Further symmetry exists in the cyclic A ~ ~ n and branching D 3 , E ~ ~ 6 , and D ~ ~ 4 diagrams. A ~ ~ n has order 2 n symmetry of a regular n -gon, { n }, and is represented by [ n [3 ]]. D 3 and E ~ ~ 6 are represented by [3[3 ]] = [3,4,3] and [3[3 ]] respectively while D ~ ~ 4 by [(3,3)[3 ]] = [3,3,4,3], with the diagram containing the order 24 symmetry of the regular tetrahedron, {3,3}. The paracompact hyperbolic group L ¯ ¯ 5 = [3 ], , contains the symmetry of a 5-cell, {3,3,3}, and thus is represented by [(3,3,3)[3 ]] = [3,4,3,3,3].