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|+ Graphs of three [[List of regular polytopes#Convex 4|regular]] and related [[uniform polytope]]s
| || || || || || || || || || ||
|-  style="vertical-align:top; text-align:center;"
|colspan=4|[[File:7-simplex t0.svg|100px]]<br/>[[7-simplex]]
|colspan=4|[[File:7-simplex t1.svg|100px]]<br/>[[Rectified 7-simplex]]
|colspan=4|[[File:7-simplex t01.svg|100px]]<br/>[[Truncated 7-simplex]]
|- style="vertical-align:top; text-align:center;"
|colspan=4|[[File:7-simplex t02.svg|100px]]<br/>[[Cantellated 7-simplex]]
|colspan=4|[[File:7-simplex t03.svg|100px]]<br/>[[Runcinated 7-simplex]]
|colspan=4|[[File:7-simplex t04.svg|100px]]<br/>[[Stericated 7-simplex]]
|-  style="vertical-align:top; text-align:center;"
|colspan=6|[[File:7-simplex t05.svg|150px]]<br/>[[Pentellated 7-simplex]]
|colspan=6|[[File:7-simplex t06.svg|150px]]<br/>[[Hexicated 7-simplex]]
|- style="vertical-align:top; text-align:center;"
|colspan=4|[[File:7-cube t6.svg|100px]]<br/>[[7-orthoplex]]
|colspan=4|[[File:7-cube t56.svg|100px]]<br/>[[Truncated 7-orthoplex]]
|colspan=4|[[File:7-cube t5.svg|100px]]<br/>[[Rectified 7-orthoplex]]
|-  style="vertical-align:top; text-align:center;"
|colspan=4|[[File:7-cube t46.svg|100px]]<br/>[[Cantellated 7-orthoplex]]
|colspan=4|[[File:7-cube t36.svg|100px]]<br/>[[Runcinated 7-orthoplex]]
|colspan=4|[[File:7-cube t26.svg|100px]]<br/>[[Stericated 7-orthoplex]]
|-  style="vertical-align:top; text-align:center;"
|colspan=4|[[File:7-cube t16.svg|100px]]<br/>[[Pentellated 7-orthoplex]]
|colspan=4|[[File:7-cube t06.svg|100px]]<br/>[[Hexicated 7-cube]]
|colspan=4|[[File:7-cube t05.svg|100px]]<br/>[[Pentellated 7-cube]]
|-  style="vertical-align:top; text-align:center;"
|colspan=4|[[File:7-cube t04.svg|100px]]<br/>[[Stericated 7-cube]]
|colspan=4|[[File:7-cube t02.svg|100px]]<br/>[[Cantellated 7-cube]]
|colspan=4|[[File:7-cube t03.svg|100px]]<br/>[[Runcinated 7-cube]]
|-  style="vertical-align:top; text-align:center;"
|colspan=4|[[File:7-cube t0.svg|100px]]<br/>[[7-cube]]
|colspan=4|[[File:7-cube t01.svg|100px]]<br/>[[Truncated 7-cube]]
|colspan=4|[[File:7-cube t1.svg|100px]]<br/>[[Rectified 7-cube]]
|-  style="vertical-align:top; text-align:center;"
|colspan=4|[[File:7-demicube t0 D7.svg|100px]]<br/>[[7-demicube]]
|colspan=4|[[File:7-demicube t01 D7.svg|100px]]<br/>[[Truncated 7-demicube]]
|colspan=4|[[File:7-demicube t02 D7.svg|100px]]<br/>[[Cantellated 7-demicube]]
|-  style="vertical-align:top; text-align:center;"
|colspan=4|[[File:7-demicube t03 D7.svg|100px]]<br/>[[Runcinated 7-demicube]]
|colspan=4|[[File:7-demicube t04 D7.svg|100px]]<br/>[[Stericated 7-demicube]]
|colspan=4|[[File:7-demicube t05 D7.svg|100px]]<br/>[[Pentellated 7-demicube]]
|-  style="vertical-align:top; text-align:center;"
|colspan=4|[[File:E7 graph.svg|100px]]<br/>[[3 21 polytope|3<sub>21</sub>]]
|colspan=4|[[File:Gosset 2 31 polytope.svg|100px]]<br/>[[2 31 polytope|2<sub>31</sub>]]
|colspan=4|[[File:Gosset 1 32 petrie.svg|100px]]<br/>[[1 32 polytope|1<sub>32</sub>]]
|}
In [[seven-dimensional space|seven-dimensional]] [[geometry]], a '''7-polytope''' is a [[polytope]] contained by 6-polytope facets. Each [[5-polytope]] [[Ridge (geometry)|ridge]] being shared by exactly two [[6-polytope]] [[Facet (mathematics)|facets]].
 
A '''uniform 7-polytope''' is one which is [[vertex-transitive]], and constructed from uniform [[6-polytope]] facets.
 
A proposed name for 7-polytopes is '''polyexon''' or '''polyecton'''.
 
== 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 [[Facet (mathematics)|facets]] around each 4-face.
 
There are exactly three such [[List of regular polytopes#Convex 4|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.
 
== Euler characteristic ==
 
The [[Euler characteristic]] for 7-polytopes that are topological [[6-sphere]]s (including all convex 7-polytopes) is two. χ=V-E+F-C+f<sub>4</sub>-f<sub>5</sub>+f<sub>6</sub>=2.
 
== 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 diagram]]s:
 
{| class="wikitable"
|-
!#
!colspan=2|[[Coxeter group]]
![[Coxeter-Dynkin diagram|Coxeter-Dynkin<BR>diagram]]
!Regular and semiregular forms
!Uniform count
|-
|1||A<sub>7</sub>|| [3<sup>6</sup>]||{{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|
* [[7-simplex]] - {3<sup>6</sup>}, {{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}
|71
|-
|2||B<sub>7</sub>||[4,3<sup>5</sup>]||{{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node}}
|
* [[7-cube]] - {4,3<sup>5</sup>}, {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node}}
* [[7-orthoplex]] - {3<sup>5</sup>,4}, {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node_1}}
|127
|-
|3||D<sub>7</sub>||[3<sup>4,1,1</sup>]||{{CDD|nodes|split2|node|3|node|3|node|3|node|3|node}}
|
* [[7-demicube]], {3<sup>1,4,1</sup>}, {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node}}
* [[7-orthoplex]], {3<sup>4,1,1</sup>}, {{CDD|nodes|split2|node|3|node|3|node|3|node|3|node_1}}
|95 (32 unique)
|-
|4||[[E7 (mathematics)|E<sub>7</sub>]]||[3<sup>3,2,1</sup>]||{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}
|
* '''[[Gosset 3 21 polytope|3<sub>21</sub>]]''' - {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea_1}}
* '''[[Gosset 1 32 polytope|1<sub>32</sub>]]''' - {{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea}}
* '''[[Gosset 2 31 polytope|2<sub>31</sub>]]''' - {{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}
|127
|}
 
