the corners of the elementary cell and potentially also in certain central positions (if the lattice is not a primitive one). In actual X-ray structure determination, these perfect Bravais lattice spheres are typically not spheres. Instead, we should consider each sphere a ‘hook’ and a molecule hung to that hook. Since not all molecules are symmetric with respect to all symmetry operations a certain Bravais lattice has, the symmetry of the crystal is markedly reduced. For example, an enantiopure chiral compound cannot form crystals that contain planes of symmetry, so even if that compound were to crystallise in the orthorhombic lattice, there would be no planes in spite of them being required in simple orthorhombic crystals.

This reduction in symmetry leads to a large number of lower-symmetry sub-divisions of the Bravais lattices called space groups. Strictly speaking, the Bravais lattices are merely highly symmetrical space groups. There is a total of 219 or 230 — depending on whether you count enantiomeric space groups as two or one.

The shorthand to unambiguously label these space groups basically describes which symmetry operations are present in a certain direction. Orthorhombic is the easiest to understand: the primitive Bravais lattice has:


in $x$ direction:


a two-fold axis
a plane of symmetry perpendicular to that axis.

in $y$ direction:


a two-fold axis
a plane of symmetry perpendicular to that axis.

in $z$ direction:


a two-fold axis
a plane of symmetry perpendicular to that axis.



You could label this is $\mathrm{P\ \frac2m \frac2m \frac2m}$, but it is tiresome to always include the ‘2 over m’ part. Hence, it is shortened to $\mathrm{P\ mmm}$, the shorthand being ‘m replaces $\frac2m$’. If there is no plane of symmetry, then only the two-fold axis remains and that dimension is given a 2 label instead. Thus, $\mathrm{P\ 222}$ is the same lattice but there are no planes of symmetry in the actual crystal structure. The crystal could also include a screw axis, in which case the symbol $2_1$ would be used, or a glide plane in which a different letter, depending on the orientation of the glide plane, is used.