An example of the hexagonal crystals, beryl Beryl The mineral beryl is a beryllium aluminium cyclosilicate with the chemical formula Be3Al26. The hexagonal crystals of beryl may be very small or range to several meters in size. Terminated crystals are relatively rare...
Hexagonal Hanksite Hanksite Hanksite is a sulfate mineral, distinguished as one of only a handful that contain both carbonate and sulfate ion groups. It has the chemical formula: Na22K92Cl.-Occurrence:...  crystal	Hexagonal lattice cell

In crystallography

Crystallography

Crystallography is the experimental science of the arrangement of atoms in solids. The word "crystallography" derives from the Greek words crystallon = cold drop / frozen drop, with its meaning extending to all solids with some degree of transparency, and grapho = write.Before the development of...

, the hexagonal crystal system is one of the 7 crystal system

Crystal system

In crystallography, the terms crystal system, crystal family, and lattice system each refer to one of several classes of space groups, lattices, point groups, or crystals...

s, the hexagonal lattice system is one of the 7 lattice systems, and the hexagonal crystal family is one of the 6 crystal families. They are closely related and often confused with each other, but they are not the same.
The hexagonal lattice system consists of just one Bravais lattice type: the hexagonal one. The hexagonal crystal system consists of the 7 point groups such that all their space groups have the hexagonal lattice as underlying lattice. The hexagonal crystal family consists of the 12 point groups such that at least one of their space groups has the hexagonal lattice as underlying lattice, and is the union of the hexagonal crystal system and the trigonal crystal system.

In some cases, it is useful or instructive to redraw a hexagonal structure with orthohexagonal axes, wherein the b axis is redrawn at 90° to the a and c axes.

Graphite

Graphite

The mineral graphite is one of the allotropes of carbon. It was named by Abraham Gottlob Werner in 1789 from the Ancient Greek γράφω , "to draw/write", for its use in pencils, where it is commonly called lead . Unlike diamond , graphite is an electrical conductor, a semimetal...

 is an example of a crystal

Crystal

A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...

 that crystallizes in the hexagonal crystal system.

Hexagonal lattice system

The hexagonal lattice system is one of the seven lattice systems, consisting of the hexagonal Bravais lattice. It is associated with 45 space group

Space group

In mathematics and geometry, a space group is a symmetry group, usually for three dimensions, that divides space into discrete repeatable domains.In three dimensions, there are 219 unique types, or counted as 230 if chiral copies are considered distinct...

s
whose underlying lattice has point group

Point group

In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O...

 of order 24. It is often confused with the smaller hexagonal crystal system, which consists of the 27 space groups such that all space groups with the same point group are in the hexagonal lattice system, or with the larger hexagonal crystal family, consisting of the 52 space groups in either the hexagonal or rhombohedral lattice systems.

Hexagonal crystal system

The point groups

Crystallographic point group

In crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a central point fixed while moving other directions and faces of the crystal to the positions of features of the same kind...

 (crystal classes) in this crystal system are listed below, followed by their representations in Hermann-Mauguin

Hermann-Mauguin notation

Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann and the French mineralogist Charles-Victor Mauguin...

 or international notation and Schoenflies notation

Schoenflies notation

The Schoenflies notation or Schönflies notation, named after the German mathematician Arthur Moritz Schoenflies, is one of two conventions commonly used to describe Point groups. This notation is used in spectroscopy. The other convention is the Hermann–Mauguin notation, also known as the...

, and mineral

Mineral

A mineral is a naturally occurring solid chemical substance formed through biogeochemical processes, having characteristic chemical composition, highly ordered atomic structure, and specific physical properties. By comparison, a rock is an aggregate of minerals and/or mineraloids and does not...

 examples, if they exist.