{| class="wikitable"
|+ Prismatic finite Coxeter groups
|-
!#
!colspan=2|[[Coxeter group]]
![[Coxeter-Dynkin diagram]]
|-
!colspan=4|6+1
|-
|1 ||A<sub>6</sub>×A<sub>1</sub>|| [3<sup>5</sup>]×[&nbsp;]|| {{CDD|node|3|node|3|node|3|node|3|node|3|node|2|node}}
|-
|2 ||BC<sub>6</sub>×A<sub>1</sub>|| [4,3<sup>4</sup>]×[&nbsp;]|| {{CDD|node|4|node|3|node|3|node|3|node|3|node|2|node}}
|-
|3 ||D<sub>6</sub>×A<sub>1</sub>|| [3<sup>3,1,1</sup>]×[&nbsp;]|| {{CDD|nodes|split2|node|3|node|3|node|3|node|2|node}}
|-
|4 ||E<sub>6</sub>×A<sub>1</sub>|| [3<sup>2,2,1</sup>]×[&nbsp;]|| {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|2|nodea}}
|-
!colspan=4|5+2
|-
|1 ||A<sub>5</sub>×I<sub>2</sub>(p)|| [3,3,3]×[p]|| {{CDD|node|3|node|3|node|3|node|3|node|2|node|p|node}}
|-
|2 ||BC<sub>5</sub>×I<sub>2</sub>(p)|| [4,3,3]×[p]|| {{CDD|node|4|node|3|node|3|node|3|node|2|node|p|node}}
|-
|3 ||D<sub>5</sub>×I<sub>2</sub>(p)|| [3<sup>2,1,1</sup>]×[p]|| {{CDD|nodes|split2|node|3|node|3|node|2|node|p|node}}
|-
!colspan=4|5+1+1
|-
|1 ||A<sub>5</sub>×A<sub>1</sub><sup>2</sup>|| [3,3,3]×[&nbsp;]<sup>2</sup>|| {{CDD|node|3|node|3|node|3|node|3|node|2|node|2|node}}
|-
|2 ||BC<sub>5</sub>×A<sub>1</sub><sup>2</sup>|| [4,3,3]×[&nbsp;]<sup>2</sup>|| {{CDD|node|4|node|3|node|3|node|3|node|2|node|2|node}}
|-
|3 ||D<sub>5</sub>×A<sub>1</sub><sup>2</sup>|| [3<sup>2,1,1</sup>]×[&nbsp;]<sup>2</sup>|| {{CDD|nodes|split2|node|3|node|3|node|2|node|2|node}}
|-
!colspan=4|4+3
|-
|4 ||A<sub>4</sub>×A<sub>3</sub>|| [3,3,3]×[3,3]|| {{CDD|node|3|node|3|node|3|node|2|node|3|node|3|node}}
|-
|5 ||A<sub>4</sub>×BC<sub>3</sub>|| [3,3,3]×[4,3]|| {{CDD|node|3|node|3|node|3|node|2|node|4|node|3|node}}
|-
|6 ||A<sub>4</sub>×H<sub>3</sub>|| [3,3,3]×[5,3]|| {{CDD|node|3|node|3|node|3|node|2|node|5|node|3|node}}
|-
|7 ||BC<sub>4</sub>×A<sub>3</sub>|| [4,3,3]×[3,3]|| {{CDD|node|4|node|3|node|3|node|2|node|3|node|3|node}}
|-
|8 ||BC<sub>4</sub>×BC<sub>3</sub>|| [4,3,3]×[4,3]|| {{CDD|node|4|node|3|node|3|node|2|node|4|node|3|node}}
|-
|9 ||BC<sub>4</sub>×H<sub>3</sub>|| [4,3,3]×[5,3]|| {{CDD|node|4|node|3|node|3|node|2|node|5|node|3|node}}
|-
|10 ||H<sub>4</sub>×A<sub>3</sub>|| [5,3,3]×[3,3]|| {{CDD|node|5|node|3|node|3|node|2|node|3|node|3|node}}
|-
|11 ||H<sub>4</sub>×BC<sub>3</sub>|| [5,3,3]×[4,3]|| {{CDD|node|5|node|3|node|3|node|2|node|4|node|3|node}}
|-
|12 ||H<sub>4</sub>×H<sub>3</sub>|| [5,3,3]×[5,3]|| {{CDD|node|5|node|3|node|3|node|2|node|5|node|3|node}}
|-
|13 ||F<sub>4</sub>×A<sub>3</sub>|| [3,4,3]×[3,3]|| {{CDD|node|3|node|4|node|3|node|2|node|3|node|3|node}}
|-
|14 ||F<sub>4</sub>×BC<sub>3</sub>|| [3,4,3]×[4,3]|| {{CDD|node|3|node|4|node|3|node|2|node|4|node|3|node}}
|-
|15 ||F<sub>4</sub>×H<sub>3</sub>|| [3,4,3]×[5,3]|| {{CDD|node|3|node|4|node|3|node|2|node|5|node|3|node}}
|-
|16 ||D<sub>4</sub>×A<sub>3</sub>|| [3<sup>1,1,1</sup>]×[3,3]|| {{CDD|nodes|split2|node|3|node|2|node|3|node|3|node}}
|-
|17 ||D<sub>4</sub>×BC<sub>3</sub>|| [3<sup>1,1,1</sup>]×[4,3]|| {{CDD|nodes|split2|node|3|node|2|node|4|node|3|node}}
|-
|18 ||D<sub>4</sub>×H<sub>3</sub>|| [3<sup>1,1,1</sup>]×[5,3]|| {{CDD|nodes|split2|node|3|node|2|node|5|node|3|node}}
|-
!colspan=4|4+2+1
|-
|5 ||A<sub>4</sub>×I<sub>2</sub>(p)×A<sub>1</sub>|| [3,3,3]×[p]×[&nbsp;]|| {{CDD|node|3|node|3|node|3|node|2|node|p|node|2|node}}
|-
|6 ||BC<sub>4</sub>×I<sub>2</sub>(p)×A<sub>1</sub>|| [4,3,3]×[p]×[&nbsp;]|| {{CDD|node|4|node|3|node|3|node|2|node|p|node|2|node}}
|-
|7 ||F<sub>4</sub>×I<sub>2</sub>(p)×A<sub>1</sub>|| [3,4,3]×[p]×[&nbsp;]|| {{CDD|node|3|node|4|node|3|node|2|node|p|node|2|node}}
|-
|8 ||H<sub>4</sub>×I<sub>2</sub>(p)×A<sub>1</sub>|| [5,3,3]×[p]×[&nbsp;]|| {{CDD|node|5|node|3|node|3|node|2|node|p|node|2|node}}
|-
|9 ||D<sub>4</sub>×I<sub>2</sub>(p)×A<sub>1</sub>|| [3<sup>1,1,1</sup>]×[p]×[&nbsp;]|| {{CDD|nodes|split2|node|3|node|2|node|p|node|2|node}}
|-
!colspan=4|4+1+1+1
|-
|5 ||A<sub>4</sub>×A<sub>1</sub><sup>3</sup>|| [3,3,3]×[&nbsp;]<sup>3</sup>|| {{CDD|node|3|node|3|node|3|node|2|node|2|node|2|node}}
|-
|6 ||BC<sub>4</sub>×A<sub>1</sub><sup>3</sup>|| [4,3,3]×[&nbsp;]<sup>3</sup>|| {{CDD|node|4|node|3|node|3|node|2|node|2|node|2|node}}
|-
|7 ||F<sub>4</sub>×A<sub>1</sub><sup>3</sup>|| [3,4,3]×[&nbsp;]<sup>3</sup>|| {{CDD|node|3|node|4|node|3|node|2|node|2|node|2|node}}
|-
|8 ||H<sub>4</sub>×A<sub>1</sub><sup>3</sup>|| [5,3,3]×[&nbsp;]<sup>3</sup>|| {{CDD|node|5|node|3|node|3|node|2|node|2|node|2|node}}
|-
|9 ||D<sub>4</sub>×A<sub>1</sub><sup>3</sup>|| [3<sup>1,1,1</sup>]×[&nbsp;]<sup>3</sup>|| {{CDD|nodes|split2|node|3|node|2|node|2|node|2|node}}
|-
!colspan=4|3+3+1
|-
|10 ||A<sub>3</sub>×A<sub>3</sub>×A<sub>1</sub>|| [3,3]×[3,3]×[&nbsp;]|| {{CDD|node|3|node|3|node|2|node|3|node|3|node|2|node}}
|-
|11 ||A<sub>3</sub>×BC<sub>3</sub>×A<sub>1</sub>|| [3,3]×[4,3]×[&nbsp;]|| {{CDD|node|3|node|3|node|2|node|4|node|3|node|2|node}}
|-
|12 ||A<sub>3</sub>×H<sub>3</sub>×A<sub>1</sub>|| [3,3]×[5,3]×[&nbsp;]|| {{CDD|node|3|node|3|node|2|node|5|node|3|node|2|node}}
|-
|13 ||BC<sub>3</sub>×BC<sub>3</sub>×A<sub>1</sub>|| [4,3]×[4,3]×[&nbsp;]|| {{CDD|node|4|node|3|node|2|node|4|node|3|node|2|node}}
|-
|14 ||BC<sub>3</sub>×H<sub>3</sub>×A<sub>1</sub>|| [4,3]×[5,3]×[&nbsp;]|| {{CDD|node|4|node|3|node|2|node|5|node|3|node|2|node}}
|-
|15 ||H<sub>3</sub>×A<sub>3</sub>×A<sub>1</sub>|| [5,3]×[5,3]×[&nbsp;]|| {{CDD|node|5|node|3|node|2|node|5|node|3|node|2|node}}
|-
!colspan=4|3+2+2
|-
|1 ||A<sub>3</sub>×I<sub>2</sub>(p)×I<sub>2</sub>(q)|| [3,3]×[p]×[q]|| {{CDD|node|3|node|3|node|2|node|p|node|2|node|q|node}}
|-
|2 ||BC<sub>3</sub>×I<sub>2</sub>(p)×I<sub>2</sub>(q)|| [4,3]×[p]×[q]|| {{CDD|node|4|node|3|node|2|node|p|node|2|node|q|node}}
|-
|3 ||H<sub>3</sub>×I<sub>2</sub>(p)×I<sub>2</sub>(q)|| [5,3]×[p]×[q]|| {{CDD|node|5|node|3|node|2|node|p|node|2|node|q|node}}
|-
!colspan=4|3+2+1+1
|-
|1 ||A<sub>3</sub>×I<sub>2</sub>(p)×A<sub>1</sub><sup>2</sup>|| [3,3]×[p]×[&nbsp;]<sup>2</sup>|| {{CDD|node|3|node|3|node|2|node|p|node|2|node|2|node}}
|-
|2 ||BC<sub>3</sub>×I<sub>2</sub>(p)×A<sub>1</sub><sup>2</sup>|| [4,3]×[p]×[&nbsp;]<sup>2</sup>|| {{CDD|node|4|node|3|node|2|node|p|node|2|node|2|node}}
|-
|3 ||H<sub>3</sub>×I<sub>2</sub>(p)×A<sub>1</sub><sup>2</sup>|| [5,3]×[p]×[&nbsp;]<sup>2</sup>|| {{CDD|node|5|node|3|node|2|node|p|node|2|node|2|node}}
|-
!colspan=4|3+1+1+1+1
|-
|1 ||A<sub>3</sub>×A<sub>1</sub><sup>4</sup>|| [3,3]×[&nbsp;]<sup>4</sup>|| {{CDD|node|3|node|3|node|2|node|2|node|2|node|2|node}}
|-
|2 ||BC<sub>3</sub>×A<sub>1</sub><sup>4</sup>|| [4,3]×[&nbsp;]<sup>4</sup>|| {{CDD|node|4|node|3|node|2|node|2|node|2|node|2|node}}
|-
|3 ||H<sub>3</sub>×A<sub>1</sub><sup>4</sup>|| [5,3]×[&nbsp;]<sup>4</sup>|| {{CDD|node|5|node|3|node|2|node|2|node|2|node|2|node}}
|-
!colspan=4|2+2+2+1
|-
|1 ||I<sub>2</sub>(p)×I<sub>2</sub>(q)×I<sub>2</sub>(r)×A<sub>1</sub>|| [p]×[q]×[r]×[&nbsp;]|| {{CDD|node|p|node|2|node|q|node|2|node|r|node|2|node}}
|-
!colspan=4|2+2+1+1+1
|-
|1 ||I<sub>2</sub>(p)×I<sub>2</sub>(q)×A<sub>1</sub><sup>3</sup>|| [p]×[q]×[&nbsp;]<sup>3</sup>|| {{CDD|node|p|node|2|node|q|node|2|node|2|node|2|node}}
|-
!colspan=4|2+1+1+1+1+1
|-
|1 ||I<sub>2</sub>(p)×A<sub>1</sub><sup>5</sup>|| [p]×[&nbsp;]<sup>5</sup>|| {{CDD|node|p|node|2|node|2|node|2|node|2|node|2|node}}
|-
!colspan=4|1+1+1+1+1+1+1
|-
|1 ||A<sub>1</sub><sup>7</sup>|| [&nbsp;]<sup>7</sup>|| {{CDD|node|2|node|2|node|2|node|2|node|2|node|2|node}}
|}
 
== The A<sub>7</sub> family ==
 
The A<sub>7</sub> family has symmetry of order 40320 (8 [[factorial]]).
 
There are 71 (64+8-1) forms based on all permutations of the [[Coxeter-Dynkin diagram]]s with one or more rings. All 71 are enumerated below. [[Norman Johnson (mathematician)|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.
 
