# Uniform 7-polytope

Graphs of three regular and related uniform polytopes :
| 7-simplex | 7-simplex | 7-simplex | 7-simplex | Rectified 7-simplex | Rectified 7-simplex | Rectified 7-simplex | Rectified 7-simplex | Truncated 7-simplex | Truncated 7-simplex | Truncated 7-simplex | Truncated 7-simplex |
| Cantellated 7-simplex | Cantellated 7-simplex | Cantellated 7-simplex | Cantellated 7-simplex | Runcinated 7-simplex | Runcinated 7-simplex | Runcinated 7-simplex | Runcinated 7-simplex | Stericated 7-simplex | Stericated 7-simplex | Stericated 7-simplex | Stericated 7-simplex |
| Pentellated 7-simplex | Pentellated 7-simplex | Pentellated 7-simplex | Pentellated 7-simplex | Pentellated 7-simplex | Pentellated 7-simplex | Hexicated 7-simplex | Hexicated 7-simplex | Hexicated 7-simplex | Hexicated 7-simplex | Hexicated 7-simplex | Hexicated 7-simplex |
| 7-orthoplex | 7-orthoplex | 7-orthoplex | 7-orthoplex | Truncated 7-orthoplex | Truncated 7-orthoplex | Truncated 7-orthoplex | Truncated 7-orthoplex | Rectified 7-orthoplex | Rectified 7-orthoplex | Rectified 7-orthoplex | Rectified 7-orthoplex |
| Cantellated 7-orthoplex | Cantellated 7-orthoplex | Cantellated 7-orthoplex | Cantellated 7-orthoplex | Runcinated 7-orthoplex | Runcinated 7-orthoplex | Runcinated 7-orthoplex | Runcinated 7-orthoplex | Stericated 7-orthoplex | Stericated 7-orthoplex | Stericated 7-orthoplex | Stericated 7-orthoplex |
| Pentellated 7-orthoplex | Pentellated 7-orthoplex | Pentellated 7-orthoplex | Pentellated 7-orthoplex | Hexicated 7-cube | Hexicated 7-cube | Hexicated 7-cube | Hexicated 7-cube | Pentellated 7-cube | Pentellated 7-cube | Pentellated 7-cube | Pentellated 7-cube |
| Stericated 7-cube | Stericated 7-cube | Stericated 7-cube | Stericated 7-cube | Cantellated 7-cube | Cantellated 7-cube | Cantellated 7-cube | Cantellated 7-cube | Runcinated 7-cube | Runcinated 7-cube | Runcinated 7-cube | Runcinated 7-cube |
| 7-cube | 7-cube | 7-cube | 7-cube | Truncated 7-cube | Truncated 7-cube | Truncated 7-cube | Truncated 7-cube | Rectified 7-cube | Rectified 7-cube | Rectified 7-cube | Rectified 7-cube |
| 7-demicube | 7-demicube | 7-demicube | 7-demicube | Cantic 7-cube | Cantic 7-cube | Cantic 7-cube | Cantic 7-cube | Runcic 7-cube | Runcic 7-cube | Runcic 7-cube | Runcic 7-cube |
| Steric 7-cube | Steric 7-cube | Steric 7-cube | Steric 7-cube | Pentic 7-cube | Pentic 7-cube | Pentic 7-cube | Pentic 7-cube | Hexic 7-cube | Hexic 7-cube | Hexic 7-cube | Hexic 7-cube |
| 3$_{21}$ | 3$_{21}$ | 3$_{21}$ | 3$_{21}$ | 2$_{31}$ | 2$_{31}$ | 2$_{31}$ | 2$_{31}$ | 1$_{32}$ | 1$_{32}$ | 1$_{32}$ | 1$_{32}$ |
In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.
A uniform 7-polytope is one whose symmetry group is transitive on vertices and whose facets are uniform 6-polytopes.

## Regular 7-polytopes

Regular 7-polytopes are represented by the Schläfli symbol {p,q,r,s,t,u} with u {p,q,r,s,t} 6-polytopes facets around each 4-face.
There are exactly three such convex regular 7-polytopes:
* {3,3,3,3,3,3} - 7-simplex
* {4,3,3,3,3,3} - 7-cube
* {3,3,3,3,3,4} - 7-orthoplex
There are no nonconvex regular 7-polytopes.

## Characteristics

The topology of any given 7-polytope is defined by its Betti numbers and torsion coefficients.
The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.
Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.

## Uniform 7-polytopes by fundamental Coxeter groups

Uniform 7-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams:
| # | Coxeter group | Coxeter group | Regular and semiregular forms | Uniform count |
| 1 | A$_7$ | [3$^6$] | 7-simplex - {3$^6$}, | 71 |
| 2 | B$_7$ | [4,3$^5$] | 7-cube - {4,3$^5$}, 7-orthoplex - {3$^5$,4}, 7-demicube - h{4,3$^5$}, | 127 + 32 |
| 3 | D$_7$ | [3$^{3,1,1}$] | 7-demicube, {3,3$^{4,1}$}, 7-orthoplex, {3$^4$,3$^{1,1}$}, | 95 (0 unique) |
| 4 | E$_7$ | [3$^{3,2,1}$] | 3$_{21}$ - 1$_{32}$ - 2$_{31}$ - | 127 |
Prismatic finite Coxeter groups :
| # | Coxeter group | Coxeter group |
| 6+1 ||||
| 1 | A$_6$A$_1$ | [3$^5$]×[ ] |
| 2 | BC$_6$A$_1$ | [4,3$^4$]×[ ] |
| 3 | D$_6$A$_1$ | [3$^{3,1,1}$]×[ ] |
| 4 | E$_6$A$_1$ | [3$^{2,2,1}$]×[ ] |
| 5+2 ||||
| 1 | A$_5$I$_2$(p) | [3,3,3]×[p] |
| 2 | BC$_5$I$_2$(p) | [4,3,3]×[p] |
| 3 | D$_5$I$_2$(p) | [3$^{2,1,1}$]×[p] |
| 5+1+1 ||||
| 1 | A$_5$A$_1$$^2$ | [3,3,3]×[ ]$^2$ |
| 2 | BC$_5$A$_1$$^2$ | [4,3,3]×[ ]$^2$ |
| 3 | D$_5$A$_1$$^2$ | [3$^{2,1,1}$]×[ ]$^2$ |
| 4+3 ||||
| 1 | A$_4$A$_3$ | [3,3,3]×[3,3] |
| 2 | A$_4$B$_3$ | [3,3,3]×[4,3] |
| 3 | A$_4$H$_3$ | [3,3,3]×[5,3] |
| 4 | BC$_4$A$_3$ | [4,3,3]×[3,3] |
| 5 | BC$_4$B$_3$ | [4,3,3]×[4,3] |
| 6 | BC$_4$H$_3$ | [4,3,3]×[5,3] |
| 7 | H$_4$A$_3$ | [5,3,3]×[3,3] |
| 8 | H$_4$B$_3$ | [5,3,3]×[4,3] |
| 9 | H$_4$H$_3$ | [5,3,3]×[5,3] |
| 10 | F$_4$A$_3$ | [3,4,3]×[3,3] |
| 11 | F$_4$B$_3$ | [3,4,3]×[4,3] |
| 12 | F$_4$H$_3$ | [3,4,3]×[5,3] |
| 13 | D$_4$A$_3$ | [3$^{1,1,1}$]×[3,3] |
| 14 | D$_4$B$_3$ | [3$^{1,1,1}$]×[4,3] |
| 15 | D$_4$H$_3$ | [3$^{1,1,1}$]×[5,3] |
| 4+2+1 ||||
| 1 | A$_4$I$_2$(p)A$_1$ | [3,3,3]×[p]×[ ] |
| 2 | BC$_4$I$_2$(p)A$_1$ | [4,3,3]×[p]×[ ] |
| 3 | F$_4$I$_2$(p)A$_1$ | [3,4,3]×[p]×[ ] |
| 4 | H$_4$I$_2$(p)A$_1$ | [5,3,3]×[p]×[ ] |
| 5 | D$_4$I$_2$(p)A$_1$ | [3$^{1,1,1}$]×[p]×[ ] |
| 4+1+1+1 ||||
| 1 | A$_4$A$_1$$^3$ | [3,3,3]×[ ]$^3$ |
| 2 | BC$_4$A$_1$$^3$ | [4,3,3]×[ ]$^3$ |
| 3 | F$_4$A$_1$$^3$ | [3,4,3]×[ ]$^3$ |
| 4 | H$_4$A$_1$$^3$ | [5,3,3]×[ ]$^3$ |
| 5 | D$_4$A$_1$$^3$ | [3$^{1,1,1}$]×[ ]$^3$ |
| 3+3+1 ||||
| 1 | A$_3$A$_3$A$_1$ | [3,3]×[3,3]×[ ] |
| 2 | A$_3$B$_3$A$_1$ | [3,3]×[4,3]×[ ] |
| 3 | A$_3$H$_3$A$_1$ | [3,3]×[5,3]×[ ] |
| 4 | BC$_3$B$_3$A$_1$ | [4,3]×[4,3]×[ ] |
| 5 | BC$_3$H$_3$A$_1$ | [4,3]×[5,3]×[ ] |
| 6 | H$_3$A$_3$A$_1$ | [5,3]×[5,3]×[ ] |
| 3+2+2 ||||
| 1 | A$_3$I$_2$(p)I$_2$(q) | [3,3]×[p]×[q] |
| 2 | BC$_3$I$_2$(p)I$_2$(q) | [4,3]×[p]×[q] |
| 3 | H$_3$I$_2$(p)I$_2$(q) | [5,3]×[p]×[q] |
| 3+2+1+1 ||||
| 1 | A$_3$I$_2$(p)A$_1$$^2$ | [3,3]×[p]×[ ]$^2$ |
| 2 | BC$_3$I$_2$(p)A$_1$$^2$ | [4,3]×[p]×[ ]$^2$ |
| 3 | H$_3$I$_2$(p)A$_1$$^2$ | [5,3]×[p]×[ ]$^2$ |
| 3+1+1+1+1 ||||
| 1 | A$_3$A$_1$$^4$ | [3,3]×[ ]$^4$ |
| 2 | BC$_3$A$_1$$^4$ | [4,3]×[ ]$^4$ |
| 3 | H$_3$A$_1$$^4$ | [5,3]×[ ]$^4$ |
| 2+2+2+1 ||||
| 1 | I$_2$(p)I$_2$(q)I$_2$(r)A$_1$ | [p]×[q]×[r]×[ ] |
| 2+2+1+1+1 ||||
| 1 | I$_2$(p)I$_2$(q)A$_1$$^3$ | [p]×[q]×[ ]$^3$ |
| 2+1+1+1+1+1 ||||
| 1 | I$_2$(p)A$_1$$^5$ | [p]×[ ]$^5$ |
| 1+1+1+1+1+1+1 ||||
| 1 | A$_1$$^7$ | [ ]$^7$ |

