Space group
A space group is a mathematical symmetry category of n-dimensional structures with translational symmetry in n independent directions, such as, for n=3, a crystal. The word 'group' in the name comes from the mathematical notion of a group: a space group is a category of symmetry groups.
In 1D there are two space groups: those with and without mirror image symmetry. In 2D there are 17: the wallpaper groups.
There are 230 possible space groups in 3D, which reduces to 219 if space-group types differing only for the enantiomorphous character (e.g. P3112 and P3212) are not distinguished (affine space-group types). The set is made from the combination of the 32 crystallographic point groups with the 14 Bravais lattices which belong to one of 7 crystal systems. This results in a space group being a combination of a unit cell with some form of motif centering, along with the point operations of reflection, rotation and improper-rotation. In addition, there are the translational symmetry elements. The basic translation is covered by the lattice type, leaving combinations of reflections and rotations with translation:
Screw axis:
A rotation of 360°/n about an axis, combined with a translation along the axis. A subscript indicates the translation distance, as a multiple of the distance obtained by dividing that of the translational symmetry by n. So, 63 is a rotation of 60° combined with a translation of 1/2 of the lattice vector, implying that there is also 3-fold rotational symmetry about this axis. The possibilities are 21, 31, 41, 42, 61, 62, and 63, and the enantiomorphous 32, 43, 64, and 65.
Glide plane:
A reflection in a plane, followed by a translation parallel with that plane. This is noted by a, b or c, depending on which axis the glide is along. There is also the n glide, which is a glide along the half of a diagonal of a face, and the d glide, which is along the fourth of a diagonal of a face of the unit cell.
It is easily noted that not all of the possible combinations of the Bravais lattices, crystal systems and point groups are apparent in the space groups ( 32 * 14 = 448 > 230). This is because a number of different combinations are isomorphic with each other (that is, they turn out to be the same thing). This was proved using group theory, and is the source of the word 'group' in the title.
There are a number of methods of identifying space groups. The International Union of Crystallography publishes a table (more correctly, a hefty tome of tables) of all space groups, and assigns each a unique number. Other than this numbering schemes there are two main forms of notation, Paterson notation and Schoenflies.
Paterson notation consists of a set of four symbols. The first describes the centering of the Bravais lattice (P, C, I or F). The next three describe the most prominent symmetry operation visible when projected from the a, b and c face respectively. These symbols are the same as used in point groups, with the addition of glide planes and screw axis, described above. By way of example, the space group for quartz is P3121, showing that it exhibits primitive centering of the motif (i.e. once per unit cell), with a threefold screw axis projecting on one face, and two fold rotation axis another. Note that it does not explicitly contain the crystal system, although this is unique to each space group (in the case of P3121, it is trigonal).