{| class="wikitable"
|-
!rowspan=2|#
!rowspan=2|[[Coxeter-Dynkin diagram]]
!rowspan=2|Truncation<br />indices
!rowspan=2|[[Norman Johnson (mathematician)|Johnson name]]<br />Bowers name (and acronym)
!rowspan=2|Basepoint
!colspan=7|Element counts
|-
! 6|| 5|| 4|| 3|| 2|| 1|| 0
|- style="text-align:center;"
|1||{{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}||t<sub>0</sub>||[[7-simplex]] (oca)||(0,0,0,0,0,0,0,1)||8||28||56||70||56||28||8
|- style="text-align:center;"
|2||{{CDD|node|3|node|3|node|3|node|3|node|3|node_1|3|node}}||t<sub>1</sub>||[[Rectified 7-simplex]] (roc)||(0,0,0,0,0,0,1,1)||16||84||224||350||336||168||28
|- style="text-align:center;"
|3||{{CDD|node|3|node|3|node|3|node|3|node_1|3|node|3|node}}||t<sub>2</sub>||[[Birectified 7-simplex]] (broc)||(0,0,0,0,0,1,1,1)||16||112||392||770||840||420||56
|-  style="text-align:center; background:#e0f0e0;"
|4||{{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node}}||t<sub>3</sub>||[[Trirectified 7-simplex]] (he)||(0,0,0,0,1,1,1,1)||16||112||448||980||1120||560||70
|- style="text-align:center;"
|5||{{CDD|node|3|node|3|node|3|node|3|node|3|node_1|3|node_1}}||t<sub>0,1</sub>||[[Truncated 7-simplex]] (toc)||(0,0,0,0,0,0,1,2)||16||84||224||350||336||196||56
|- style="text-align:center;"
|6||{{CDD|node|3|node|3|node|3|node|3|node_1|3|node|3|node_1}}||t<sub>0,2</sub>||[[Cantellated 7-simplex]] (saro)||(0,0,0,0,0,1,1,2)||44||308||980||1750||1876||1008||168
|- style="text-align:center;"
|7||{{CDD|node|3|node|3|node|3|node|3|node_1|3|node_1|3|node}}||t<sub>1,2</sub>||[[Bitruncated 7-simplex]] (bittoc)||(0,0,0,0,0,1,2,2)||||||||||||588||168
|- style="text-align:center;"
|8||{{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node_1}}||t<sub>0,3</sub>||[[Runcinated 7-simplex]] (spo)||(0,0,0,0,1,1,1,2)||100||756||2548||4830||4760||2100||280
|- style="text-align:center;"
|9||{{CDD|node|3|node|3|node|3|node_1|3|node|3|node_1|3|node}}||t<sub>1,3</sub>||[[Bicantellated 7-simplex]] (sabro)||(0,0,0,0,1,1,2,2)||||||||||||2520||420
|- style="text-align:center;"
|10||{{CDD|node|3|node|3|node|3|node_1|3|node_1|3|node|3|node}}||t<sub>2,3</sub>||[[Tritruncated 7-simplex]] (tattoc)||(0,0,0,0,1,2,2,2)||||||||||||980||280
|- style="text-align:center;"
|11||{{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node_1}}||t<sub>0,4</sub>||[[Stericated 7-simplex]] (sco)||(0,0,0,1,1,1,1,2)||||||||||||2240||280
|- style="text-align:center;"
|12||{{CDD|node|3|node|3|node_1|3|node|3|node|3|node_1|3|node}}||t<sub>1,4</sub>||[[Biruncinated 7-simplex]] (sibpo)||(0,0,0,1,1,1,2,2)||||||||||||4200||560
|-  style="text-align:center; background:#e0f0e0;"
|13||{{CDD|node|3|node|3|node_1|3|node|3|node_1|3|node|3|node}}||t<sub>2,4</sub>||[[Tricantellated 7-simplex]] (stiroh)||(0,0,0,1,1,2,2,2)||||||||||||3360||560
|- style="text-align:center;"
|14||{{CDD|node|3|node_1|3|node|3|node|3|node|3|node|3|node_1}}||t<sub>0,5</sub>||[[Pentellated 7-simplex]] (seto)||(0,0,1,1,1,1,1,2)||||||||||||1260||168
|-  style="text-align:center; background:#e0f0e0;"
|15||{{CDD|node|3|node_1|3|node|3|node|3|node|3|node_1|3|node}}||t<sub>1,5</sub>||[[Bistericated 7-simplex]] (sabach)||(0,0,1,1,1,1,2,2)||||||||||||3360||420
|-  style="text-align:center; background:#e0f0e0;"
|16||{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node_1}}||t<sub>0,6</sub>||[[Hexicated 7-simplex]] (suph)||(0,1,1,1,1,1,1,2)||||||||||||336||56
|- style="text-align:center;"
|17||{{CDD|node|3|node|3|node|3|node|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2</sub>||[[Cantitruncated 7-simplex]] (garo)||(0,0,0,0,0,1,2,3)||||||||||||1176||336
|- style="text-align:center;"
|18||{{CDD|node|3|node|3|node|3|node_1|3|node|3|node_1|3|node_1}}||t<sub>0,1,3</sub>||[[Runcitruncated 7-simplex]] (patto)||(0,0,0,0,1,1,2,3)||||||||||||4620||840
|- style="text-align:center;"
|19||{{CDD|node|3|node|3|node|3|node_1|3|node_1|3|node|3|node_1}}||t<sub>0,2,3</sub>||[[Runcicantellated 7-simplex]] (paro)||(0,0,0,0,1,2,2,3)||||||||||||3360||840
|- style="text-align:center;"
|20||{{CDD|node|3|node|3|node|3|node_1|3|node_1|3|node_1|3|node}}||t<sub>1,2,3</sub>||[[Bicantitruncated 7-simplex]] (gabro)||(0,0,0,0,1,2,3,3)||||||||||||2940||840
|- style="text-align:center;"
|21||{{CDD|node|3|node|3|node_1|3|node|3|node|3|node_1|3|node_1}}||t<sub>0,1,4</sub>||[[Steritruncated 7-simplex]] (cato)||(0,0,0,1,1,1,2,3)||||||||||||7280||1120
|- style="text-align:center;"
|22||{{CDD|node|3|node|3|node_1|3|node|3|node_1|3|node|3|node_1}}||t<sub>0,2,4</sub>||[[Stericantellated 7-simplex]] (caro)||(0,0,0,1,1,2,2,3)||||||||||||10080||1680
|- style="text-align:center;"
|23||{{CDD|node|3|node|3|node_1|3|node|3|node_1|3|node_1|3|node}}||t<sub>1,2,4</sub>||[[Biruncitruncated 7-simplex]] (bipto)||(0,0,0,1,1,2,3,3)||||||||||||8400||1680
|- style="text-align:center;"
|24||{{CDD|node|3|node|3|node_1|3|node_1|3|node|3|node|3|node_1}}||t<sub>0,3,4</sub>||[[Steriruncinated 7-simplex]] (cepo)||(0,0,0,1,2,2,2,3)||||||||||||5040||1120
|- style="text-align:center;"
|25||{{CDD|node|3|node|3|node_1|3|node_1|3|node|3|node_1|3|node}}||t<sub>1,3,4</sub>||[[Biruncicantellated 7-simplex]] (bipro)||(0,0,0,1,2,2,3,3)||||||||||||7560||1680
|-  style="text-align:center; background:#e0f0e0;"
|26||{{CDD|node|3|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}||t<sub>2,3,4</sub>||[[Tricantitruncated 7-simplex]] (gatroh)||(0,0,0,1,2,3,3,3)||||||||||||3920||1120
|- style="text-align:center;"
|27||{{CDD|node|3|node_1|3|node|3|node|3|node|3|node_1|3|node_1}}||t<sub>0,1,5</sub>||[[Pentitruncated 7-simplex]] (teto)||(0,0,1,1,1,1,2,3)||||||||||||5460||840
|- style="text-align:center;"
|28||{{CDD|node|3|node_1|3|node|3|node|3|node_1|3|node|3|node_1}}||t<sub>0,2,5</sub>||[[Penticantellated 7-simplex]] (tero)||(0,0,1,1,1,2,2,3)||||||||||||11760||1680
|- style="text-align:center;"
|29||{{CDD|node|3|node_1|3|node|3|node|3|node_1|3|node_1|3|node}}||t<sub>1,2,5</sub>||[[Bisteritruncated 7-simplex]] (bacto)||(0,0,1,1,1,2,3,3)||||||||||||9240||1680
|- style="text-align:center;"
|30||{{CDD|node|3|node_1|3|node|3|node_1|3|node|3|node|3|node_1}}||t<sub>0,3,5</sub>||[[Pentiruncinated 7-simplex]] (tepo)||(0,0,1,1,2,2,2,3)||||||||||||10920||1680
|-  style="text-align:center; background:#e0f0e0;"
|31||{{CDD|node|3|node_1|3|node|3|node_1|3|node|3|node_1|3|node}}||t<sub>1,3,5</sub>||[[Bistericantellated 7-simplex]] (bacroh)||(0,0,1,1,2,2,3,3)||||||||||||15120||2520
|- style="text-align:center;"
|32||{{CDD|node|3|node_1|3|node_1|3|node|3|node|3|node|3|node_1}}||t<sub>0,4,5</sub>||[[Pentistericated 7-simplex]] (teco)||(0,0,1,2,2,2,2,3)||||||||||||4200||840
|- style="text-align:center;"
|33||{{CDD|node_1|3|node|3|node|3|node|3|node|3|node_1|3|node_1}}||t<sub>0,1,6</sub>||[[Hexitruncated 7-simplex]] (puto)||(0,1,1,1,1,1,2,3)||||||||||||1848||336
|- style="text-align:center;"
|34||{{CDD|node_1|3|node|3|node|3|node|3|node_1|3|node|3|node_1}}||t<sub>0,2,6</sub>||[[Hexicantellated 7-simplex]] (puro)||(0,1,1,1,1,2,2,3)||||||||||||5880||840
|-  style="text-align:center; background:#e0f0e0;"
|35||{{CDD|node_1|3|node|3|node|3|node_1|3|node|3|node|3|node_1}}||t<sub>0,3,6</sub>||[[Hexiruncinated 7-simplex]] (puph)||(0,1,1,1,2,2,2,3)||||||||||||8400||1120
|- style="text-align:center;"
|36||{{CDD|node|3|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,3</sub>||[[Runcicantitruncated 7-simplex]] (gapo)||(0,0,0,0,1,2,3,4)||||||||||||5880||1680
|- style="text-align:center;"