## The A$_7$ family

The A$_7$ family has symmetry of order 40320 (8 factorial).
There are 71 (64+8-1) forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. All 71 are enumerated below. Norman Johnson's truncation names are given. Bowers names and acronym are also given for cross-referencing.
See also a list of A7 polytopes for symmetric Coxeter plane graphs of these polytopes.
A$_7$ uniform polytopes :
| # | Coxeter-Dynkin diagram | Truncation indices | Johnson name Bowers name (and acronym) | Basepoint | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts |
| # | Coxeter-Dynkin diagram | Truncation indices | Johnson name Bowers name (and acronym) | Basepoint | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | | t$_0$ | 7-simplex (oca) | (0,0,0,0,0,0,0,1) | 8 | 28 | 56 | 70 | 56 | 28 | 8 |
| 2 | | t$_1$ | Rectified 7-simplex (roc) | (0,0,0,0,0,0,1,1) | 16 | 84 | 224 | 350 | 336 | 168 | 28 |
| 3 | | t$_2$ | Birectified 7-simplex (broc) | (0,0,0,0,0,1,1,1) | 16 | 112 | 392 | 770 | 840 | 420 | 56 |
| 4 | | t$_3$ | Trirectified 7-simplex (he) | (0,0,0,0,1,1,1,1) | 16 | 112 | 448 | 980 | 1120 | 560 | 70 |
| 5 | | t$_{0,1}$ | Truncated 7-simplex (toc) | (0,0,0,0,0,0,1,2) | 16 | 84 | 224 | 350 | 336 | 196 | 56 |
| 6 | | t$_{0,2}$ | Cantellated 7-simplex (saro) | (0,0,0,0,0,1,1,2) | 44 | 308 | 980 | 1750 | 1876 | 1008 | 168 |
| 7 | | t$_{1,2}$ | Bitruncated 7-simplex (bittoc) | (0,0,0,0,0,1,2,2) | | | | | | 588 | 168 |
| 8 | | t$_{0,3}$ | Runcinated 7-simplex (spo) | (0,0,0,0,1,1,1,2) | 100 | 756 | 2548 | 4830 | 4760 | 2100 | 280 |
| 9 | | t$_{1,3}$ | Bicantellated 7-simplex (sabro) | (0,0,0,0,1,1,2,2) | | | | | | 2520 | 420 |
| 10 | | t$_{2,3}$ | Tritruncated 7-simplex (tattoc) | (0,0,0,0,1,2,2,2) | | | | | | 980 | 280 |
| 11 | | t$_{0,4}$ | Stericated 7-simplex (sco) | (0,0,0,1,1,1,1,2) | | | | | | 2240 | 280 |
| 12 | | t$_{1,4}$ | Biruncinated 7-simplex (sibpo) | (0,0,0,1,1,1,2,2) | | | | | | 4200 | 560 |
| 13 | | t$_{2,4}$ | Tricantellated 7-simplex (stiroh) | (0,0,0,1,1,2,2,2) | | | | | | 3360 | 560 |
| 14 | | t$_{0,5}$ | Pentellated 7-simplex (seto) | (0,0,1,1,1,1,1,2) | | | | | | 1260 | 168 |
| 15 | | t$_{1,5}$ | Bistericated 7-simplex (sabach) | (0,0,1,1,1,1,2,2) | | | | | | 3360 | 420 |
| 16 | | t$_{0,6}$ | Hexicated 7-simplex (suph) | (0,1,1,1,1,1,1,2) | | | | | | 336 | 56 |
| 17 | | t$_{0,1,2}$ | Cantitruncated 7-simplex (garo) | (0,0,0,0,0,1,2,3) | | | | | | 1176 | 336 |
| 18 | | t$_{0,1,3}$ | Runcitruncated 7-simplex (patto) | (0,0,0,0,1,1,2,3) | | | | | | 4620 | 840 |
| 19 | | t$_{0,2,3}$ | Runcicantellated 7-simplex (paro) | (0,0,0,0,1,2,2,3) | | | | | | 3360 | 840 |
| 20 | | t$_{1,2,3}$ | Bicantitruncated 7-simplex (gabro) | (0,0,0,0,1,2,3,3) | | | | | | 2940 | 840 |
| 21 | | t$_{0,1,4}$ | Steritruncated 7-simplex (cato) | (0,0,0,1,1,1,2,3) | | | | | | 7280 | 1120 |
| 22 | | t$_{0,2,4}$ | Stericantellated 7-simplex (caro) | (0,0,0,1,1,2,2,3) | | | | | | 10080 | 1680 |
| 23 | | t$_{1,2,4}$ | Biruncitruncated 7-simplex (bipto) | (0,0,0,1,1,2,3,3) | | | | | | 8400 | 1680 |
| 24 | | t$_{0,3,4}$ | Steriruncinated 7-simplex (cepo) | (0,0,0,1,2,2,2,3) | | | | | | 5040 | 1120 |
| 25 | | t$_{1,3,4}$ | Biruncicantellated 7-simplex (bipro) | (0,0,0,1,2,2,3,3) | | | | | | 7560 | 1680 |
| 26 | | t$_{2,3,4}$ | Tricantitruncated 7-simplex (gatroh) | (0,0,0,1,2,3,3,3) | | | | | | 3920 | 1120 |
| 27 | | t$_{0,1,5}$ | Pentitruncated 7-simplex (teto) | (0,0,1,1,1,1,2,3) | | | | | | 5460 | 840 |
| 28 | | t$_{0,2,5}$ | Penticantellated 7-simplex (tero) | (0,0,1,1,1,2,2,3) | | | | | | 11760 | 1680 |
| 29 | | t$_{1,2,5}$ | Bisteritruncated 7-simplex (bacto) | (0,0,1,1,1,2,3,3) | | | | | | 9240 | 1680 |
| 30 | | t$_{0,3,5}$ | Pentiruncinated 7-simplex (tepo) | (0,0,1,1,2,2,2,3) | | | | | | 10920 | 1680 |
| 31 | | t$_{1,3,5}$ | Bistericantellated 7-simplex (bacroh) | (0,0,1,1,2,2,3,3) | | | | | | 15120 | 2520 |
| 32 | | t$_{0,4,5}$ | Pentistericated 7-simplex (teco) | (0,0,1,2,2,2,2,3) | | | | | | 4200 | 840 |
| 33 | | t$_{0,1,6}$ | Hexitruncated 7-simplex (puto) | (0,1,1,1,1,1,2,3) | | | | | | 1848 | 336 |
| 34 | | t$_{0,2,6}$ | Hexicantellated 7-simplex (puro) | (0,1,1,1,1,2,2,3) | | | | | | 5880 | 840 |
| 35 | | t$_{0,3,6}$ | Hexiruncinated 7-simplex (puph) | (0,1,1,1,2,2,2,3) | | | | | | 8400 | 1120 |
| 36 | | t$_{0,1,2,3}$ | Runcicantitruncated 7-simplex (gapo) | (0,0,0,0,1,2,3,4) | | | | | | 5880 | 1680 |
| 37 | | t$_{0,1,2,4}$ | Stericantitruncated 7-simplex (cagro) | (0,0,0,1,1,2,3,4) | | | | | | 16800 | 3360 |
| 38 | | t$_{0,1,3,4}$ | Steriruncitruncated 7-simplex (capto) | (0,0,0,1,2,2,3,4) | | | | | | 13440 | 3360 |
| 39 | | t$_{0,2,3,4}$ | Steriruncicantellated 7-simplex (capro) | (0,0,0,1,2,3,3,4) | | | | | | 13440 | 3360 |
| 40 | | t$_{1,2,3,4}$ | Biruncicantitruncated 7-simplex (gibpo) | (0,0,0,1,2,3,4,4) | | | | | | 11760 | 3360 |
| 41 | | t$_{0,1,2,5}$ | Penticantitruncated 7-simplex (tegro) | (0,0,1,1,1,2,3,4) | | | | | | 18480 | 3360 |
| 42 | | t$_{0,1,3,5}$ | Pentiruncitruncated 7-simplex (tapto) | (0,0,1,1,2,2,3,4) | | | | | | 27720 | 5040 |
| 43 | | t$_{0,2,3,5}$ | Pentiruncicantellated 7-simplex (tapro) | (0,0,1,1,2,3,3,4) | | | | | | 25200 | 5040 |
| 44 | | t$_{1,2,3,5}$ | Bistericantitruncated 7-simplex (bacogro) | (0,0,1,1,2,3,4,4) | | | | | | 22680 | 5040 |
| 45 | | t$_{0,1,4,5}$ | Pentisteritruncated 7-simplex (tecto) | (0,0,1,2,2,2,3,4) | | | | | | 15120 | 3360 |
| 46 | | t$_{0,2,4,5}$ | Pentistericantellated 7-simplex (tecro) | (0,0,1,2,2,3,3,4) | | | | | | 25200 | 5040 |
| 47 | | t$_{1,2,4,5}$ | Bisteriruncitruncated 7-simplex (bicpath) | (0,0,1,2,2,3,4,4) | | | | | | 20160 | 5040 |
| 48 | | t$_{0,3,4,5}$ | Pentisteriruncinated 7-simplex (tacpo) | (0,0,1,2,3,3,3,4) | | | | | | 15120 | 3360 |
| 49 | | t$_{0,1,2,6}$ | Hexicantitruncated 7-simplex (pugro) | (0,1,1,1,1,2,3,4) | | | | | | 8400 | 1680 |
| 50 | | t$_{0,1,3,6}$ | Hexiruncitruncated 7-simplex (pugato) | (0,1,1,1,2,2,3,4) | | | | | | 20160 | 3360 |
| 51 | | t$_{0,2,3,6}$ | Hexiruncicantellated 7-simplex (pugro) | (0,1,1,1,2,3,3,4) | | | | | | 16800 | 3360 |
| 52 | | t$_{0,1,4,6}$ | Hexisteritruncated 7-simplex (pucto) | (0,1,1,2,2,2,3,4) | | | | | | 20160 | 3360 |
| 53 | | t$_{0,2,4,6}$ | Hexistericantellated 7-simplex (pucroh) | (0,1,1,2,2,3,3,4) | | | | | | 30240 | 5040 |
| 54 | | t$_{0,1,5,6}$ | Hexipentitruncated 7-simplex (putath) | (0,1,2,2,2,2,3,4) | | | | | | 8400 | 1680 |
| 55 | | t$_{0,1,2,3,4}$ | Steriruncicantitruncated 7-simplex (gecco) | (0,0,0,1,2,3,4,5) | | | | | | 23520 | 6720 |
| 56 | | t$_{0,1,2,3,5}$ | Pentiruncicantitruncated 7-simplex (tegapo) | (0,0,1,1,2,3,4,5) | | | | | | 45360 | 10080 |
| 57 | | t$_{0,1,2,4,5}$ | Pentistericantitruncated 7-simplex (tecagro) | (0,0,1,2,2,3,4,5) | | | | | | 40320 | 10080 |
| 58 | | t$_{0,1,3,4,5}$ | Pentisteriruncitruncated 7-simplex (tacpeto) | (0,0,1,2,3,3,4,5) | | | | | | 40320 | 10080 |
| 59 | | t$_{0,2,3,4,5}$ | Pentisteriruncicantellated 7-simplex (tacpro) | (0,0,1,2,3,4,4,5) | | | | | | 40320 | 10080 |
| 60 | | t$_{1,2,3,4,5}$ | Bisteriruncicantitruncated 7-simplex (gabach) | (0,0,1,2,3,4,5,5) | | | | | | 35280 | 10080 |
| 61 | | t$_{0,1,2,3,6}$ | Hexiruncicantitruncated 7-simplex (pugopo) | (0,1,1,1,2,3,4,5) | | | | | | 30240 | 6720 |
| 62 | | t$_{0,1,2,4,6}$ | Hexistericantitruncated 7-simplex (pucagro) | (0,1,1,2,2,3,4,5) | | | | | | 50400 | 10080 |
| 63 | | t$_{0,1,3,4,6}$ | Hexisteriruncitruncated 7-simplex (pucpato) | (0,1,1,2,3,3,4,5) | | | | | | 45360 | 10080 |
| 64 | | t$_{0,2,3,4,6}$ | Hexisteriruncicantellated 7-simplex (pucproh) | (0,1,1,2,3,4,4,5) | | | | | | 45360 | 10080 |
| 65 | | t$_{0,1,2,5,6}$ | Hexipenticantitruncated 7-simplex (putagro) | (0,1,2,2,2,3,4,5) | | | | | | 30240 | 6720 |
| 66 | | t$_{0,1,3,5,6}$ | Hexipentiruncitruncated 7-simplex (putpath) | (0,1,2,2,3,3,4,5) | | | | | | 50400 | 10080 |
| 67 | | t$_{0,1,2,3,4,5}$ | Pentisteriruncicantitruncated 7-simplex (geto) | (0,0,1,2,3,4,5,6) | | | | | | 70560 | 20160 |
| 68 | | t$_{0,1,2,3,4,6}$ | Hexisteriruncicantitruncated 7-simplex (pugaco) | (0,1,1,2,3,4,5,6) | | | | | | 80640 | 20160 |
| 69 | | t$_{0,1,2,3,5,6}$ | Hexipentiruncicantitruncated 7-simplex (putgapo) | (0,1,2,2,3,4,5,6) | | | | | | 80640 | 20160 |
| 70 | | t$_{0,1,2,4,5,6}$ | Hexipentistericantitruncated 7-simplex (putcagroh) | (0,1,2,3,3,4,5,6) | | | | | | 80640 | 20160 |
| 71 | | t$_{0,1,2,3,4,5,6}$ | Omnitruncated 7-simplex (guph) | (0,1,2,3,4,5,6,7) | | | | | | 141120 | 40320 |

