The term "polychoron" (plural "polychora", adjective "polychoric"), from the Greek roots "poly" ("many") and "choros" ("room" or "space") and is advocated by Norman Johnson and George Olshevsky in the context of uniform polychora (4-polytopes), and their related 4-dimensional symmetry groups. Rank 4 Coxeter groups allow a set of 4 mirrors to span 4-space, and divides the 3-sphere into tetrahedral fundamental domains. Lower rank Coxeter groups can only bound hosohedron or hosotope fundamental domains on the 3-sphere. Like the 3D polyhedral groups, the names of the 4D polychoric groups given are constructed by the Greek prefixes of the cell counts of the corresponding triangle-faced regular polytopes. Extended symmetries exist in uniform polychora with symmetric ring-patterns within the Coxeter diagram construct. Chiral symmetries exist in alternated uniform polychora. Only irreducible groups have Coxeter numbers, but duoprismatic groups [p,2,p] can be doubled to p,2,p by adding a 2-fold gyration to the fundamental domain, and this gives an effective Coxeter number of 2"p", for example the [4,2,4] and its full symmetry B4, [4,3,3] group with Coxeter number 8. The symmetry order is equal to the number of cells of the regular polychoron times the symmetry of its cells. The omnitruncated dual polychora have cells that match the fundamental domains of the symmetry group. Direct subgroups of the reflective 4-dimensional point groups are:        This is a summary of 4-dimensional [[point group]]s in [[Coxeter notation]]. 227 of them are crystallographic point groups (for particular values of p and q). (nc) is given for non-crystallographic groups. Some crystallographic group have their orders indexed (order.index) by their abstract group structure. [[Category:4-polytopes]]