|37||{{CDD|node|3|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,4</sub>||[[Stericantitruncated 7-simplex]] (cagro)||(0,0,0,1,1,2,3,4)||||||||||||16800||3360
|- style="text-align:center;"
|38||{{CDD|node|3|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}||t<sub>0,1,3,4</sub>||[[Steriruncitruncated 7-simplex]] (capto)||(0,0,0,1,2,2,3,4)||||||||||||13440||3360
|- style="text-align:center;"
|39||{{CDD|node|3|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}||t<sub>0,2,3,4</sub>||[[Steriruncicantellated 7-simplex]] (capro)||(0,0,0,1,2,3,3,4)||||||||||||13440||3360
|- style="text-align:center;"
|40||{{CDD|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}||t<sub>1,2,3,4</sub>||[[Biruncicantitruncated 7-simplex]] (gibpo)||(0,0,0,1,2,3,4,4)||||||||||||11760||3360
|- style="text-align:center;"
|41||{{CDD|node|3|node_1|3|node|3|node|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,5</sub>||[[Penticantitruncated 7-simplex]] (tegro)||(0,0,1,1,1,2,3,4)||||||||||||18480||3360
|- style="text-align:center;"
|42||{{CDD|node|3|node_1|3|node|3|node_1|3|node|3|node_1|3|node_1}}||t<sub>0,1,3,5</sub>||[[Pentiruncitruncated 7-simplex]] (tapto)||(0,0,1,1,2,2,3,4)||||||||||||27720||5040
|- style="text-align:center;"
|43||{{CDD|node|3|node_1|3|node|3|node_1|3|node_1|3|node|3|node_1}}||t<sub>0,2,3,5</sub>||[[Pentiruncicantellated 7-simplex]] (tapro)||(0,0,1,1,2,3,3,4)||||||||||||25200||5040
|- style="text-align:center;"
|44||{{CDD|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node}}||t<sub>1,2,3,5</sub>||[[Bistericantitruncated 7-simplex]] (bacogro)||(0,0,1,1,2,3,4,4)||||||||||||22680||5040
|- style="text-align:center;"
|45||{{CDD|node|3|node_1|3|node_1|3|node|3|node|3|node_1|3|node_1}}||t<sub>0,1,4,5</sub>||[[Pentisteritruncated 7-simplex]] (tecto)||(0,0,1,2,2,2,3,4)||||||||||||15120||3360
|- style="text-align:center;"
|46||{{CDD|node|3|node_1|3|node_1|3|node|3|node_1|3|node|3|node_1}}||t<sub>0,2,4,5</sub>||[[Pentistericantellated 7-simplex]] (tecro)||(0,0,1,2,2,3,3,4)||||||||||||25200||5040
|-  style="text-align:center; background:#e0f0e0;"
|47||{{CDD|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node}}||t<sub>1,2,4,5</sub>||[[Bisteriruncitruncated 7-simplex]] (bicpath)||(0,0,1,2,2,3,4,4)||||||||||||20160||5040
|- style="text-align:center;"
|48||{{CDD|node|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node_1}}||t<sub>0,3,4,5</sub>||[[Pentisteriruncinated 7-simplex]] (tacpo)||(0,0,1,2,3,3,3,4)||||||||||||15120||3360
|- style="text-align:center;"
|49||{{CDD|node_1|3|node|3|node|3|node|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,6</sub>||[[Hexicantitruncated 7-simplex]] (pugro)||(0,1,1,1,1,2,3,4)||||||||||||8400||1680
|- style="text-align:center;"
|50||{{CDD|node_1|3|node|3|node|3|node_1|3|node|3|node_1|3|node_1}}||t<sub>0,1,3,6</sub>||[[Hexiruncitruncated 7-simplex]] (pugato)||(0,1,1,1,2,2,3,4)||||||||||||20160||3360
|- style="text-align:center;"
|51||{{CDD|node_1|3|node|3|node|3|node_1|3|node_1|3|node|3|node_1}}||t<sub>0,2,3,6</sub>||[[Hexiruncicantellated 7-simplex]] (pugro)||(0,1,1,1,2,3,3,4)||||||||||||16800||3360
|- style="text-align:center;"
|52||{{CDD|node_1|3|node|3|node_1|3|node|3|node|3|node_1|3|node_1}}||t<sub>0,1,4,6</sub>||[[Hexisteritruncated 7-simplex]] (pucto)||(0,1,1,2,2,2,3,4)||||||||||||20160||3360
|-  style="text-align:center; background:#e0f0e0;"
|53||{{CDD|node_1|3|node|3|node_1|3|node|3|node_1|3|node|3|node_1}}||t<sub>0,2,4,6</sub>||[[Hexistericantellated 7-simplex]] (pucroh)||(0,1,1,2,2,3,3,4)||||||||||||30240||5040
|-  style="text-align:center; background:#e0f0e0;"
|54||{{CDD|node_1|3|node_1|3|node|3|node|3|node|3|node_1|3|node_1}}||t<sub>0,1,5,6</sub>||[[Hexipentitruncated 7-simplex]] (putath)||(0,1,2,2,2,2,3,4)||||||||||||8400||1680
|- style="text-align:center;"
|55||{{CDD|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,3,4</sub>||[[Steriruncicantitruncated 7-simplex]] (gecco)||(0,0,0,1,2,3,4,5)||||||||||||23520||6720
|- style="text-align:center;"
|56||{{CDD|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,3,5</sub>||[[Pentiruncicantitruncated 7-simplex]] (tegapo)||(0,0,1,1,2,3,4,5)||||||||||||45360||10080
|- style="text-align:center;"
|57||{{CDD|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,4,5</sub>||[[Pentistericantitruncated 7-simplex]] (tecagro)||(0,0,1,2,2,3,4,5)||||||||||||40320||10080
|- style="text-align:center;"
|58||{{CDD|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}||t<sub>0,1,3,4,5</sub>||[[Pentisteriruncitruncated 7-simplex]] (tacpeto)||(0,0,1,2,3,3,4,5)||||||||||||40320||10080
|- style="text-align:center;"
|59||{{CDD|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}||t<sub>0,2,3,4,5</sub>||[[Pentisteriruncicantellated 7-simplex]] (tacpro)||(0,0,1,2,3,4,4,5)||||||||||||40320||10080
|-  style="text-align:center; background:#e0f0e0;"
|60||{{CDD|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}||t<sub>1,2,3,4,5</sub>||[[Bisteriruncicantitruncated 7-simplex]] (gabach)||(0,0,1,2,3,4,5,5)||||||||||||35280||10080
|- style="text-align:center;"
|61||{{CDD|node_1|3|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,3,6</sub>||[[Hexiruncicantitruncated 7-simplex]] (pugopo)||(0,1,1,1,2,3,4,5)||||||||||||30240||6720
|- style="text-align:center;"
|62||{{CDD|node_1|3|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,4,6</sub>||[[Hexistericantitruncated 7-simplex]] (pucagro)||(0,1,1,2,2,3,4,5)||||||||||||50400||10080
|- style="text-align:center;"
|63||{{CDD|node_1|3|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}||t<sub>0,1,3,4,6</sub>||[[Hexisteriruncitruncated 7-simplex]] (pucpato)||(0,1,1,2,3,3,4,5)||||||||||||45360||10080
|-  style="text-align:center; background:#e0f0e0;"
|64||{{CDD|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}||t<sub>0,2,3,4,6</sub>||[[Hexisteriruncicantellated 7-simplex]] (pucproh)||(0,1,1,2,3,4,4,5)||||||||||||45360||10080
|- style="text-align:center;"
|65||{{CDD|node_1|3|node_1|3|node|3|node|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,5,6</sub>||[[Hexipenticantitruncated 7-simplex]] (putagro)||(0,1,2,2,2,3,4,5)||||||||||||30240||6720
|-  style="text-align:center; background:#e0f0e0;"
|66||{{CDD|node_1|3|node_1|3|node|3|node_1|3|node|3|node_1|3|node_1}}||t<sub>0,1,3,5,6</sub>||[[Hexipentiruncitruncated 7-simplex]] (putpath)||(0,1,2,2,3,3,4,5)||||||||||||50400||10080
|- style="text-align:center;"
|67||{{CDD|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,3,4,5</sub>||[[Pentisteriruncicantitruncated 7-simplex]] (geto)||(0,0,1,2,3,4,5,6)||||||||||||70560||20160
|- style="text-align:center;"
|68||{{CDD|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,3,4,6</sub>||[[Hexisteriruncicantitruncated 7-simplex]] (pugaco)||(0,1,1,2,3,4,5,6)||||||||||||80640||20160
|- style="text-align:center;"
|69||{{CDD|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,3,5,6</sub>||[[Hexipentiruncicantitruncated 7-simplex]] (putgapo)||(0,1,2,2,3,4,5,6)||||||||||||80640||20160
|-  style="text-align:center; background:#e0f0e0;"
|70||{{CDD|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,4,5,6</sub>||[[Hexipentistericantitruncated 7-simplex]] (putcagroh)||(0,1,2,3,3,4,5,6)||||||||||||80640||20160
|-  style="text-align:center; background:#e0f0e0;"
|71||{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,3,4,5,6</sub>||[[Omnitruncated 7-simplex]] (guph)||(0,1,2,3,4,5,6,7)||||||||||||141120||40320
|}
 