## The B$_7$ family

The B$_7$ family has symmetry of order 645120 (7 factorial x 2$^7$).
There are 127 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Johnson and Bowers names.
See also a list of B7 polytopes for symmetric Coxeter plane graphs of these polytopes.
B$_7$ uniform polytopes :
| # | Coxeter-Dynkin diagram t-notation | Name (BSA) | Base point | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts |
| # | Coxeter-Dynkin diagram t-notation | Name (BSA) | Base point | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | t$_0${3,3,3,3,3,4} | 7-orthoplex (zee) | (0,0,0,0,0,0,1)√2 | 128 | 448 | 672 | 560 | 280 | 84 | 14 |
| 2 | t$_1${3,3,3,3,3,4} | Rectified 7-orthoplex (rez) | (0,0,0,0,0,1,1)√2 | 142 | 1344 | 3360 | 3920 | 2520 | 840 | 84 |
| 3 | t$_2${3,3,3,3,3,4} | Birectified 7-orthoplex (barz) | (0,0,0,0,1,1,1)√2 | 142 | 1428 | 6048 | 10640 | 8960 | 3360 | 280 |
| 4 | t$_3${4,3,3,3,3,3} | Trirectified 7-cube (sez) | (0,0,0,1,1,1,1)√2 | 142 | 1428 | 6328 | 14560 | 15680 | 6720 | 560 |
| 5 | t$_2${4,3,3,3,3,3} | Birectified 7-cube (bersa) | (0,0,1,1,1,1,1)√2 | 142 | 1428 | 5656 | 11760 | 13440 | 6720 | 672 |
| 6 | t$_1${4,3,3,3,3,3} | Rectified 7-cube (rasa) | (0,1,1,1,1,1,1)√2 | 142 | 980 | 2968 | 5040 | 5152 | 2688 | 448 |
| 7 | t$_0${4,3,3,3,3,3} | 7-cube (hept) | (0,0,0,0,0,0,0)√2 + (1,1,1,1,1,1,1) | 14 | 84 | 280 | 560 | 672 | 448 | 128 |
| 8 | t$_{0,1}${3,3,3,3,3,4} | Truncated 7-orthoplex (Taz) | (0,0,0,0,0,1,2)√2 | 142 | 1344 | 3360 | 4760 | 2520 | 924 | 168 |
| 9 | t$_{0,2}${3,3,3,3,3,4} | Cantellated 7-orthoplex (Sarz) | (0,0,0,0,1,1,2)√2 | 226 | 4200 | 15456 | 24080 | 19320 | 7560 | 840 |
| 10 | t$_{1,2}${3,3,3,3,3,4} | Bitruncated 7-orthoplex (Botaz) | (0,0,0,0,1,2,2)√2 | | | | | | 4200 | 840 |
| 11 | t$_{0,3}${3,3,3,3,3,4} | Runcinated 7-orthoplex (Spaz) | (0,0,0,1,1,1,2)√2 | | | | | | 23520 | 2240 |
| 12 | t$_{1,3}${3,3,3,3,3,4} | Bicantellated 7-orthoplex (Sebraz) | (0,0,0,1,1,2,2)√2 | | | | | | 26880 | 3360 |
| 13 | t$_{2,3}${3,3,3,3,3,4} | Tritruncated 7-orthoplex (Totaz) | (0,0,0,1,2,2,2)√2 | | | | | | 10080 | 2240 |
| 14 | t$_{0,4}${3,3,3,3,3,4} | Stericated 7-orthoplex (Scaz) | (0,0,1,1,1,1,2)√2 | | | | | | 33600 | 3360 |
| 15 | t$_{1,4}${3,3,3,3,3,4} | Biruncinated 7-orthoplex (Sibpaz) | (0,0,1,1,1,2,2)√2 | | | | | | 60480 | 6720 |
| 16 | t$_{2,4}${4,3,3,3,3,3} | Tricantellated 7-cube (Strasaz) | (0,0,1,1,2,2,2)√2 | | | | | | 47040 | 6720 |
| 17 | t$_{2,3}${4,3,3,3,3,3} | Tritruncated 7-cube (Tatsa) | (0,0,1,2,2,2,2)√2 | | | | | | 13440 | 3360 |
| 18 | t$_{0,5}${3,3,3,3,3,4} | Pentellated 7-orthoplex (Staz) | (0,1,1,1,1,1,2)√2 | | | | | | 20160 | 2688 |
| 19 | t$_{1,5}${4,3,3,3,3,3} | Bistericated 7-cube (Sabcosaz) | (0,1,1,1,1,2,2)√2 | | | | | | 53760 | 6720 |
| 20 | t$_{1,4}${4,3,3,3,3,3} | Biruncinated 7-cube (Sibposa) | (0,1,1,1,2,2,2)√2 | | | | | | 67200 | 8960 |
| 21 | t$_{1,3}${4,3,3,3,3,3} | Bicantellated 7-cube (Sibrosa) | (0,1,1,2,2,2,2)√2 | | | | | | 40320 | 6720 |
| 22 | t$_{1,2}${4,3,3,3,3,3} | Bitruncated 7-cube (Betsa) | (0,1,2,2,2,2,2)√2 | | | | | | 9408 | 2688 |
| 23 | t$_{0,6}${4,3,3,3,3,3} | Hexicated 7-cube (Supposaz) | (0,0,0,0,0,0,1)√2 + (1,1,1,1,1,1,1) | | | | | | 5376 | 896 |
| 24 | t$_{0,5}${4,3,3,3,3,3} | Pentellated 7-cube (Stesa) | (0,0,0,0,0,1,1)√2 + (1,1,1,1,1,1,1) | | | | | | 20160 | 2688 |
| 25 | t$_{0,4}${4,3,3,3,3,3} | Stericated 7-cube (Scosa) | (0,0,0,0,1,1,1)√2 + (1,1,1,1,1,1,1) | | | | | | 35840 | 4480 |
| 26 | t$_{0,3}${4,3,3,3,3,3} | Runcinated 7-cube (Spesa) | (0,0,0,1,1,1,1)√2 + (1,1,1,1,1,1,1) | | | | | | 33600 | 4480 |
| 27 | t$_{0,2}${4,3,3,3,3,3} | Cantellated 7-cube (Sersa) | (0,0,1,1,1,1,1)√2 + (1,1,1,1,1,1,1) | | | | | | 16128 | 2688 |
| 28 | t$_{0,1}${4,3,3,3,3,3} | Truncated 7-cube (Tasa) | (0,1,1,1,1,1,1)√2 + (1,1,1,1,1,1,1) | 142 | 980 | 2968 | 5040 | 5152 | 3136 | 896 |
| 29 | t$_{0,1,2}${3,3,3,3,3,4} | Cantitruncated 7-orthoplex (Garz) | (0,1,2,3,3,3,3)√2 | | | | | | 8400 | 1680 |
| 30 | t$_{0,1,3}${3,3,3,3,3,4} | Runcitruncated 7-orthoplex (Potaz) | (0,1,2,2,3,3,3)√2 | | | | | | 50400 | 6720 |
| 31 | t$_{0,2,3}${3,3,3,3,3,4} | Runcicantellated 7-orthoplex (Parz) | (0,1,1,2,3,3,3)√2 | | | | | | 33600 | 6720 |
| 32 | t$_{1,2,3}${3,3,3,3,3,4} | Bicantitruncated 7-orthoplex (Gebraz) | (0,0,1,2,3,3,3)√2 | | | | | | 30240 | 6720 |
| 33 | t$_{0,1,4}${3,3,3,3,3,4} | Steritruncated 7-orthoplex (Catz) | (0,0,1,1,1,2,3)√2 | | | | | | 107520 | 13440 |
| 34 | t$_{0,2,4}${3,3,3,3,3,4} | Stericantellated 7-orthoplex (Craze) | (0,0,1,1,2,2,3)√2 | | | | | | 141120 | 20160 |
| 35 | t$_{1,2,4}${3,3,3,3,3,4} | Biruncitruncated 7-orthoplex (Baptize) | (0,0,1,1,2,3,3)√2 | | | | | | 120960 | 20160 |
| 36 | t$_{0,3,4}${3,3,3,3,3,4} | Steriruncinated 7-orthoplex (Copaz) | (0,1,1,1,2,3,3)√2 | | | | | | 67200 | 13440 |
| 37 | t$_{1,3,4}${3,3,3,3,3,4} | Biruncicantellated 7-orthoplex (Boparz) | (0,0,1,2,2,3,3)√2 | | | | | | 100800 | 20160 |
| 38 | t$_{2,3,4}${4,3,3,3,3,3} | Tricantitruncated 7-cube (Gotrasaz) | (0,0,0,1,2,3,3)√2 | | | | | | 53760 | 13440 |
| 39 | t$_{0,1,5}${3,3,3,3,3,4} | Pentitruncated 7-orthoplex (Tetaz) | (0,1,1,1,1,2,3)√2 | | | | | | 87360 | 13440 |
| 40 | t$_{0,2,5}${3,3,3,3,3,4} | Penticantellated 7-orthoplex (Teroz) | (0,1,1,1,2,2,3)√2 | | | | | | 188160 | 26880 |
| 41 | t$_{1,2,5}${3,3,3,3,3,4} | Bisteritruncated 7-orthoplex (Boctaz) | (0,1,1,1,2,3,3)√2 | | | | | | 147840 | 26880 |
| 42 | t$_{0,3,5}${3,3,3,3,3,4} | Pentiruncinated 7-orthoplex (Topaz) | (0,1,1,2,2,2,3)√2 | | | | | | 174720 | 26880 |
| 43 | t$_{1,3,5}${4,3,3,3,3,3} | Bistericantellated 7-cube (Bacresaz) | (0,1,1,2,2,3,3)√2 | | | | | | 241920 | 40320 |
| 44 | t$_{1,3,4}${4,3,3,3,3,3} | Biruncicantellated 7-cube (Bopresa) | (0,1,1,2,3,3,3)√2 | | | | | | 120960 | 26880 |