#	Point group					Example	Space groups
	Class	Intl Hermann-Mauguin notation Hermann–Mauguin notation is used to represent the symmetry elements in point groups, plane groups and space groups. It is named after the German crystallographer Carl Hermann and the French mineralogist Charles-Victor Mauguin...	Schoenflies Schoenflies notation The Schoenflies notation or Schönflies notation, named after the German mathematician Arthur Moritz Schoenflies, is one of two conventions commonly used to describe Point groups. This notation is used in spectroscopy. The other convention is the Hermann–Mauguin notation, also known as the...	Orbifold	Coxeter Coxeter notation In geometry, Coxeter notation is a system of classifying symmetry groups, describing the angles between with fundamental reflections of a Coxeter group. It uses a bracketed notation, with modifiers to indicate certain subgroups. The notation is named after H. S. M...
168-173	Hexagonal pyramidal	6	C6	66	[6]+	nepheline Nepheline Nepheline, also called nephelite , is a feldspathoid: a silica-undersaturated aluminosilicate, Na3KAl4Si4O16, that occurs in intrusive and volcanic rocks with low silica, and in their associated pegmatites... , cancrinite Cancrinite Cancrinite is a complex carbonate and silicate of sodium, calcium and aluminium with the formula Na6Ca2[2|Al6Si6O24]·2H2O. It is classed as a member of the feldspathoid group of minerals; the alkali feldspars that are poor in silica. Yellow, orange, pink, white or even blue, it has a vitreous or...	P6, P61, P65, P62, P64, P63
174	Trigonal dipyramidal		C3h	3*	[2,3+]	Laurelite	P
175-176	Hexagonal dipyramidal	6/m	C6h	6*	[2,6+]	apatite Apatite Apatite is a group of phosphate minerals, usually referring to hydroxylapatite, fluorapatite, chlorapatite and bromapatite, named for high concentrations of OH−, F−, Cl− or Br− ions, respectively, in the crystal... , vanadinite Vanadinite Vanadinite is a mineral belonging to the apatite group of phosphates, with the chemical formula Pb53Cl. It is one of the main industrial ores of the metal vanadium and a minor source of lead. A dense, brittle mineral, it is usually found in the form of red hexagonal crystals. It is an uncommon...	P6/m, P63/m
177-182	Hexagonal trapezohedral	622	D6	226	[2,6]+	kalsilite Kalsilite Kalsilite is a vitreous white to grey feldspathoidal mineral that is found in some potassium-rich lavas, such as from Chamengo Crater in Uganda. It has a relative hardness of 5.5.- References :* * *...  and high quartz Quartz Quartz is the second-most-abundant mineral in the Earth's continental crust, after feldspar. It is made up of a continuous framework of SiO4 silicon–oxygen tetrahedra, with each oxygen being shared between two tetrahedra, giving an overall formula SiO2. There are many different varieties of quartz,...	P622, P6122, P6522, P6222, P6422, P6322
183-186	Dihexagonal pyramidal	6mm	C6v	*66	[6]	greenockite Greenockite Greenockite is a rare cadmium mineral that consists of cadmium sulfide, CdS, in crystalline form. Greenockite crystallizes in the hexagonal system. It occurs as massive encrustations and as six-sided pyramidal crystals which vary in color from a honey yellow through shades of red to brown... , wurtzite Wurtzite Wurtzite is a zinc iron sulfide mineral a less frequently encountered mineral form of sphalerite. The iron content is variable up to eight percent. It is trimorphous with matraite and sphalerite....	P6mm, P6cc, P63cm, P63mc
187-190	Ditrigonal dipyramidal	m2	D3h	*223	[2,3]	benitoite Benitoite Benitoite is a rare blue barium titanium silicate mineral, found in hydrothermally altered serpentinite. Benitoite fluoresces under short wave ultraviolet light, appearing bright blue to bluish white in color. The more rarely seen clear to white benitoite crystals fluoresce red under long-wave UV...	Pm2, Pc2, P2m, P2c
191-194	Dihexagonal dipyramidal	6/mmm	D6h	*226	[2,6]	beryl Beryl The mineral beryl is a beryllium aluminium cyclosilicate with the chemical formula Be3Al26. The hexagonal crystals of beryl may be very small or range to several meters in size. Terminated crystals are relatively rare...	P6/mmm, P6/mcc, P63/mcm, P63/mmc

Hexagonal crystal family

The hexagonal crystal family consists of the 12 point groups such that at least one of their space groups has the hexagonal lattice as underlying lattice, and is the union of the hexagonal crystal system and the trigonal crystal system. There are 50 space groups associated with it, which are exactly those whose Bravais lattice is either hexagonal or rhombohedral.