== The B<sub>7</sub> family ==
 
The B<sub>7</sub> family has symmetry of order 645120 (7 [[factorial]] x 2<sup>7</sup>).
 
There are 127 forms based on all permutations of the [[Coxeter-Dynkin diagram]]s with one or more rings. Johnson and Bowers names.
 
See also a [[list of B7 polytopes]] for symmetric [[Coxeter plane]] graphs of these polytopes.
 
{| class=wikitable
!rowspan=2|#
!rowspan=2|[[Coxeter-Dynkin diagram]]<BR>t-notation
!rowspan=2|Name (BSA)
!rowspan=2|Base point
!colspan=7|Element counts
|-
!6||5||4||3||2||1||0
|-  style="text-align:center; background:#f0e0e0;"
!1
|<!-- [x3o3o3o3o3o4o] -->{{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node_1}}<BR>t<sub>0</sub>{3,3,3,3,3,4}||[[7-orthoplex]] (zee)|||(0,0,0,0,0,0,1)√2||128||448||672||560||280||84||14
|-  style="text-align:center; background:#f0e0e0;"
!2
|<!-- [o3x3o3o3o3o4o] -->{{CDD|node|4|node|3|node|3|node|3|node|3|node_1|3|node}}<BR>t<sub>1</sub>{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
|-  style="text-align:center; background:#f0e0e0;"
!3
|<!-- [o3o3x3o3o3o4o] -->{{CDD|node|4|node|3|node|3|node|3|node_1|3|node|3|node}}<BR>t<sub>2</sub>{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
|-  style="text-align:center; background:#e0f0e0;"
!4
|<!-- [o3o3o3x3o3o4o] -->{{CDD|node|4|node|3|node|3|node_1|3|node|3|node|3|node}}<BR>t<sub>3</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!5
|<!-- [o3o3o3o3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node|3|node|3|node|3|node}}<BR>t<sub>2</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!6
|<!-- [o3o3o3o3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node|3|node|3|node|3|node}}<BR>t<sub>1</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!7
|<!-- [o3o3o3o3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node}}<BR>t<sub>0</sub>{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
|-  style="text-align:center; background:#f0e0e0;"
!8
|<!-- [x3x3o3o3o3o4o] -->{{CDD|node|4|node|3|node|3|node|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1</sub>{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
|-  style="text-align:center; background:#f0e0e0;"
!9
|<!-- [x3o3x3o3o3o4o] -->{{CDD|node|4|node|3|node|3|node|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2</sub>{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
|-  style="text-align:center; background:#f0e0e0;"
!10
|<!-- [o3x3x3o3o3o4o] -->{{CDD|node|4|node|3|node|3|node|3|node_1|3|node_1|3|node}}<BR>t<sub>1,2</sub>{3,3,3,3,3,4}||[[Bitruncated 7-orthoplex]] (Botaz)|||(0,0,0,0,1,2,2)√2|||||| || || ||4200||840
|-  style="text-align:center; background:#f0e0e0;"
!11
|<!-- [x3o3o3x3o3o4o] -->{{CDD|node|4|node|3|node|3|node_1|3|node|3|node|3|node_1}}<BR>t<sub>0,3</sub>{3,3,3,3,3,4}||[[Runcinated 7-orthoplex]] (Spaz)|||(0,0,0,1,1,1,2)√2|||||| || || ||23520||2240
|-  style="text-align:center; background:#f0e0e0;"
!12
|<!-- [o3x3o3x3o3o4o] -->{{CDD|node|4|node|3|node|3|node_1|3|node|3|node_1|3|node}}<BR>t<sub>1,3</sub>{3,3,3,3,3,4}||[[Bicantellated 7-orthoplex]] (Sebraz)|||(0,0,0,1,1,2,2)√2|||||| || || ||26880||3360
|-  style="text-align:center; background:#f0e0e0;"
!13
|<!-- [o3o3x3x3o3o4o] -->{{CDD|node|4|node|3|node|3|node_1|3|node_1|3|node|3|node}}<BR>t<sub>2,3</sub>{3,3,3,3,3,4}||[[Tritruncated 7-orthoplex]] (Totaz)|||(0,0,0,1,2,2,2)√2|||||| || || ||10080||2240
|-  style="text-align:center; background:#f0e0e0;"
!14
|<!-- [x3o3o3o3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node|3|node|3|node|3|node_1}}<BR>t<sub>0,4</sub>{3,3,3,3,3,4}||[[Stericated 7-orthoplex]] (Scaz)|||(0,0,1,1,1,1,2)√2|||||| || || ||33600||3360
|-  style="text-align:center; background:#f0e0e0;"
!15
|<!-- [o3x3o3o3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node|3|node|3|node_1|3|node}}<BR>t<sub>1,4</sub>{3,3,3,3,3,4}||[[Biruncinated 7-orthoplex]] (Sibpaz)|||(0,0,1,1,1,2,2)√2|||||| || || ||60480||6720
|-  style="text-align:center; background:#e0f0e0;"
!16
|<!-- [o3o3x3o3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node|3|node_1|3|node|3|node}}<BR>t<sub>2,4</sub>{4,3,3,3,3,3}||[[Tricantellated 7-cube]] (Strasaz)|||(0,0,1,1,2,2,2)√2|||||| || || ||47040||6720
|-  style="text-align:center; background:#e0e0f0;"
!17
|<!-- [o3o3o3x3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node_1|3|node|3|node|3|node}}<BR>t<sub>2,3</sub>{4,3,3,3,3,3}||[[Tritruncated 7-cube]] (Tatsa)|||(0,0,1,2,2,2,2)√2|||||| || || ||13440||3360
|-  style="text-align:center; background:#f0e0e0;"
!18
|<!-- [x3o3o3o3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node|3|node|3|node|3|node_1}}<BR>t<sub>0,5</sub>{3,3,3,3,3,4}||[[Pentellated 7-orthoplex]] (Staz)|||(0,1,1,1,1,1,2)√2|||||| || || ||20160||2688
|-  style="text-align:center; background:#e0f0e0;"
!19
|<!-- [o3x3o3o3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node|3|node|3|node_1|3|node}}<BR>t<sub>1,5</sub>{4,3,3,3,3,3}||[[Bistericated 7-cube]] (Sabcosaz)|||(0,1,1,1,1,2,2)√2|||||| || || ||53760||6720
|-  style="text-align:center; background:#e0e0f0;"
!20
|<!-- [o3o3x3o3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node|3|node_1|3|node|3|node}}<BR>t<sub>1,4</sub>{4,3,3,3,3,3}||[[Biruncinated 7-cube]] (Sibposa)|||(0,1,1,1,2,2,2)√2|||||| || || ||67200||8960
|-  style="text-align:center; background:#e0e0f0;"
!21
|<!-- [o3o3o3x3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node|3|node}}<BR>t<sub>1,3</sub>{4,3,3,3,3,3}||[[Bicantellated 7-cube]] (Sibrosa)|||(0,1,1,2,2,2,2)√2|||||| || || ||40320||6720
|-  style="text-align:center; background:#e0e0f0;"
!22
|<!-- [o3o3o3o3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node|3|node}}<BR>t<sub>1,2</sub>{4,3,3,3,3,3}||[[Bitruncated 7-cube]] (Betsa)|||(0,1,2,2,2,2,2)√2|||||| || || ||9408||2688
|-  style="text-align:center; background:#e0f0e0;"
!23
|<!-- [x3o3o3o3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node_1}}<BR>t<sub>0,6</sub>{4,3,3,3,3,3}||[[Hexicated 7-cube]] (Suposaz)|||(0,0,0,0,0,0,1)√2 + (1,1,1,1,1,1,1)|||||| || || ||5376||896
|-  style="text-align:center; background:#e0e0f0;"
!24
|<!-- [o3x3o3o3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node|3|node|3|node_1|3|node}}<BR>t<sub>0,5</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!25
|<!-- [o3o3x3o3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node|3|node_1|3|node|3|node}}<BR>t<sub>0,4</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!26
|<!-- [o3o3o3x3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node|3|node}}<BR>t<sub>0,3</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!27
|<!-- [o3o3o3o3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node|3|node}}<BR>t<sub>0,2</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!28
|<!-- [o3o3o3o3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node|3|node}}<BR>t<sub>0,1</sub>{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
|-  style="text-align:center; background:#f0e0e0;"
!29
|<!-- [x3x3x3o3o3o4o] -->{{CDD|node|4|node|3|node|3|node|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2</sub>{3,3,3,3,3,4}||[[Cantitruncated 7-orthoplex]] (Garz)|||(0,1,2,3,3,3,3)√2|||||| || || ||8400||1680
|-  style="text-align:center; background:#f0e0e0;"
!30
|<!-- [x3x3o3x3o3o4o] -->{{CDD|node|4|node|3|node|3|node_1|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,3</sub>{3,3,3,3,3,4}||[[Runcitruncated 7-orthoplex]] (Potaz)|||(0,1,2,2,3,3,3)√2|||||| || || ||50400||6720
|-  style="text-align:center; background:#f0e0e0;"
!31
|<!-- [x3o3x3x3o3o4o] -->{{CDD|node|4|node|3|node|3|node_1|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,3</sub>{3,3,3,3,3,4}||[[Runcicantellated 7-orthoplex]] (Parz)|||(0,1,1,2,3,3,3)√2|||||| || || ||33600||6720
|-  style="text-align:center; background:#f0e0e0;"
!32
|<!-- [o3x3x3x3o3o4o] -->{{CDD|node|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node}}<BR>t<sub>1,2,3</sub>{3,3,3,3,3,4}||[[Bicantitruncated 7-orthoplex]] (Gebraz)|||(0,0,1,2,3,3,3)√2|||||| || || ||30240||6720
|-  style="text-align:center; background:#f0e0e0;"
!33
|<!-- [x3x3o3o3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,4</sub>{3,3,3,3,3,4}||[[Steritruncated 7-orthoplex]] (Catz)|||(0,0,1,1,1,2,3)√2|||||| || || ||107520||13440
|-  style="text-align:center; background:#f0e0e0;"
!34
|<!-- [x3o3x3o3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,4</sub>{3,3,3,3,3,4}||[[Stericantellated 7-orthoplex]] (Craze)|||(0,0,1,1,2,2,3)√2|||||| || || ||141120||20160
|-  style="text-align:center; background:#f0e0e0;"
!35
|<!-- [o3x3x3o3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node}}<BR>t<sub>1,2,4</sub>{3,3,3,3,3,4}||[[Biruncitruncated 7-orthoplex]] (Baptize)|||(0,0,1,1,2,3,3)√2|||||| || || ||120960||20160
|-  style="text-align:center; background:#f0e0e0;"
!36
|<!-- [x3o3o3x3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node_1|3|node|3|node|3|node_1}}<BR>t<sub>0,3,4</sub>{3,3,3,3,3,4}||[[Steriruncinated 7-orthoplex]] (Copaz)|||(0,1,1,1,2,3,3)√2|||||| || || ||67200||13440
|-  style="text-align:center; background:#f0e0e0;"
!37
|<!-- [o3x3o3x3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node}}<BR>t<sub>1,3,4</sub>{3,3,3,3,3,4}||[[Biruncicantellated 7-orthoplex]] (Boparz)|||(0,0,1,2,2,3,3)√2|||||| || || ||100800||20160
|-  style="text-align:center; background:#e0f0e0;"
!38
|<!-- [o3o3x3x3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}<BR>t<sub>2,3,4</sub>{4,3,3,3,3,3}||[[Tricantitruncated 7-cube]] (Gotrasaz)|||(0,0,0,1,2,3,3)√2|||||| || || ||53760||13440
|-  style="text-align:center; background:#f0e0e0;"
!39
|<!-- [x3x3o3o3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,5</sub>{3,3,3,3,3,4}||[[Pentitruncated 7-orthoplex]] (Tetaz)|||(0,1,1,1,1,2,3)√2|||||| || || ||87360||13440
|-  style="text-align:center; background:#f0e0e0;"
!40
|<!-- [x3o3x3o3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,5</sub>{3,3,3,3,3,4}||[[Penticantellated 7-orthoplex]] (Teroz)|||(0,1,1,1,2,2,3)√2|||||| || || ||188160||26880
|-  style="text-align:center; background:#f0e0e0;"
!41
|<!-- [o3x3x3o3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node|3|node_1|3|node_1|3|node}}<BR>t<sub>1,2,5</sub>{3,3,3,3,3,4}||[[Bisteritruncated 7-orthoplex]] (Boctaz)|||(0,1,1,1,2,3,3)√2|||||| || || ||147840||26880
|-  style="text-align:center; background:#f0e0e0;"
!42
|<!-- [x3o3o3x3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node|3|node_1}}<BR>t<sub>0,3,5</sub>{3,3,3,3,3,4}||[[Pentiruncinated 7-orthoplex]] (Topaz)|||(0,1,1,2,2,2,3)√2|||||| || || ||174720||26880
|-  style="text-align:center; background:#e0f0e0;"
!43
|<!-- [o3x3o3x3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node_1|3|node}}<BR>t<sub>1,3,5</sub>{4,3,3,3,3,3}||[[Bistericantellated 7-cube]] (Bacresaz)|||(0,1,1,2,2,3,3)√2|||||| || || ||241920||40320
|-  style="text-align:center; background:#e0e0f0;"
!44
|<!-- [o3o3x3x3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node_1|3|node_1|3|node|3|node}}<BR>t<sub>1,3,4</sub>{4,3,3,3,3,3}||[[Biruncicantellated 7-cube]] (Bopresa)|||(0,1,1,2,3,3,3)√2|||||| || || ||120960||26880
|-  style="text-align:center; background:#f0e0e0;"
!45
|<!-- [x3o3o3o3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node|3|node_1}}<BR>t<sub>0,4,5</sub>{3,3,3,3,3,4}||[[Pentistericated 7-orthoplex]] (Tocaz)|||(0,1,2,2,2,2,3)√2|||||| || || ||67200||13440
|-  style="text-align:center; background:#e0e0f0;"
!46
|<!-- [o3x3o3o3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node_1|3|node}}<BR>t<sub>1,2,5</sub>{4,3,3,3,3,3}||[[Bisteritruncated 7-cube]] (Bactasa)|||(0,1,2,2,2,3,3)√2|||||| || || ||147840||26880
|-  style="text-align:center; background:#e0e0f0;"
!47
|<!-- [o3o3x3o3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node|3|node}}<BR>t<sub>1,2,4</sub>{4,3,3,3,3,3}||[[Biruncitruncated 7-cube]] (Biptesa)|||(0,1,2,2,3,3,3)√2|||||| || || ||134400||26880
|-  style="text-align:center; background:#e0e0f0;"
!48
|<!-- [o3o3o3x3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node|3|node}}<BR>t<sub>1,2,3</sub>{4,3,3,3,3,3}||[[Bicantitruncated 7-cube]] (Gibrosa)|||(0,1,2,3,3,3,3)√2|||||| || || ||47040||13440
|-  style="text-align:center; background:#f0e0e0;"
!49
|<!-- [x3x3o3o3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,6</sub>{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
|-  style="text-align:center; background:#f0e0e0;"
!50
|<!-- [x3o3x3o3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,6</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!51
|<!-- [o3x3x3o3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node|3|node_1|3|node_1|3|node}}<BR>t<sub>0,4,5</sub>{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
|-  style="text-align:center; background:#e0f0e0;"
!52
|<!-- [x3o3o3x3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node|3|node_1}}<BR>t<sub>0,3,6</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!53
|<!-- [o3x3o3x3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node_1|3|node}}<BR>t<sub>0,3,5</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!54
|<!-- [o3o3x3x3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node_1|3|node_1|3|node|3|node}}<BR>t<sub>0,3,4</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!55
|<!-- [x3o3o3o3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node|3|node_1}}<BR>t<sub>0,2,6</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!56
|<!-- [o3x3o3o3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node_1|3|node}}<BR>t<sub>0,2,5</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!57
|<!-- [o3o3x3o3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node|3|node_1|3|node|3|node}}<BR>t<sub>0,2,4</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!58
|<!-- [o3o3o3x3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node|3|node}}<BR>t<sub>0,2,3</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!59