| 45 | t$_{0,4,5}${3,3,3,3,3,4} | Pentistericated 7-orthoplex (Tocaz) | (0,1,2,2,2,2,3)√2 | | | | | | 67200 | 13440 |
| 46 | t$_{1,2,5}${4,3,3,3,3,3} | Bisteritruncated 7-cube (Bactasa) | (0,1,2,2,2,3,3)√2 | | | | | | 147840 | 26880 |
| 47 | t$_{1,2,4}${4,3,3,3,3,3} | Biruncitruncated 7-cube (Biptesa) | (0,1,2,2,3,3,3)√2 | | | | | | 134400 | 26880 |
| 48 | t$_{1,2,3}${4,3,3,3,3,3} | Bicantitruncated 7-cube (Gibrosa) | (0,1,2,3,3,3,3)√2 | | | | | | 47040 | 13440 |
| 49 | t$_{0,1,6}${3,3,3,3,3,4} | Hexitruncated 7-orthoplex (Putaz) | (0,0,0,0,0,1,2)√2 + (1,1,1,1,1,1,1) | | | | | | 29568 | 5376 |
| 50 | t$_{0,2,6}${3,3,3,3,3,4} | Hexicantellated 7-orthoplex (Puraz) | (0,0,0,0,1,1,2)√2 + (1,1,1,1,1,1,1) | | | | | | 94080 | 13440 |
| 51 | t$_{0,4,5}${4,3,3,3,3,3} | Pentistericated 7-cube (Tacosa) | (0,0,0,0,1,2,2)√2 + (1,1,1,1,1,1,1) | | | | | | 67200 | 13440 |
| 52 | t$_{0,3,6}${4,3,3,3,3,3} | Hexiruncinated 7-cube (Pupsez) | (0,0,0,1,1,1,2)√2 + (1,1,1,1,1,1,1) | | | | | | 134400 | 17920 |
| 53 | t$_{0,3,5}${4,3,3,3,3,3} | Pentiruncinated 7-cube (Tapsa) | (0,0,0,1,1,2,2)√2 + (1,1,1,1,1,1,1) | | | | | | 174720 | 26880 |
| 54 | t$_{0,3,4}${4,3,3,3,3,3} | Steriruncinated 7-cube (Capsa) | (0,0,0,1,2,2,2)√2 + (1,1,1,1,1,1,1) | | | | | | 80640 | 17920 |
| 55 | t$_{0,2,6}${4,3,3,3,3,3} | Hexicantellated 7-cube (Purosa) | (0,0,1,1,1,1,2)√2 + (1,1,1,1,1,1,1) | | | | | | 94080 | 13440 |
| 56 | t$_{0,2,5}${4,3,3,3,3,3} | Penticantellated 7-cube (Tersa) | (0,0,1,1,1,2,2)√2 + (1,1,1,1,1,1,1) | | | | | | 188160 | 26880 |
| 57 | t$_{0,2,4}${4,3,3,3,3,3} | Stericantellated 7-cube (Carsa) | (0,0,1,1,2,2,2)√2 + (1,1,1,1,1,1,1) | | | | | | 161280 | 26880 |
| 58 | t$_{0,2,3}${4,3,3,3,3,3} | Runcicantellated 7-cube (Parsa) | (0,0,1,2,2,2,2)√2 + (1,1,1,1,1,1,1) | | | | | | 53760 | 13440 |
| 59 | t$_{0,1,6}${4,3,3,3,3,3} | Hexitruncated 7-cube (Putsa) | (0,1,1,1,1,1,2)√2 + (1,1,1,1,1,1,1) | | | | | | 29568 | 5376 |
| 60 | t$_{0,1,5}${4,3,3,3,3,3} | Pentitruncated 7-cube (Tetsa) | (0,1,1,1,1,2,2)√2 + (1,1,1,1,1,1,1) | | | | | | 87360 | 13440 |
| 61 | t$_{0,1,4}${4,3,3,3,3,3} | Steritruncated 7-cube (Catsa) | (0,1,1,1,2,2,2)√2 + (1,1,1,1,1,1,1) | | | | | | 116480 | 17920 |
| 62 | t$_{0,1,3}${4,3,3,3,3,3} | Runcitruncated 7-cube (Petsa) | (0,1,1,2,2,2,2)√2 + (1,1,1,1,1,1,1) | | | | | | 73920 | 13440 |
| 63 | t$_{0,1,2}${4,3,3,3,3,3} | Cantitruncated 7-cube (Gersa) | (0,1,2,2,2,2,2)√2 + (1,1,1,1,1,1,1) | | | | | | 18816 | 5376 |
| 64 | t$_{0,1,2,3}${3,3,3,3,3,4} | Runcicantitruncated 7-orthoplex (Gopaz) | (0,1,2,3,4,4,4)√2 | | | | | | 60480 | 13440 |
| 65 | t$_{0,1,2,4}${3,3,3,3,3,4} | Stericantitruncated 7-orthoplex (Cogarz) | (0,0,1,1,2,3,4)√2 | | | | | | 241920 | 40320 |
| 66 | t$_{0,1,3,4}${3,3,3,3,3,4} | Steriruncitruncated 7-orthoplex (Captaz) | (0,0,1,2,2,3,4)√2 | | | | | | 181440 | 40320 |
| 67 | t$_{0,2,3,4}${3,3,3,3,3,4} | Steriruncicantellated 7-orthoplex (Caparz) | (0,0,1,2,3,3,4)√2 | | | | | | 181440 | 40320 |
| 68 | t$_{1,2,3,4}${3,3,3,3,3,4} | Biruncicantitruncated 7-orthoplex (Gibpaz) | (0,0,1,2,3,4,4)√2 | | | | | | 161280 | 40320 |
| 69 | t$_{0,1,2,5}${3,3,3,3,3,4} | Penticantitruncated 7-orthoplex (Tograz) | (0,1,1,1,2,3,4)√2 | | | | | | 295680 | 53760 |
| 70 | t$_{0,1,3,5}${3,3,3,3,3,4} | Pentiruncitruncated 7-orthoplex (Toptaz) | (0,1,1,2,2,3,4)√2 | | | | | | 443520 | 80640 |
| 71 | t$_{0,2,3,5}${3,3,3,3,3,4} | Pentiruncicantellated 7-orthoplex (Toparz) | (0,1,1,2,3,3,4)√2 | | | | | | 403200 | 80640 |
| 72 | t$_{1,2,3,5}${3,3,3,3,3,4} | Bistericantitruncated 7-orthoplex (Becogarz) | (0,1,1,2,3,4,4)√2 | | | | | | 362880 | 80640 |
| 73 | t$_{0,1,4,5}${3,3,3,3,3,4} | Pentisteritruncated 7-orthoplex (Tacotaz) | (0,1,2,2,2,3,4)√2 | | | | | | 241920 | 53760 |
| 74 | t$_{0,2,4,5}${3,3,3,3,3,4} | Pentistericantellated 7-orthoplex (Tocarz) | (0,1,2,2,3,3,4)√2 | | | | | | 403200 | 80640 |
| 75 | t$_{1,2,4,5}${4,3,3,3,3,3} | Bisteriruncitruncated 7-cube (Bocaptosaz) | (0,1,2,2,3,4,4)√2 | | | | | | 322560 | 80640 |
| 76 | t$_{0,3,4,5}${3,3,3,3,3,4} | Pentisteriruncinated 7-orthoplex (Tecpaz) | (0,1,2,3,3,3,4)√2 | | | | | | 241920 | 53760 |
| 77 | t$_{1,2,3,5}${4,3,3,3,3,3} | Bistericantitruncated 7-cube (Becgresa) | (0,1,2,3,3,4,4)√2 | | | | | | 362880 | 80640 |
| 78 | t$_{1,2,3,4}${4,3,3,3,3,3} | Biruncicantitruncated 7-cube (Gibposa) | (0,1,2,3,4,4,4)√2 | | | | | | 188160 | 53760 |
| 79 | t$_{0,1,2,6}${3,3,3,3,3,4} | Hexicantitruncated 7-orthoplex (Pugarez) | (0,0,0,0,1,2,3)√2 + (1,1,1,1,1,1,1) | | | | | | 134400 | 26880 |
| 80 | t$_{0,1,3,6}${3,3,3,3,3,4} | Hexiruncitruncated 7-orthoplex (Papataz) | (0,0,0,1,1,2,3)√2 + (1,1,1,1,1,1,1) | | | | | | 322560 | 53760 |
| 81 | t$_{0,2,3,6}${3,3,3,3,3,4} | Hexiruncicantellated 7-orthoplex (Puparez) | (0,0,0,1,2,2,3)√2 + (1,1,1,1,1,1,1) | | | | | | 268800 | 53760 |
| 82 | t$_{0,3,4,5}${4,3,3,3,3,3} | Pentisteriruncinated 7-cube (Tecpasa) | (0,0,0,1,2,3,3)√2 + (1,1,1,1,1,1,1) | | | | | | 241920 | 53760 |
| 83 | t$_{0,1,4,6}${3,3,3,3,3,4} | Hexisteritruncated 7-orthoplex (Pucotaz) | (0,0,1,1,1,2,3)√2 + (1,1,1,1,1,1,1) | | | | | | 322560 | 53760 |
| 84 | t$_{0,2,4,6}${4,3,3,3,3,3} | Hexistericantellated 7-cube (Pucrosaz) | (0,0,1,1,2,2,3)√2 + (1,1,1,1,1,1,1) | | | | | | 483840 | 80640 |
| 85 | t$_{0,2,4,5}${4,3,3,3,3,3} | Pentistericantellated 7-cube (Tecresa) | (0,0,1,1,2,3,3)√2 + (1,1,1,1,1,1,1) | | | | | | 403200 | 80640 |
| 86 | t$_{0,2,3,6}${4,3,3,3,3,3} | Hexiruncicantellated 7-cube (Pupresa) | (0,0,1,2,2,2,3)√2 + (1,1,1,1,1,1,1) | | | | | | 268800 | 53760 |
| 87 | t$_{0,2,3,5}${4,3,3,3,3,3} | Pentiruncicantellated 7-cube (Topresa) | (0,0,1,2,2,3,3)√2 + (1,1,1,1,1,1,1) | | | | | | 403200 | 80640 |
| 88 | t$_{0,2,3,4}${4,3,3,3,3,3} | Steriruncicantellated 7-cube (Copresa) | (0,0,1,2,3,3,3)√2 + (1,1,1,1,1,1,1) | | | | | | 215040 | 53760 |
