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The prismatic groups are denoted by D nh. These groups are characterized by i) an n-fold proper rotation axis C n; ii) n 2-fold proper rotation axes C 2 normal to C n; iii) a mirror plane σ h normal to C n and containing the C 2 s. The D 1h group is the same as the C 2v group in the pyramidal groups section.
The two octahedral cells project onto the entire volume of this envelope, while the 8 triangular prismic cells project onto its 8 triangular faces. The triangular-prism-first orthographic projection of the octahedral prism into 3D space has a hexagonal prismic envelope. The two octahedral cells project onto the two hexagonal faces.
A polychoric group is one of five symmetry groups of the 4-dimensional regular polytopes. There are also three polyhedral prismatic groups, and an infinite set of duoprismatic groups. Each group defined by a Goursat tetrahedron fundamental domain bounded by mirror planes. The dihedral angles between the mirrors determine order of dihedral symmetry.
The octahedral symmetry is represented by a fundamental triangle (4 3 2) counting the mirrors at each vertex. It can also be represented by the Coxeter group B 2 or [4,3], as well as a Coxeter diagram: . There are 48 triangles, visible in the faces of the disdyakis dodecahedron, and in the alternately colored triangles on a sphere: #
The octahedron is one of the Platonic solids, although octahedral molecules typically have an atom in their centre and no bonds between the ligand atoms. A perfect octahedron belongs to the point group O h. Examples of octahedral compounds are sulfur hexafluoride SF 6 and molybdenum hexacarbonyl Mo(CO) 6. The term "octahedral" is used somewhat ...
In some sense this reduces the classification of these groups to the classification of p-groups. Some of the small groups that do not have a normal p-complement include: Order 24: The symmetric group S 4; Order 48: The binary octahedral group and the product S 4 × Z 2; Order 60: The alternating group A 5.
A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to a lattice system. Of the 32 point groups that exist in three dimensions, most are assigned to only one lattice system, in which case the crystal system and lattice system both have the same name.
The table below organizes the space groups of the monoclinic crystal system by crystal class. It lists the International Tables for Crystallography space group numbers, [ 2 ] followed by the crystal class name, its point group in Schoenflies notation , Hermann–Mauguin (international) notation , orbifold notation, and Coxeter notation, type ...