|<!-- [x3o3o3o3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node|3|node_1}}<BR>t<sub>0,1,6</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!60
|<!-- [o3x3o3o3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node_1|3|node}}<BR>t<sub>0,1,5</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!61
|<!-- [o3o3x3o3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node|3|node_1|3|node|3|node}}<BR>t<sub>0,1,4</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!62
|<!-- [o3o3o3x3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node|3|node}}<BR>t<sub>0,1,3</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!63
|<!-- [o3o3o3o3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node|3|node}}<BR>t<sub>0,1,2</sub>{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
|-  style="text-align:center; background:#f0e0e0;"
!64
|<!-- [x3x3x3x3o3o4o] -->{{CDD|node|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,3</sub>{3,3,3,3,3,4}||[[Runcicantitruncated 7-orthoplex]] (Gopaz)|||(0,1,2,3,4,4,4)√2|||||| || || ||60480||13440
|-  style="text-align:center; background:#f0e0e0;"
!65
|<!-- [x3x3x3o3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,4</sub>{3,3,3,3,3,4}||[[Stericantitruncated 7-orthoplex]] (Cogarz)|||(0,0,1,1,2,3,4)√2|||||| || || ||241920||40320
|-  style="text-align:center; background:#f0e0e0;"
!66
|<!-- [x3x3o3x3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,3,4</sub>{3,3,3,3,3,4}||[[Steriruncitruncated 7-orthoplex]] (Captaz)|||(0,0,1,2,2,3,4)√2|||||| || || ||181440||40320
|-  style="text-align:center; background:#f0e0e0;"
!67
|<!-- [x3o3x3x3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,3,4</sub>{3,3,3,3,3,4}||[[Steriruncicantellated 7-orthoplex]] (Caparz)|||(0,0,1,2,3,3,4)√2|||||| || || ||181440||40320
|-  style="text-align:center; background:#f0e0e0;"
!68
|<!-- [o3x3x3x3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}<BR>t<sub>1,2,3,4</sub>{3,3,3,3,3,4}||[[Biruncicantitruncated 7-orthoplex]] (Gibpaz)|||(0,0,1,2,3,4,4)√2|||||| || || ||161280||40320
|-  style="text-align:center; background:#f0e0e0;"
!69
|<!-- [x3x3x3o3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,5</sub>{3,3,3,3,3,4}||[[Penticantitruncated 7-orthoplex]] (Tograz)|||(0,1,1,1,2,3,4)√2|||||| || || ||295680||53760
|-  style="text-align:center; background:#f0e0e0;"
!70
|<!-- [x3x3o3x3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,3,5</sub>{3,3,3,3,3,4}||[[Pentiruncitruncated 7-orthoplex]] (Toptaz)|||(0,1,1,2,2,3,4)√2|||||| || || ||443520||80640
|-  style="text-align:center; background:#f0e0e0;"
!71
|<!-- [x3o3x3x3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node_1|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,3,5</sub>{3,3,3,3,3,4}||[[Pentiruncicantellated 7-orthoplex]] (Toparz)|||(0,1,1,2,3,3,4)√2|||||| || || ||403200||80640
|-  style="text-align:center; background:#f0e0e0;"
!72
|<!-- [o3x3x3x3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node}}<BR>t<sub>1,2,3,5</sub>{3,3,3,3,3,4}||[[Bistericantitruncated 7-orthoplex]] (Becogarz)|||(0,1,1,2,3,4,4)√2|||||| || || ||362880||80640
|-  style="text-align:center; background:#f0e0e0;"
!73
|<!-- [x3x3o3o3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,4,5</sub>{3,3,3,3,3,4}||[[Pentisteritruncated 7-orthoplex]] (Tacotaz)|||(0,1,2,2,2,3,4)√2|||||| || || ||241920||53760
|-  style="text-align:center; background:#f0e0e0;"
!74
|<!-- [x3o3x3o3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,4,5</sub>{3,3,3,3,3,4}||[[Pentistericantellated 7-orthoplex]] (Tocarz)|||(0,1,2,2,3,3,4)√2|||||| || || ||403200||80640
|-  style="text-align:center; background:#e0f0e0;"
!75
|<!-- [o3x3x3o3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node}}<BR>t<sub>1,2,4,5</sub>{4,3,3,3,3,3}||[[Bisteriruncitruncated 7-cube]] (Bocaptosaz)|||(0,1,2,2,3,4,4)√2|||||| |||| ||322560||80640
|-  style="text-align:center; background:#f0e0e0;"
!76
|<!-- [x3o3o3x3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node|3|node_1}}<BR>t<sub>0,3,4,5</sub>{3,3,3,3,3,4}||[[Pentisteriruncinated 7-orthoplex]] (Tecpaz)|||(0,1,2,3,3,3,4)√2|||||| || || ||241920||53760
|-  style="text-align:center; background:#e0e0f0;"
!77
|<!-- [o3x3o3x3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node}}<BR>t<sub>1,2,3,5</sub>{4,3,3,3,3,3}||[[Bistericantitruncated 7-cube]] (Becgresa)|||(0,1,2,3,3,4,4)√2|||||| || || ||362880||80640
|-  style="text-align:center; background:#e0e0f0;"
!78
|<!-- [o3o3x3x3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node}}<BR>t<sub>1,2,3,4</sub>{4,3,3,3,3,3}||[[Biruncicantitruncated 7-cube]] (Gibposa)|||(0,1,2,3,4,4,4)√2|||||| || || ||188160||53760
|-  style="text-align:center; background:#f0e0e0;"
!79
|<!-- [x3x3x3o3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,6</sub>{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
|-  style="text-align:center; background:#f0e0e0;"
!80
|<!-- [x3x3o3x3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,3,6</sub>{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
|-  style="text-align:center; background:#f0e0e0;"
!81
|<!-- [x3o3x3x3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node_1|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,3,6</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!82
|<!-- [o3x3x3x3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node}}<BR>t<sub>0,3,4,5</sub>{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
|-  style="text-align:center; background:#f0e0e0;"
!83
|<!-- [x3x3o3o3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,4,6</sub>{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
|-  style="text-align:center; background:#e0f0e0;"
!84
|<!-- [x3o3x3o3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,4,6</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!85
|<!-- [o3x3x3o3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node}}<BR>t<sub>0,2,4,5</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!86
|<!-- [x3o3o3x3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node|3|node_1}}<BR>t<sub>0,2,3,6</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!87
|<!-- [o3x3o3x3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node}}<BR>t<sub>0,2,3,5</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!88
|<!-- [o3o3x3x3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}<BR>t<sub>0,2,3,4</sub>{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
|-  style="text-align:center; background:#e0f0e0;"
!89
|<!-- [x3x3o3o3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,5,6</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!90
|<!-- [x3o3x3o3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node|3|node_1|3|node|3|node_1}}<BR>t<sub>0,1,4,6</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!91
|<!-- [o3x3x3o3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node|3|node_1|3|node_1|3|node}}<BR>t<sub>0,1,4,5</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!92
|<!-- [x3o3o3x3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node|3|node_1}}<BR>t<sub>0,1,3,6</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!93
|<!-- [o3x3o3x3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node_1|3|node}}<BR>t<sub>0,1,3,5</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!94
|<!-- [o3o3x3x3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node_1|3|node_1|3|node|3|node}}<BR>t<sub>0,1,3,4</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!95
|<!-- [x3o3o3o3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node|3|node_1}}<BR>t<sub>0,1,2,6</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!96
|<!-- [o3x3o3o3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node_1|3|node}}<BR>t<sub>0,1,2,5</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!97
|<!-- [o3o3x3o3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node|3|node_1|3|node|3|node}}<BR>t<sub>0,1,2,4</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!98
|<!-- [o3o3o3x3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node|3|node}}<BR>t<sub>0,1,2,3</sub>{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
|-  style="text-align:center; background:#f0e0e0;"
!99
|<!-- [x3x3x3x3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,3,4</sub>{3,3,3,3,3,4}||[[Steriruncicantitruncated 7-orthoplex]] (Gocaz)|||(0,0,1,2,3,4,5)√2|||||| || || ||322560||80640
|-  style="text-align:center; background:#f0e0e0;"
!100
|<!-- [x3x3x3x3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,3,5</sub>{3,3,3,3,3,4}||[[Pentiruncicantitruncated 7-orthoplex]] (Tegopaz)|||(0,1,1,2,3,4,5)√2|||||| || || ||725760||161280
|-  style="text-align:center; background:#f0e0e0;"
!101
|<!-- [x3x3x3o3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,4,5</sub>{3,3,3,3,3,4}||[[Pentistericantitruncated 7-orthoplex]] (Tecagraz)|||(0,1,2,2,3,4,5)√2|||||| || || ||645120||161280
|-  style="text-align:center; background:#f0e0e0;"
!102
|<!-- [x3x3o3x3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,3,4,5</sub>{3,3,3,3,3,4}||[[Pentisteriruncitruncated 7-orthoplex]] (Tecpotaz)|||(0,1,2,3,3,4,5)√2|||||| || || ||645120||161280
|-  style="text-align:center; background:#f0e0e0;"
!103
|<!-- [x3o3x3x3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,3,4,5</sub>{3,3,3,3,3,4}||[[Pentisteriruncicantellated 7-orthoplex]] (Tacparez)|||(0,1,2,3,4,4,5)√2|||||| || || ||645120||161280
|-  style="text-align:center; background:#e0f0e0;"
!104
|<!-- [o3x3x3x3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}<BR>t<sub>1,2,3,4,5</sub>{4,3,3,3,3,3}||[[Bisteriruncicantitruncated 7-cube]] (Gabcosaz)|||(0,1,2,3,4,5,5)√2|||||| || || ||564480||161280
|-  style="text-align:center; background:#f0e0e0;"
!105
|<!-- [x3x3x3x3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,3,6</sub>{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
|-  style="text-align:center; background:#f0e0e0;"
!106
|<!-- [x3x3x3o3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,4,6</sub>{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
|-  style="text-align:center; background:#f0e0e0;"
!107
|<!-- [x3x3o3x3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,3,4,6</sub>{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
|-  style="text-align:center; background:#e0f0e0;"
!108
|<!-- [x3o3x3x3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,3,4,6</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!109
|<!-- [o3x3x3x3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}<BR>t<sub>0,2,3,4,5</sub>{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
|-  style="text-align:center; background:#f0e0e0;"
!110
|<!-- [x3x3x3o3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,5,6</sub>{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
|-  style="text-align:center; background:#e0f0e0;"
!111
|<!-- [x3x3o3x3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,3,5,6</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!112
|<!-- [x3o3x3x3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node_1|3|node_1|3|node|3|node_1}}<BR>t<sub>0,1,3,4,6</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!113
|<!-- [o3x3x3x3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node}}<BR>t<sub>0,1,3,4,5</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!114
|<!-- [x3x3o3o3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,2,5,6</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!115
|<!-- [x3o3x3o3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node|3|node_1|3|node|3|node_1}}<BR>t<sub>0,1,2,4,6</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!116
|<!-- [o3x3x3o3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node}}<BR>t<sub>0,1,2,4,5</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!117
|<!-- [x3o3o3x3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node|3|node_1}}<BR>t<sub>0,1,2,3,6</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!118
|<!-- [o3x3o3x3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node}}<BR>t<sub>0,1,2,3,5</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!119
|<!-- [o3o3x3x3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node}}<BR>t<sub>0,1,2,3,4</sub>{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
|-  style="text-align:center; background:#f0e0e0;"
!120
|<!-- [x3x3x3x3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,3,4,5</sub>{3,3,3,3,3,4}||[[Pentisteriruncicantitruncated 7-orthoplex]] (Gotaz)|||(0,1,2,3,4,5,6)√2|||||| || || ||1128960||322560
|-  style="text-align:center; background:#f0e0e0;"
!121
|<!-- [x3x3x3x3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,3,4,6</sub>{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
|-  style="text-align:center; background:#f0e0e0;"
!122
|<!-- [x3x3x3x3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,3,5,6</sub>{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
|-  style="text-align:center; background:#e0f0e0;"
!123
|<!-- [x3x3x3o3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,4,5,6</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!124
|<!-- [x3x3o3x3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,2,3,5,6</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!125
|<!-- [x3o3x3x3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}<BR>t<sub>0,1,2,3,4,6</sub>{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
|-  style="text-align:center; background:#e0e0f0;"
!126
|<!-- [o3x3x3x3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}<BR>t<sub>0,1,2,3,4,5</sub>{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
|-  style="text-align:center; background:#e0f0e0;"
!127
|<!-- [x3x3x3x3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,3,4,5,6</sub>{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<sub>7</sub> family ==
 