| 89 | t$_{0,1,5,6}${4,3,3,3,3,3} | Hexipentitruncated 7-cube (Putatosez) | (0,1,1,1,1,2,3)√2 + (1,1,1,1,1,1,1) | | | | | | 134400 | 26880 |
| 90 | t$_{0,1,4,6}${4,3,3,3,3,3} | Hexisteritruncated 7-cube (Pacutsa) | (0,1,1,1,2,2,3)√2 + (1,1,1,1,1,1,1) | | | | | | 322560 | 53760 |
| 91 | t$_{0,1,4,5}${4,3,3,3,3,3} | Pentisteritruncated 7-cube (Tecatsa) | (0,1,1,1,2,3,3)√2 + (1,1,1,1,1,1,1) | | | | | | 241920 | 53760 |
| 92 | t$_{0,1,3,6}${4,3,3,3,3,3} | Hexiruncitruncated 7-cube (Pupetsa) | (0,1,1,2,2,2,3)√2 + (1,1,1,1,1,1,1) | | | | | | 322560 | 53760 |
| 93 | t$_{0,1,3,5}${4,3,3,3,3,3} | Pentiruncitruncated 7-cube (Toptosa) | (0,1,1,2,2,3,3)√2 + (1,1,1,1,1,1,1) | | | | | | 443520 | 80640 |
| 94 | t$_{0,1,3,4}${4,3,3,3,3,3} | Steriruncitruncated 7-cube (Captesa) | (0,1,1,2,3,3,3)√2 + (1,1,1,1,1,1,1) | | | | | | 215040 | 53760 |
| 95 | t$_{0,1,2,6}${4,3,3,3,3,3} | Hexicantitruncated 7-cube (Pugrosa) | (0,1,2,2,2,2,3)√2 + (1,1,1,1,1,1,1) | | | | | | 134400 | 26880 |
| 96 | t$_{0,1,2,5}${4,3,3,3,3,3} | Penticantitruncated 7-cube (Togresa) | (0,1,2,2,2,3,3)√2 + (1,1,1,1,1,1,1) | | | | | | 295680 | 53760 |
| 97 | t$_{0,1,2,4}${4,3,3,3,3,3} | Stericantitruncated 7-cube (Cogarsa) | (0,1,2,2,3,3,3)√2 + (1,1,1,1,1,1,1) | | | | | | 268800 | 53760 |
| 98 | t$_{0,1,2,3}${4,3,3,3,3,3} | Runcicantitruncated 7-cube (Gapsa) | (0,1,2,3,3,3,3)√2 + (1,1,1,1,1,1,1) | | | | | | 94080 | 26880 |
| 99 | t$_{0,1,2,3,4}${3,3,3,3,3,4} | Steriruncicantitruncated 7-orthoplex (Gocaz) | (0,0,1,2,3,4,5)√2 | | | | | | 322560 | 80640 |
| 100 | t$_{0,1,2,3,5}${3,3,3,3,3,4} | Pentiruncicantitruncated 7-orthoplex (Tegopaz) | (0,1,1,2,3,4,5)√2 | | | | | | 725760 | 161280 |
| 101 | t$_{0,1,2,4,5}${3,3,3,3,3,4} | Pentistericantitruncated 7-orthoplex (Tecagraz) | (0,1,2,2,3,4,5)√2 | | | | | | 645120 | 161280 |
| 102 | t$_{0,1,3,4,5}${3,3,3,3,3,4} | Pentisteriruncitruncated 7-orthoplex (Tecpotaz) | (0,1,2,3,3,4,5)√2 | | | | | | 645120 | 161280 |
| 103 | t$_{0,2,3,4,5}${3,3,3,3,3,4} | Pentisteriruncicantellated 7-orthoplex (Tacparez) | (0,1,2,3,4,4,5)√2 | | | | | | 645120 | 161280 |
| 104 | t$_{1,2,3,4,5}${4,3,3,3,3,3} | Bisteriruncicantitruncated 7-cube (Gabcosaz) | (0,1,2,3,4,5,5)√2 | | | | | | 564480 | 161280 |
| 105 | t$_{0,1,2,3,6}${3,3,3,3,3,4} | Hexiruncicantitruncated 7-orthoplex (Pugopaz) | (0,0,0,1,2,3,4)√2 + (1,1,1,1,1,1,1) | | | | | | 483840 | 107520 |
| 106 | t$_{0,1,2,4,6}${3,3,3,3,3,4} | Hexistericantitruncated 7-orthoplex (Pucagraz) | (0,0,1,1,2,3,4)√2 + (1,1,1,1,1,1,1) | | | | | | 806400 | 161280 |
| 107 | t$_{0,1,3,4,6}${3,3,3,3,3,4} | Hexisteriruncitruncated 7-orthoplex (Pucpotaz) | (0,0,1,2,2,3,4)√2 + (1,1,1,1,1,1,1) | | | | | | 725760 | 161280 |
| 108 | t$_{0,2,3,4,6}${4,3,3,3,3,3} | Hexisteriruncicantellated 7-cube (Pucprosaz) | (0,0,1,2,3,3,4)√2 + (1,1,1,1,1,1,1) | | | | | | 725760 | 161280 |
| 109 | t$_{0,2,3,4,5}${4,3,3,3,3,3} | Pentisteriruncicantellated 7-cube (Tocpresa) | (0,0,1,2,3,4,4)√2 + (1,1,1,1,1,1,1) | | | | | | 645120 | 161280 |
| 110 | t$_{0,1,2,5,6}${3,3,3,3,3,4} | Hexipenticantitruncated 7-orthoplex (Putegraz) | (0,1,1,1,2,3,4)√2 + (1,1,1,1,1,1,1) | | | | | | 483840 | 107520 |
| 111 | t$_{0,1,3,5,6}${4,3,3,3,3,3} | Hexipentiruncitruncated 7-cube (Putpetsaz) | (0,1,1,2,2,3,4)√2 + (1,1,1,1,1,1,1) | | | | | | 806400 | 161280 |
| 112 | t$_{0,1,3,4,6}${4,3,3,3,3,3} | Hexisteriruncitruncated 7-cube (Pucpetsa) | (0,1,1,2,3,3,4)√2 + (1,1,1,1,1,1,1) | | | | | | 725760 | 161280 |
| 113 | t$_{0,1,3,4,5}${4,3,3,3,3,3} | Pentisteriruncitruncated 7-cube (Tecpetsa) | (0,1,1,2,3,4,4)√2 + (1,1,1,1,1,1,1) | | | | | | 645120 | 161280 |
| 114 | t$_{0,1,2,5,6}${4,3,3,3,3,3} | Hexipenticantitruncated 7-cube (Putgresa) | (0,1,2,2,2,3,4)√2 + (1,1,1,1,1,1,1) | | | | | | 483840 | 107520 |
| 115 | t$_{0,1,2,4,6}${4,3,3,3,3,3} | Hexistericantitruncated 7-cube (Pucagrosa) | (0,1,2,2,3,3,4)√2 + (1,1,1,1,1,1,1) | | | | | | 806400 | 161280 |
| 116 | t$_{0,1,2,4,5}${4,3,3,3,3,3} | Pentistericantitruncated 7-cube (Tecgresa) | (0,1,2,2,3,4,4)√2 + (1,1,1,1,1,1,1) | | | | | | 645120 | 161280 |
| 117 | t$_{0,1,2,3,6}${4,3,3,3,3,3} | Hexiruncicantitruncated 7-cube (Pugopsa) | (0,1,2,3,3,3,4)√2 + (1,1,1,1,1,1,1) | | | | | | 483840 | 107520 |
| 118 | t$_{0,1,2,3,5}${4,3,3,3,3,3} | Pentiruncicantitruncated 7-cube (Togapsa) | (0,1,2,3,3,4,4)√2 + (1,1,1,1,1,1,1) | | | | | | 725760 | 161280 |
| 119 | t$_{0,1,2,3,4}${4,3,3,3,3,3} | Steriruncicantitruncated 7-cube (Gacosa) | (0,1,2,3,4,4,4)√2 + (1,1,1,1,1,1,1) | | | | | | 376320 | 107520 |
| 120 | t$_{0,1,2,3,4,5}${3,3,3,3,3,4} | Pentisteriruncicantitruncated 7-orthoplex (Gotaz) | (0,1,2,3,4,5,6)√2 | | | | | | 1128960 | 322560 |
| 121 | t$_{0,1,2,3,4,6}${3,3,3,3,3,4} | Hexisteriruncicantitruncated 7-orthoplex (Pugacaz) | (0,0,1,2,3,4,5)√2 + (1,1,1,1,1,1,1) | | | | | | 1290240 | 322560 |
| 122 | t$_{0,1,2,3,5,6}${3,3,3,3,3,4} | Hexipentiruncicantitruncated 7-orthoplex (Putgapaz) | (0,1,1,2,3,4,5)√2 + (1,1,1,1,1,1,1) | | | | | | 1290240 | 322560 |
| 123 | t$_{0,1,2,4,5,6}${4,3,3,3,3,3} | Hexipentistericantitruncated 7-cube (Putcagrasaz) | (0,1,2,2,3,4,5)√2 + (1,1,1,1,1,1,1) | | | | | | 1290240 | 322560 |
| 124 | t$_{0,1,2,3,5,6}${4,3,3,3,3,3} | Hexipentiruncicantitruncated 7-cube (Putgapsa) | (0,1,2,3,3,4,5)√2 + (1,1,1,1,1,1,1) | | | | | | 1290240 | 322560 |
| 125 | t$_{0,1,2,3,4,6}${4,3,3,3,3,3} | Hexisteriruncicantitruncated 7-cube (Pugacasa) | (0,1,2,3,4,4,5)√2 + (1,1,1,1,1,1,1) | | | | | | 1290240 | 322560 |
| 126 | t$_{0,1,2,3,4,5}${4,3,3,3,3,3} | Pentisteriruncicantitruncated 7-cube (Gotesa) | (0,1,2,3,4,5,5)√2 + (1,1,1,1,1,1,1) | | | | | | 1128960 | 322560 |
| 127 | t$_{0,1,2,3,4,5,6}${4,3,3,3,3,3} | Omnitruncated 7-cube (Guposaz) | (0,1,2,3,4,5,6)√2 + (1,1,1,1,1,1,1) | | | | | | 2257920 | 645120 |

