= Solution
A <Bravais lattice> is the translation set $\Lambda=\{n_1\mathbf a_1+n_2\mathbf a_2+n_3\mathbf a_3:n_i\in\mathbb Z\}$ generated by <primitive lattice translation vectors>. Translating by any member of $\Lambda$ leaves its surroundings equivalent. A <crystal structure> requires both this lattice and an atomic basis attached to every lattice point. The basis can lower the symmetry of the bare point lattice; a familiar-looking geometrical arrangement of atoms is not necessarily itself a <Bravais lattice>. For example, the <diamond cubic structure> is a face-centred cubic lattice with an identical-atom basis at fractional positions $(0,0,0)$ and $(1/4,1/4,1/4)$ in conventional cubic coordinates. The displacement between those two basis atoms is not itself a <lattice translation>.
The seven three-dimensional <crystal systems> are triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal and cubic; the distinct translation types give 14 <Bravais lattices>. A conventional cell can contain several lattice points, while a <primitive unit cell> contains one. Simple cubic <primitive lattice translation vectors> can be chosen along the cube edges; for face-centred cubic they can be $(a/2)(0,1,1)$, $(a/2)(1,0,1)$ and $(a/2)(1,1,0)$. Their primitive volume is $a^3/4$, explaining the four lattice points per conventional face-centred cell.
A <crystallographic point group> comprises the orthogonal parts of the <crystal>'s spatial symmetry operations. It constrains possible rotations, reflections, inversion and their combinations, while forgetting translations. Lattice preservation restricts proper rotational orders in three dimensions to $1,2,3,4,6$; a fivefold axis cannot be a symmetry of a periodic three-dimensional lattice. One quick argument restricts a rotation's integral lattice representation through $1+2\cos\theta$, an integer trace, producing these orders. There are 32 crystallographic point-group types. Examples are the cubic full octahedral group $O_h$ with Hermann–Mauguin symbol $m\bar3m$, the tetrahedral class $T_d$ with symbol $\bar43m$, and the polar tetragonal class $C_{4v}$ with symbol $4mm$.
The <space group> retains the full <isometry>. In <Seitz notation> $\{R\mid\mathbf t\}$ acts by $\mathbf x\mapsto R\mathbf x+\mathbf t$, and multiplication is
$$
\{R\mid\mathbf t\}\{R'\mid\mathbf t'\}=\{RR'\mid R\mathbf t'+\mathbf t\}.
$$
Its <translation subgroup> is normal and the quotient is the <crystallographic point group>. A <screw axis> combines rotation with translation along the axis; for example, applying a $2_1$ screw twice gives one whole <lattice translation>. A <glide plane> combines reflection with a translation parallel to the plane; applying the glide twice likewise gives a translation. These operations can preserve the complete <crystal> without their rotation or reflection alone preserving it.
A <symmorphic space group> admits an origin and representatives for which the point-group operations have no translation component. A <nonsymmorphic space group> has essential fractional translations that cannot all be removed by moving the origin. There are 230 three-dimensional space-group types, distinguished more finely than their <point groups>. These counts refer to ordinary, nonmagnetic periodic <space groups>; magnetic and aperiodic generalizations are different classifications.<International Tables for Crystallography, Volume A: Space-group symmetry>[2016 edition, overview and Part 1 introduction to crystallographic and space-group symmetry]{ref}
For concrete comparisons, a monatomic simple-cubic <crystal> has <space group> $Pm\bar3m$ and <point group> $m\bar3m$. Rock salt has a face-centred cubic lattice, a two-species basis and <symmorphic space group> $Fm\bar3m$. Diamond has the same cubic <point group> and translation-lattice type, but its basis leads to nonsymmorphic $Fd\bar3m$. Replacing the two diamond basis sites by distinct species gives the <zinc blende structure>, exemplified by <GaAs>, with $F\bar43m$ and <point group> $\bar43m$; inversion that would interchange the species is no longer a symmetry. \b[The lattice specifies translations, the <point group> specifies the linear symmetry classes, and the <space group> specifies their complete spatial implementation.] These examples show why none alone uniquely describes an atomic structure.
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