The D<sub>7</sub> family has symmetry of order 322560 (7 [[factorial]] x 2<sup>6</sup>).
 
This family has 3×32−1=95 Wythoffian uniform polytopes, generated by marking one or more nodes of the D<sub>7</sub> [[Coxeter-Dynkin diagram]]. Of these, 63 (2×32−1) are repeated from the B<sub>7</sub> 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.
{| class="wikitable"
!rowspan=2|#
!rowspan=2|[[Coxeter diagram]]
!rowspan=2|Names
!rowspan=2|Base point<BR>(Alternately signed)
!colspan=7|Element counts
|-
!6||5||4||3||2||1||0
|- align=center
!1
||{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node}}||[[7-demicube]]<BR>Demihepteract (Hesa)||(1,1,1,1,1,1,1)||78||532||1624||2800||2240||672||64
|- align=center
!2
||{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node|3|node}}||[[Truncated 7-demicube]]<BR>Truncated demihepteract (Thesa)||(1,1,3,3,3,3,3)||142||1428||5656||11760||13440||7392||1344
|- align=center
!3
||{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node|3|node}}||[[Cantellated 7-demicube]]<BR>Small rhombated demihepteract (Sirhesa)||(1,1,1,3,3,3,3)|| || || || || ||16800||2240
|- align=center
!4
||{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node|3|node}}||[[Runcinated 7-demicube]]<BR>Small prismated demihepteract (Sphosa)||(1,1,1,1,3,3,3)|| || || || || ||20160||2240
|- align=center
!5
||{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node_1|3|node}}||[[Stericated 7-demicube]]<BR>Small cellated demihepteract (Sochesa)||(1,1,1,1,1,3,3)|| || || || || ||13440||1344
|- align=center
!6
||{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node_1}}||[[Pentellated 7-demicube]]<BR>Small terated demihepteract (Suthesa)||(1,1,1,1,1,1,3)|| || || || || ||4704||448
|- align=center
!7
||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node|3|node}}||[[Cantitruncated 7-demicube]]<BR>Great rhombated demihepteract (Girhesa)||(1,1,3,5,5,5,5)|| || || || || ||23520||6720
|- align=center
!8
||{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node|3|node}}||[[Runcitruncated 7-demicube]]<BR>Prismatotruncated demihepteract (Pothesa)||(1,1,3,3,5,5,5)|| || || || || ||73920||13440
|- align=center
!9
||{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node|3|node}}||[[Runcicantellated 7-demicube]]<BR>Prismatorhomated demihepteract (Prohesa)||(1,1,1,3,5,5,5)|| || || || || ||40320||8960
|- align=center
!10
||{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node_1|3|node}}||[[Steritruncated 7-demicube]]<BR>Cellitruncated demihepteract (Cothesa)||(1,1,3,3,3,5,5)|| || || || || ||87360||13440
|- align=center
!11
||{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node_1|3|node}}||[[Stericantellated 7-demicube]]<BR>Cellirhombated demihepteract (Crohesa)||(1,1,1,3,3,5,5)|| || || || || ||87360||13440
|- align=center
!12
||{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node_1|3|node}}||[[Steriruncinated 7-demicube]]<BR>Celliprismated demihepteract (Caphesa)||(1,1,1,1,3,5,5)|| || || || || ||40320||6720
|- align=center
!13
||{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node|3|node_1}}||[[Pentitruncated 7-demicube]]<BR>Teritruncated demihepteract (Tuthesa)||(1,1,3,3,3,3,5)|| || || || || ||43680||6720
|- align=center
!14
||{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node|3|node_1}}||[[Penticantellated 7-demicube]]<BR>Terirhombated demihepteract (Turhesa)||(1,1,1,3,3,3,5)|| || || || || ||67200||8960
|- align=center
!15
||{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node|3|node_1}}||[[Pentiruncinated 7-demicube]]<BR>Teriprismated demihepteract (Tuphesa)||(1,1,1,1,3,3,5)|| || || || || ||53760||6720
|- align=center
!16
||{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node_1|3|node_1}}||[[Pentistericated 7-demicube]]<BR>Tericellated demihepteract (Tuchesa)||(1,1,1,1,1,3,5)|| || || || || ||21504||2688
|- align=center
!17
||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}||[[Runcicantitruncated 7-demicube]]<BR>Great prismated demihepteract (Gephosa)||(1,1,3,5,7,7,7)|| || || || || ||94080||26880
|- align=center
!18
||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node}}||[[Stericantitruncated 7-demicube]]<BR>Celligreatorhombated demihepteract (Cagrohesa)||(1,1,3,5,5,7,7)|| || || || || ||181440||40320
|- align=center
!19
||{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node}}||[[Steriruncitruncated 7-demicube]]<BR>Celliprismatotruncated demihepteract (Capthesa)||(1,1,3,3,5,7,7)|| || || || || ||181440||40320
|- align=center
!20
||{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node}}||[[Steriruncicantellated 7-demicube]]<BR>Celliprismatorhombated demihepteract (Coprahesa)||(1,1,1,3,5,7,7)|| || || || || ||120960||26880
|- align=center
!21
||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node|3|node_1}}||[[Penticantitruncated 7-demicube]]<BR>Terigreatorhombated demihepteract (Tugrohesa)||(1,1,3,5,5,5,7)|| || || || || ||120960||26880
|- align=center
!22
||{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node|3|node_1}}||[[Pentiruncitruncated 7-demicube]]<BR>Teriprismatotruncated demihepteract (Tupthesa)||(1,1,3,3,5,5,7)|| || || || || ||221760||40320
|- align=center
!23
||{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node|3|node_1}}||[[Pentiruncicantellated 7-demicube]]<BR>Teriprismatorhombated demihepteract (Tuprohesa)||(1,1,1,3,5,5,7)|| || || || || ||134400||26880
|- align=center
!24
||{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node_1|3|node_1}}||[[Pentisteritruncated 7-demicube]]<BR>Tericellitruncated demihepteract (Tucothesa)||(1,1,3,3,3,5,7)|| || || || || ||147840||26880
|- align=center
!25
||{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node_1|3|node_1}}||[[Pentistericantellated 7-demicube]]<BR>Tericellirhombated demihepteract (Tucrohesa)||(1,1,1,3,3,5,7)|| || || || || ||161280||26880
|- align=center
!26
||{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node_1|3|node_1}}||[[Pentisteriruncinated 7-demicube]]<BR>Tericelliprismated demihepteract (Tucophesa)||(1,1,1,1,3,5,7)|| || || || || ||80640||13440
|- align=center
!27
||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}||[[Steriruncicantitruncated 7-demicube]]<BR>Great cellated demihepteract (Gochesa)||(1,1,3,5,7,9,9)|| || || || || ||282240||80640
|- align=center
!28
||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}||[[Pentiruncicantitruncated 7-demicube]]<BR>Terigreatoprimated demihepteract (Tugphesa)||(1,1,3,5,7,7,9)|| || || || || ||322560||80640
|- align=center
!29
||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}||[[Pentistericantitruncated 7-demicube]]<BR>Tericelligreatorhombated demihepteract (Tucagrohesa)||(1,1,3,5,5,7,9)|| || || || || ||322560||80640
|- align=center
!30
||{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}||[[Pentisteriruncitruncated 7-demicube]]<BR>Tericelliprismatotruncated demihepteract (Tucpathesa)||(1,1,3,3,5,7,9)|| || || || || ||362880||80640
|- align=center
!31
||{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}||[[Pentisteriruncicantellated 7-demicube]]<BR>Tericellprismatorhombated demihepteract (Tucprohesa)||(1,1,1,3,5,7,9)|| || || || || ||241920||53760
|- align=center
!32
||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}||[[Pentisteriruncicantitruncated 7-demicube]]<BR>Great terated demihepteract (Guthesa)||(1,1,3,5,7,9,11)|| || || || || ||564480||161280
|}
 
== The E<sub>7</sub> family ==
 
The E<sub>7</sub> [[Coxeter group]] has order 2,903,040.
 