## The D$_7$ family

The D$_7$ family has symmetry of order 322560 (7 factorial x 2$^6$).
This family has 3×32−1=95 Wythoffian uniform polytopes, generated by marking one or more nodes of the D$_7$ Coxeter-Dynkin diagram. Of these, 63 (2×32−1) are repeated from the B$_7$ family and 32 are unique to this family, listed below. Bowers names and acronym are given for cross-referencing.
See also list of D7 polytopes for Coxeter plane graphs of these polytopes.
D$_7$ uniform polytopes :
| # | Coxeter diagram | Names | Base point (Alternately signed) | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts |
| # | Coxeter diagram | Names | Base point (Alternately signed) | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | = | 7-cube demihepteract (hesa) | (1,1,1,1,1,1,1) | 78 | 532 | 1624 | 2800 | 2240 | 672 | 64 |
| 2 | = | cantic 7-cube truncated demihepteract (thesa) | (1,1,3,3,3,3,3) | 142 | 1428 | 5656 | 11760 | 13440 | 7392 | 1344 |
| 3 | = | runcic 7-cube small rhombated demihepteract (sirhesa) | (1,1,1,3,3,3,3) | | | | | | 16800 | 2240 |
| 4 | = | steric 7-cube small prismated demihepteract (sphosa) | (1,1,1,1,3,3,3) | | | | | | 20160 | 2240 |
| 5 | = | pentic 7-cube small cellated demihepteract (sochesa) | (1,1,1,1,1,3,3) | | | | | | 13440 | 1344 |
| 6 | = | hexic 7-cube small terated demihepteract (suthesa) | (1,1,1,1,1,1,3) | | | | | | 4704 | 448 |
| 7 | = | runcicantic 7-cube great rhombated demihepteract (Girhesa) | (1,1,3,5,5,5,5) | | | | | | 23520 | 6720 |
| 8 | = | stericantic 7-cube prismatotruncated demihepteract (pothesa) | (1,1,3,3,5,5,5) | | | | | | 73920 | 13440 |
| 9 | = | steriruncic 7-cube prismatorhomated demihepteract (prohesa) | (1,1,1,3,5,5,5) | | | | | | 40320 | 8960 |
| 10 | = | penticantic 7-cube cellitruncated demihepteract (cothesa) | (1,1,3,3,3,5,5) | | | | | | 87360 | 13440 |
| 11 | = | pentiruncic 7-cube cellirhombated demihepteract (crohesa) | (1,1,1,3,3,5,5) | | | | | | 87360 | 13440 |
| 12 | = | pentisteric 7-cube celliprismated demihepteract (caphesa) | (1,1,1,1,3,5,5) | | | | | | 40320 | 6720 |
| 13 | = | hexicantic 7-cube tericantic demihepteract (tuthesa) | (1,1,3,3,3,3,5) | | | | | | 43680 | 6720 |
| 14 | = | hexiruncic 7-cube terirhombated demihepteract (turhesa) | (1,1,1,3,3,3,5) | | | | | | 67200 | 8960 |
| 15 | = | hexisteric 7-cube teriprismated demihepteract (tuphesa) | (1,1,1,1,3,3,5) | | | | | | 53760 | 6720 |
| 16 | = | hexipentic 7-cube tericellated demihepteract (tuchesa) | (1,1,1,1,1,3,5) | | | | | | 21504 | 2688 |
| 17 | = | steriruncicantic 7-cube great prismated demihepteract (Gephosa) | (1,1,3,5,7,7,7) | | | | | | 94080 | 26880 |
| 18 | = | pentiruncicantic 7-cube celligreatorhombated demihepteract (cagrohesa) | (1,1,3,5,5,7,7) | | | | | | 181440 | 40320 |
| 19 | = | pentistericantic 7-cube celliprismatotruncated demihepteract (capthesa) | (1,1,3,3,5,7,7) | | | | | | 181440 | 40320 |
| 20 | = | pentisteriruncic 7-cube celliprismatorhombated demihepteract (coprahesa) | (1,1,1,3,5,7,7) | | | | | | 120960 | 26880 |
| 21 | = | hexiruncicantic 7-cube terigreatorhombated demihepteract (tugrohesa) | (1,1,3,5,5,5,7) | | | | | | 120960 | 26880 |
| 22 | = | hexistericantic 7-cube teriprismatotruncated demihepteract (tupthesa) | (1,1,3,3,5,5,7) | | | | | | 221760 | 40320 |
| 23 | = | hexisteriruncic 7-cube teriprismatorhombated demihepteract (tuprohesa) | (1,1,1,3,5,5,7) | | | | | | 134400 | 26880 |
| 24 | = | hexipenticantic 7-cube teriCellitruncated demihepteract (tucothesa) | (1,1,3,3,3,5,7) | | | | | | 147840 | 26880 |
| 25 | = | hexipentiruncic 7-cube tericellirhombated demihepteract (tucrohesa) | (1,1,1,3,3,5,7) | | | | | | 161280 | 26880 |
| 26 | = | hexipentisteric 7-cube tericelliprismated demihepteract (tucophesa) | (1,1,1,1,3,5,7) | | | | | | 80640 | 13440 |
| 27 | = | pentisteriruncicantic 7-cube great cellated demihepteract (gochesa) | (1,1,3,5,7,9,9) | | | | | | 282240 | 80640 |
| 28 | = | hexisteriruncicantic 7-cube terigreatoprimated demihepteract (tugphesa) | (1,1,3,5,7,7,9) | | | | | | 322560 | 80640 |
| 29 | = | hexipentiruncicantic 7-cube tericelligreatorhombated demihepteract (tucagrohesa) | (1,1,3,5,5,7,9) | | | | | | 322560 | 80640 |
| 30 | = | hexipentistericantic 7-cube tericelliprismatotruncated demihepteract (tucpathesa) | (1,1,3,3,5,7,9) | | | | | | 362880 | 80640 |
| 31 | = | hexipentisteriruncic 7-cube tericellprismatorhombated demihepteract (tucprohesa) | (1,1,1,3,5,7,9) | | | | | | 241920 | 53760 |
| 32 | = | hexipentisteriruncicantic 7-cube great terated demihepteract (guthesa) | (1,1,3,5,7,9,11) | | | | | | 564480 | 161280 |

