Uniform 6-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams.
There are four fundamental reflective symmetry groups which generate 153 unique uniform 6-polytopes.
These fundamental families generate 153 nonprismatic convex uniform polypeta.
The coordinates of uniform 6-polytopes with 6-simplex symmetry can be generated as permutations of simple integers in 7-space, all in hyperplanes with normal vector (1,1,1,1,1,1,1).
There are 63 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.
They are named by Norman Johnson from the Wythoff construction operations upon the regular 6-cube and 6-orthoplex. Bowers names and acronym names are given for cross-referencing.
There are four fundamental affine Coxeter groups and 27 prismatic groups that generate regular and uniform tessellations in 5-space:
Here's the primary operators available for constructing and naming the uniform 6-polytopes.
The prismatic forms and bifurcating graphs can use the same truncation indexing notation, but require an explicit numbering system on the nodes for clarity.