Point groups in three dimensions the finite subgroups of SO(3), are: the cyclic groups "C" (the rotation group of a canonical pyramid), the dihedral groups "D" (the rotation group of a uniform prism, or canonical bipyramid), and the rotation groups "T", "O" and "I" of a regular tetrahedron, octahedron/cube and icosahedron/dodecahedron. In particular, the dihedral groups "D", "D" etc. are the rotation groups of plane regular polygons embedded in three-dimensional space, and such a figure may be considered as a degenerate regular prism. Therefore, it is also called a "dihedron" (Greek: solid with two faces), which explains the name "dihedral group". The rotation group of
Smooth projective plane then formula_7 is a translation plane, or a dual translation plane, or a Hughes plane. The latter can be characterized as follows: formula_44 leaves some classical complex subplane formula_45 invariant and induces on formula_45 the connected component of its full automorphism group. The Hughes planes are not smooth. This yields a result similar to the case of 4-dimensional planes: Theorem. "If formula_7 is a smooth 8-dimensional plane, then formula_7 is the classical quaternion plane or formula_49". Let formula_44 denote the automorphism group of a compact 16-dimensional topological projective plane formula_7. Either formula_7 is the smooth classical octonion plane or formula_53.