## The E$_7$ family

The E$_7$ Coxeter group has order 2,903,040.
There are 127 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.
See also a list of E7 polytopes for symmetric Coxeter plane graphs of these polytopes.
E$_7$ uniform polytopes :
| # | Coxeter-Dynkin diagram Schläfli symbol | Names | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts | Element counts |
| # | Coxeter-Dynkin diagram Schläfli symbol | Names | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
| 1 | | 2$_{31}$ (laq) | 632 | 4788 | 16128 | 20160 | 10080 | 2016 | 126 |
| 2 | | Rectified 2$_{31}$ (rolaq) | 758 | 10332 | 47880 | 100800 | 90720 | 30240 | 2016 |
| 3 | | Rectified 1$_{32}$ (rolin) | 758 | 12348 | 72072 | 191520 | 241920 | 120960 | 10080 |
| 4 | | 1$_{32}$ (lin) | 182 | 4284 | 23688 | 50400 | 40320 | 10080 | 576 |
| 5 | | Birectified 3$_{21}$ (branq) | 758 | 12348 | 68040 | 161280 | 161280 | 60480 | 4032 |
| 6 | | Rectified 3$_{21}$ (ranq) | 758 | 44352 | 70560 | 48384 | 11592 | 12096 | 756 |
| 7 | | 3$_{21}$ (naq) | 702 | 6048 | 12096 | 10080 | 4032 | 756 | 56 |
| 8 | | Truncated 2$_{31}$ (talq) | 758 | 10332 | 47880 | 100800 | 90720 | 32256 | 4032 |
| 9 | | Cantellated 2$_{31}$ (sirlaq) | | | | | | 131040 | 20160 |
| 10 | | Bitruncated 2$_{31}$ (botlaq) | | | | | | | 30240 |
| 11 | | small demified 2$_{31}$ (shilq) | 2774 | 22428 | 78120 | 151200 | 131040 | 42336 | 4032 |
| 12 | | demirectified 2$_{31}$ (hirlaq) | | | | | | | 12096 |
| 13 | | truncated 1$_{32}$ (tolin) | | | | | | | 20160 |
| 14 | | small demiprismated 2$_{31}$ (shiplaq) | | | | | | | 20160 |
| 15 | | birectified 1$_{32}$ (berlin) | 758 | 22428 | 142632 | 403200 | 544320 | 302400 | 40320 |
| 16 | | tritruncated 3$_{21}$ (totanq) | | | | | | | 40320 |
| 17 | | demibirectified 3$_{21}$ (hobranq) | | | | | | | 20160 |
| 18 | | small cellated 2$_{31}$ (scalq) | | | | | | | 7560 |
| 19 | | small biprismated 2$_{31}$ (sobpalq) | | | | | | | 30240 |
| 20 | | small birhombated 3$_{21}$ (sabranq) | | | | | | | 60480 |
| 21 | | demirectified 3$_{21}$ (harnaq) | | | | | | | 12096 |
| 22 | | bitruncated 3$_{21}$ (botnaq) | | | | | | | 12096 |
| 23 | | small terated 3$_{21}$ (stanq) | | | | | | | 1512 |
| 24 | | small demicellated 3$_{21}$ (shocanq) | | | | | | | 12096 |
| 25 | | small prismated 3$_{21}$ (spanq) | | | | | | | 40320 |
| 26 | | small demified 3$_{21}$ (shanq) | | | | | | | 4032 |
| 27 | | small rhombated 3$_{21}$ (sranq) | | | | | | | 12096 |
| 28 | | Truncated 3$_{21}$ (tanq) | 758 | 11592 | 48384 | 70560 | 44352 | 12852 | 1512 |
| 29 | | great rhombated 2$_{31}$ (girlaq) | | | | | | | 60480 |
| 30 | | demitruncated 2$_{31}$ (hotlaq) | | | | | | | 24192 |
| 31 | | small demirhombated 2$_{31}$ (sherlaq) | | | | | | | 60480 |
| 32 | | demibitruncated 2$_{31}$ (hobtalq) | | | | | | | 60480 |
| 33 | | demiprismated 2$_{31}$ (hiptalq) | | | | | | | 80640 |
| 34 | | demiprismatorhombated 2$_{31}$ (hiprolaq) | | | | | | | 120960 |
| 35 | | bitruncated 1$_{32}$ (batlin) | | | | | | | 120960 |
| 36 | | small prismated 2$_{31}$ (spalq) | | | | | | | 80640 |
| 37 | | small rhombated 1$_{32}$ (sirlin) | | | | | | | 120960 |
| 38 | | tritruncated 2$_{31}$ (tatilq) | | | | | | | 80640 |
| 39 | | cellitruncated 2$_{31}$ (catalaq) | | | | | | | 60480 |
| 40 | | cellirhombated 2$_{31}$ (crilq) | | | | | | | 362880 |
| 41 | | biprismatotruncated 2$_{31}$ (biptalq) | | | | | | | 181440 |
| 42 | | small prismated 1$_{32}$ (seplin) | | | | | | | 60480 |
| 43 | | small biprismated 3$_{21}$ (sabipnaq) | | | | | | | 120960 |
| 44 | | small demibirhombated 3$_{21}$ (shobranq) | | | | | | | 120960 |
| 45 | | cellidemiprismated 2$_{31}$ (chaplaq) | | | | | | | 60480 |
| 46 | | demibiprismatotruncated 3$_{21}$ (hobpotanq) | | | | | | | 120960 |
| 47 | | great birhombated 3$_{21}$ (gobranq) | | | | | | | 120960 |
| 48 | | demibitruncated 3$_{21}$ (hobtanq) | | | | | | | 60480 |
| 49 | | teritruncated 2$_{31}$ (totalq) | | | | | | | 24192 |
| 50 | | terirhombated 2$_{31}$ (trilq) | | | | | | | 120960 |
| 51 | | demicelliprismated 3$_{21}$ (hicpanq) | | | | | | | 120960 |
| 52 | | small teridemified 2$_{31}$ (sethalq) | | | | | | | 24192 |
| 53 | | small cellated 3$_{21}$ (scanq) | | | | | | | 60480 |
| 54 | | demiprismated 3$_{21}$ (hipnaq) | | | | | | | 80640 |
| 55 | | terirhombated 3$_{21}$ (tranq) | | | | | | | 60480 |
| 56 | | demicellirhombated 3$_{21}$ (hocranq) | | | | | | | 120960 |
| 57 | | prismatorhombated 3$_{21}$ (pranq) | | | | | | | 120960 |
| 58 | | small demirhombated 3$_{21}$ (sharnaq) | | | | | | | 60480 |
| 59 | | teritruncated 3$_{21}$ (tetanq) | | | | | | | 15120 |
| 60 | | demicellitruncated 3$_{21}$ (hictanq) | | | | | | | 60480 |
| 61 | | prismatotruncated 3$_{21}$ (potanq) | | | | | | | 120960 |
| 62 | | demitruncated 3$_{21}$ (hotnaq) | | | | | | | 24192 |
| 63 | | great rhombated 3$_{21}$ (granq) | | | | | | | 24192 |
| 64 | | great demified 2$_{31}$ (gahlaq) | | | | | | | 120960 |
| 65 | | great demiprismated 2$_{31}$ (gahplaq) | | | | | | | 241920 |
| 66 | | prismatotruncated 2$_{31}$ (potlaq) | | | | | | | 241920 |
| 67 | | prismatorhombated 2$_{31}$ (prolaq) | | | | | | | 241920 |
| 68 | | great rhombated 1$_{32}$ (girlin) | | | | | | | 241920 |
| 69 | | celligreatorhombated 2$_{31}$ (cagrilq) | | | | | | | 362880 |
| 70 | | cellidemitruncated 2$_{31}$ (chotalq) | | | | | | | 241920 |
| 71 | | prismatotruncated 1$_{32}$ (patlin) | | | | | | | 362880 |
| 72 | | biprismatorhombated 3$_{21}$ (bipirnaq) | | | | | | | 362880 |
| 73 | | tritruncated 1$_{32}$ (tatlin) | | | | | | | 241920 |
| 74 | | cellidemiprismatorhombated 2$_{31}$ (chopralq) | | | | | | | 362880 |
| 75 | | great demibiprismated 3$_{21}$ (ghobipnaq) | | | | | | | 362880 |
| 76 | | celliprismated 2$_{31}$ (caplaq) | | | | | | | 241920 |
| 77 | | biprismatotruncated 3$_{21}$ (boptanq) | | | | | | | 362880 |
| 78 | | great trirhombated 2$_{31}$ (gatralaq) | | | | | | | 241920 |
| 79 | | terigreatorhombated 2$_{31}$ (togrilq) | | | | | | | 241920 |
| 80 | | teridemitruncated 2$_{31}$ (thotalq) | | | | | | | 120960 |
| 81 | | teridemirhombated 2$_{31}$ (thorlaq) | | | | | | | 241920 |
| 82 | | celliprismated 3$_{21}$ (capnaq) | | | | | | | 241920 |
| 83 | | teridemiprismatotruncated 2$_{31}$ (thoptalq) | | | | | | | 241920 |
| 84 | | teriprismatorhombated 3$_{21}$ (tapronaq) | | | | | | | 362880 |
| 85 | | demicelliprismatorhombated 3$_{21}$ (hacpranq) | | | | | | | 362880 |
| 86 | | teriprismated 2$_{31}$ (toplaq) | | | | | | | 241920 |
| 87 | | cellirhombated 3$_{21}$ (cranq) | | | | | | | 362880 |
| 88 | | demiprismatorhombated 3$_{21}$ (hapranq) | | | | | | | 241920 |
| 89 | | tericellitruncated 2$_{31}$ (tectalq) | | | | | | | 120960 |
| 90 | | teriprismatotruncated 3$_{21}$ (toptanq) | | | | | | | 362880 |
| 91 | | demicelliprismatotruncated 3$_{21}$ (hecpotanq) | | | | | | | 362880 |
| 92 | | teridemitruncated 3$_{21}$ (thotanq) | | | | | | | 120960 |
| 93 | | cellitruncated 3$_{21}$ (catnaq) | | | | | | | 241920 |
| 94 | | demiprismatotruncated 3$_{21}$ (hiptanq) | | | | | | | 241920 |
| 95 | | terigreatorhombated 3$_{21}$ (tagranq) | | | | | | | 120960 |
| 96 | | demicelligreatorhombated 3$_{21}$ (hicgarnq) | | | | | | | 241920 |
| 97 | | great prismated 3$_{21}$ (gopanq) | | | | | | | 241920 |
| 98 | | great demirhombated 3$_{21}$ (gahranq) | | | | | | | 120960 |
| 99 | | great prismated 2$_{31}$ (gopalq) | | | | | | | 483840 |
| 100 | | great cellidemified 2$_{31}$ (gechalq) | | | | | | | 725760 |
| 101 | | great birhombated 1$_{32}$ (gebrolin) | | | | | | | 725760 |
| 102 | | prismatorhombated 1$_{32}$ (prolin) | | | | | | | 725760 |
| 103 | | celliprismatorhombated 2$_{31}$ (caprolaq) | | | | | | | 725760 |
| 104 | | great biprismated 2$_{31}$ (gobpalq) | | | | | | | 725760 |
| 105 | | tericelliprismated 3$_{21}$ (ticpanq) | | | | | | | 483840 |
| 106 | | teridemigreatoprismated 2$_{31}$ (thegpalq) | | | | | | | 725760 |
| 107 | | teriprismatotruncated 2$_{31}$ (teptalq) | | | | | | | 725760 |
| 108 | | teriprismatorhombated 2$_{31}$ (topralq) | | | | | | | 725760 |
| 109 | | cellipriemsatorhombated 3$_{21}$ (copranq) | | | | | | | 725760 |
| 110 | | tericelligreatorhombated 2$_{31}$ (tecgrolaq) | | | | | | | 725760 |
| 111 | | tericellitruncated 3$_{21}$ (tectanq) | | | | | | | 483840 |
| 112 | | teridemiprismatotruncated 3$_{21}$ (thoptanq) | | | | | | | 725760 |
| 113 | | celliprismatotruncated 3$_{21}$ (coptanq) | | | | | | | 725760 |
| 114 | | teridemicelligreatorhombated 3$_{21}$ (thocgranq) | | | | | | | 483840 |
| 115 | | terigreatoprismated 3$_{21}$ (tagpanq) | | | | | | | 725760 |
| 116 | | great demicellated 3$_{21}$ (gahcnaq) | | | | | | | 725760 |
| 117 | | tericelliprismated laq (tecpalq) | | | | | | | 483840 |
| 118 | | celligreatorhombated 3$_{21}$ (cogranq) | | | | | | | 725760 |
| 119 | | great demified 3$_{21}$ (gahnq) | | | | | | | 483840 |
| 120 | | great cellated 2$_{31}$ (gocalq) | | | | | | | 1451520 |
| 121 | | terigreatoprismated 2$_{31}$ (tegpalq) | | | | | | | 1451520 |
| 122 | | tericelliprismatotruncated 3$_{21}$ (tecpotniq) | | | | | | | 1451520 |
| 123 | | tericellidemigreatoprismated 2$_{31}$ (techogaplaq) | | | | | | | 1451520 |
| 124 | | tericelligreatorhombated 3$_{21}$ (tacgarnq) | | | | | | | 1451520 |
| 125 | | tericelliprismatorhombated 2$_{31}$ (tecprolaq) | | | | | | | 1451520 |
| 126 | | great cellated 3$_{21}$ (gocanq) | | | | | | | 1451520 |
| 127 | | great terated 3$_{21}$ (gotanq) | | | | | | | 2903040 |

## Regular and uniform honeycombs

There are five fundamental affine Coxeter groups and sixteen prismatic groups that generate regular and uniform tessellations in 6-space:
| # | Coxeter group | Coxeter group | Forms |
| 1 | ${\tilde {A}}_{6}$ | [3$^{[7]}$] | 17 |
| 2 | ${\tilde {C}}_{6}$ | [4,3$^4$,4] | 71 |
| 3 | ${\tilde {B}}_{6}$ | h[4,3$^4$,4] [4,3$^3$,3$^{1,1}$] | 95 (32 new) |
| 4 | ${\tilde {D}}_{6}$ | q[4,3$^4$,4] [3$^{1,1}$,3$^2$,3$^{1,1}$] | 41 (6 new) |
| 5 | ${\tilde {E}}_{6}$ | [3$^{2,2,2}$] | 39 |
Regular and uniform tessellations include:
* ${\tilde {A}}_{6}$, 17 forms
** Uniform 6-simplex honeycomb: {3$^{[7]}$}
** Uniform Cyclotruncated 6-simplex honeycomb: t$_{0,1}${3$^{[7]}$}
** Uniform Omnitruncated 6-simplex honeycomb: t$_{0,1,2,3,4,5,6,7}${3$^{[7]}$}
* ${\tilde {C}}_{6}$, [4,3$^4$,4], 71 forms
** Regular 6-cube honeycomb, represented by symbols {4,3$^4$,4},
* ${\tilde {B}}_{6}$, [3$^{1,1}$,3$^3$,4], 95 forms, 64 shared with ${\tilde {C}}_{6}$, 32 new
** Uniform 6-demicube honeycomb, represented by symbols h{4,3$^4$,4} = {3$^{1,1}$,3$^3$,4}, =
* ${\tilde {D}}_{6}$, [3$^{1,1}$,3$^2$,3$^{1,1}$], 41 unique ringed permutations, most shared with ${\tilde {B}}_{6}$ and ${\tilde {C}}_{6}$, and 6 are new. Coxeter calls the first one a quarter 6-cubic honeycomb.
* ${\tilde {E}}_{6}$: [3$^{2,2,2}$], 39 forms
** Uniform 2$_{22}$ honeycomb: represented by symbols {3,3,3$^{2,2}$},
** Uniform t$_4$(2$_{22}$) honeycomb: 4r{3,3,3$^{2,2}$},
** Uniform 0$_{222}$ honeycomb: {3$^{2,2,2}$},
** Uniform t$_2$(0$_{222}$) honeycomb: 2r{3$^{2,2,2}$},
Prismatic groups :
| # | Coxeter group | Coxeter group |
| 1 | ${\tilde {A}}_{5}$x${\tilde {I}}_{1}$ | [3$^{[6]}$,2,∞] |
| 2 | ${\tilde {B}}_{5}$x${\tilde {I}}_{1}$ | [4,3,3$^{1,1}$,2,∞] |
| 3 | ${\tilde {C}}_{5}$x${\tilde {I}}_{1}$ | [4,3$^3$,4,2,∞] |
| 4 | ${\tilde {D}}_{5}$x${\tilde {I}}_{1}$ | [3$^{1,1}$,3,3$^{1,1}$,2,∞] |
| 5 | ${\tilde {A}}_{4}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$ | [3$^{[5]}$,2,∞,2,∞,2,∞] |
| 6 | ${\tilde {B}}_{4}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$ | [4,3,3$^{1,1}$,2,∞,2,∞] |
| 7 | ${\tilde {C}}_{4}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$ | [4,3,3,4,2,∞,2,∞] |
| 8 | ${\tilde {D}}_{4}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$ | [3$^{1,1,1,1}$,2,∞,2,∞] |
| 9 | ${\tilde {F}}_{4}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$ | [3,4,3,3,2,∞,2,∞] |
| 10 | ${\tilde {C}}_{3}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$ | [4,3,4,2,∞,2,∞,2,∞] |
| 11 | ${\tilde {B}}_{3}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$ | [4,3$^{1,1}$,2,∞,2,∞,2,∞] |
| 12 | ${\tilde {A}}_{3}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$ | [3$^{[4]}$,2,∞,2,∞,2,∞] |
| 13 | ${\tilde {C}}_{2}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$ | [4,4,2,∞,2,∞,2,∞,2,∞] |
| 14 | ${\tilde {H}}_{2}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$ | [6,3,2,∞,2,∞,2,∞,2,∞] |
| 15 | ${\tilde {A}}_{2}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$ | [3$^{[3]}$,2,∞,2,∞,2,∞,2,∞] |
| 16 | ${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$x${\tilde {I}}_{1}$ | [∞,2,∞,2,∞,2,∞,2,∞] |

### Regular and uniform hyperbolic honeycombs

There are no compact hyperbolic Coxeter groups of rank 7, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However, there are 3 paracompact hyperbolic Coxeter groups of rank 7, each generating uniform honeycombs in 6-space as permutations of rings of the Coxeter diagrams.
| ${\bar {P}}_{6}$ = [3,3$^{[6]}$]: | ${\bar {Q}}_{6}$ = [3$^{1,1}$,3,3$^{2,1}$]: | ${\bar {S}}_{6}$ = [4,3,3,3$^{2,1}$]: |

## Notes on the Wythoff construction for the uniform 7-polytopes

The reflective 7-dimensional uniform polytopes are constructed through a Wythoff construction process, and represented by a Coxeter-Dynkin diagram, where each node represents a mirror. An active mirror is represented by a ringed node. Each combination of active mirrors generates a unique uniform polytope. Uniform polytopes are named in relation to the regular polytopes in each family. Some families have two regular constructors and thus may be named in two equally valid ways.
Here are the primary operators available for constructing and naming the uniform 7-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.
| Operation | Extended Schläfli symbol | Description |
| Parent | t$_0${p,q,r,s,t,u} | Any regular 7-polytope |
| Rectified | t$_1${p,q,r,s,t,u} | The edges are fully truncated into single points. The 7-polytope now has the combined faces of the parent and dual. |
| Birectified | t$_2${p,q,r,s,t,u} | Birectification reduces cells to their duals. |
| Truncated | t$_{0,1}${p,q,r,s,t,u} | Each original vertex is cut off, with a new face filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated 7-polytope. The 7-polytope has its original faces doubled in sides, and contains the faces of the dual. |
| Bitruncated | t$_{1,2}${p,q,r,s,t,u} | Bitrunction transforms cells to their dual truncation. |
| Tritruncated | t$_{2,3}${p,q,r,s,t,u} | Tritruncation transforms 4-faces to their dual truncation. |
| Cantellated | t$_{0,2}${p,q,r,s,t,u} | In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is halfway between both the parent and dual forms. |
| Bicantellated | t$_{1,3}${p,q,r,s,t,u} | In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is halfway between both the parent and dual forms. |
| Runcinated | t$_{0,3}${p,q,r,s,t,u} | Runcination reduces cells and creates new cells at the vertices and edges. |
| Biruncinated | t$_{1,4}${p,q,r,s,t,u} | Runcination reduces cells and creates new cells at the vertices and edges. |
| Stericated | t$_{0,4}${p,q,r,s,t,u} | Sterication reduces 4-faces and creates new 4-faces at the vertices, edges, and faces in the gaps. |
| Pentellated | t$_{0,5}${p,q,r,s,t,u} | Pentellation reduces 5-faces and creates new 5-faces at the vertices, edges, faces, and cells in the gaps. |
| Hexicated | t$_{0,6}${p,q,r,s,t,u} | Hexication reduces 6-faces and creates new 6-faces at the vertices, edges, faces, cells, and 4-faces in the gaps. (expansion operation for 7-polytopes) |
| Omnitruncated | t$_{0,1,2,3,4,5,6}${p,q,r,s,t,u} | All six operators, truncation, cantellation, runcination, sterication, pentellation, and hexication are applied. |