There are 127 forms based on all permutations of the [[Coxeter-Dynkin diagram]]s with one or more rings.  
 
See also a [[list of E7 polytopes]] for symmetric Coxeter plane graphs of these polytopes.
 
{| class="wikitable"
|-
!rowspan=2|#
!rowspan=2|[[Coxeter-Dynkin diagram]]<br />[[Schläfli symbol]]
!rowspan=2|Names
!colspan=7|Element counts
|-
! 6|| 5|| 4|| 3|| 2|| 1|| 0
 
|- style="text-align:center;"
|1||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}||[[Gosset 2 31 polytope|2<sub>31</sub>]] (laq)||632||4788||16128||20160||10080||2016||126
|- style="text-align:center;"
|2||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea_1|3a|nodea}}||[[Rectified 2 31 polytope|Rectified 2<sub>31</sub>]] (rolaq)||758||10332||47880||100800||90720||30240||2016
|- style="text-align:center;"
|3||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}||[[Rectified 1 32 polytope|Rectified 1<sub>32</sub>]] (rolin)||758||12348||72072||191520||241920||120960||10080
|- style="text-align:center;"
|4||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}||[[1 32 polytope|1<sub>32</sub>]] (lin)||182||4284||23688||50400||40320||10080||576
|- style="text-align:center;"
|5||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}}||[[Birectified 3 21 polytope|Birectified 3<sub>21</sub>]] (branq)||758||12348||68040||161280||161280||60480||4032
|- style="text-align:center;"
|6||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea}}||[[Rectified 3 21 polytope|Rectified 3<sub>21</sub>]] (ranq)||758||44352||70560||48384||11592||12096||756
|- style="text-align:center;"
|7||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}||[[Gosset 3 21 polytope|3<sub>21</sub>]] (naq)||702||6048||12096||10080||4032||756||56
|- align=center
|8||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea_1|3a|nodea_1}}||[[Truncated 2 31 polytope|Truncated 2<sub>31</sub>]] (talq)||758||10332||47880||100800||90720||32256||4032
|- align=center
|9||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea_1}}||[[Cantellated 2 31 polytope|Cantellated 2<sub>31</sub>]] (sirlaq)|| || || || || ||131040||20160
|- align=center
|10||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea}}||Bitruncated 2<sub>31</sub> (botlaq)||||||||||||||30240
|- align=center
|11||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea_1}}||small demified 2<sub>31</sub> (shilq)||2774||22428||78120||151200||131040||42336||4032
|- align=center
|12||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea}}||demirectified 2<sub>31</sub> (hirlaq)||||||||||||||12096
|- align=center
|13||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}||truncated 1<sub>32</sub> (tolin)||||||||||||||20160
|- align=center
|14||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea_1}}||small demiprismated 2<sub>31</sub> (shiplaq)||||||||||||||20160
|- align=center
|15||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea}}||birectified 1<sub>32</sub> (berlin)||758||22428||142632||403200||544320||302400||40320
|- align=center
|16||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea}}||tritruncated 3<sub>21</sub> (totanq)||||||||||||||40320
|- align=center
|17||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea}}||demibirectified 3<sub>21</sub> (hobranq)||||||||||||||20160
|- align=center
|18||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}||small cellated 2<sub>31</sub> (scalq)||||||||||||||7560
|- align=center
|19||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch|3a|nodea_1|3a|nodea}}||small biprismated 2<sub>31</sub> (sobpalq)||||||||||||||30240
|- align=center
|20||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}||small birhombated 3<sub>21</sub> (sabranq)||||||||||||||60480
|- align=center
|21||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}||demirectified 3<sub>21</sub> (harnaq)||||||||||||||12096
|- align=center
|22||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}}||bitruncated 3<sub>21</sub> (botnaq)||||||||||||||12096
|- align=center
|23||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}||small terated 3<sub>21</sub> (stanq)||||||||||||||1512
|- align=center
|24||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea_1|3a|nodea}}||small demicellated 3<sub>21</sub> (shocanq)||||||||||||||12096
|- align=center
|25||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}||small prismated 3<sub>21</sub> (spanq)||||||||||||||40320
|- align=center
|26||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}||small demified 3<sub>21</sub> (shanq)||||||||||||||4032
|- align=center
|27||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}}||small rhombated 3<sub>21</sub> (sranq)||||||||||||||12096
|- align=center
|28||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea}}||[[Truncated 3 21 polytope|Truncated 3<sub>21</sub>]] (tanq)||758||11592||48384||70560||44352||12852||1512
|- align=center
|29||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea_1}}||great rhombated 2<sub>31</sub> (girlaq)||||||||||||||60480
|- align=center
|30||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||demitruncated 2<sub>31</sub> (hotlaq)||||||||||||||24192
|- align=center
|31||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_11|3a|nodea|3a|nodea_1}}||small demirhombated 2<sub>31</sub> (sherlaq)||||||||||||||60480
|- align=center
|32||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea}}||demibitruncated 2<sub>31</sub> (hobtalq)||||||||||||||60480
|- align=center
|33||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea_1}}||demiprismated 2<sub>31</sub> (hiptalq)||||||||||||||80640
|- align=center
|34||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea_1}}||demiprismatorhombated 2<sub>31</sub> (hiprolaq)||||||||||||||120960
|- align=center
|35||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea}}||bitruncated 1<sub>32</sub> (batlin)||||||||||||||120960
|- align=center
|36||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea_1}}||small prismated 2<sub>31</sub> (spalq)||||||||||||||80640
|- align=center
|37||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea}}||small rhombated 1<sub>32</sub> (sirlin)||||||||||||||120960
|- align=center
|38||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea}}||tritruncated 2<sub>31</sub> (tatilq)||||||||||||||80640
|- align=center
|39||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch|3a|nodea_1|3a|nodea_1}}||cellitruncated 2<sub>31</sub> (catalaq)||||||||||||||60480
|- align=center
|40||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea|3a|nodea_1}}||cellirhombated 2<sub>31</sub> (crilq)||||||||||||||362880
|- align=center
|41||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea}}||biprismatotruncated 2<sub>31</sub> (biptalq)||||||||||||||181440
|- align=center
|42||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea_1}}||small prismated 1<sub>32</sub> (seplin)||||||||||||||60480
|- align=center
|43||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea}}||small biprismated 3<sub>21</sub> (sabipnaq)||||||||||||||120960
|- align=center
|44||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}||small demibirhombated 3<sub>21</sub> (shobranq)||||||||||||||120960
|- align=center
|45||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea|3a|nodea_1}}||cellidemiprismated 2<sub>31</sub> (chaplaq)||||||||||||||60480
|- align=center
|46||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea}}||demibiprismatotruncated 3<sub>21</sub> (hobpotanq)||||||||||||||120960
|- align=center
|47||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea}}||great birhombated 3<sub>21</sub> (gobranq)||||||||||||||120960
|- align=center
|48||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea}}||demibitruncated 3<sub>21</sub> (hobtanq)||||||||||||||60480
|- align=center
|49||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea_1|3a|nodea_1}}||teritruncated 2<sub>31</sub> (totalq)||||||||||||||24192
|- align=center
|50||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea_1}}||terirhombated 2<sub>31</sub> (trilq)||||||||||||||120960
|- align=center
|51||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea}}||demicelliprismated 3<sub>21</sub> (hicpanq)||||||||||||||120960
|- align=center
|52||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea_1}}||small teridemified 2<sub>31</sub> (sethalq)||||||||||||||24192
|- align=center
|53||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea}}||small cellated 3<sub>21</sub> (scanq)||||||||||||||60480
|- align=center
|54||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}||demiprismated 3<sub>21</sub> (hipnaq)||||||||||||||80640
|- align=center
|55||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea_1}}||terirhombated 3<sub>21</sub> (tranq)||||||||||||||60480
|- align=center
|56||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea}}||demicellirhombated 3<sub>21</sub> (hocranq)||||||||||||||120960
|- align=center
|57||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea}}||prismatorhombated 3<sub>21</sub> (pranq)||||||||||||||120960
|- align=center
|58||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea}}||small demirhombated 3<sub>21</sub> (sharnaq)||||||||||||||60480
|- align=center
|59||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}||teritruncated 3<sub>21</sub> (tetanq)||||||||||||||15120
|- align=center
|60||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch|3a|nodea_1|3a|nodea}}||demicellitruncated 3<sub>21</sub> (hictanq)||||||||||||||60480
|- align=center
|61||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}||prismatotruncated 3<sub>21</sub> (potanq)||||||||||||||120960
|- align=center
|62||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}||demitruncated 3<sub>21</sub> (hotnaq)||||||||||||||24192
|- align=center
|63||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}}||great rhombated 3<sub>21</sub> (granq)||||||||||||||24192
|- align=center
|64||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea_1}}||great demified 2<sub>31</sub> (gahlaq)||||||||||||||120960
|- align=center
|65||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea_1}}||great demiprismated 2<sub>31</sub> (gahplaq)||||||||||||||241920
|- align=center
|66||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||prismatotruncated 2<sub>31</sub> (potlaq)||||||||||||||241920
|- align=center
|67||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea_1}}||prismatorhombated 2<sub>31</sub> (prolaq)||||||||||||||241920
|- align=center
|68||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea}}||great rhombated 1<sub>32</sub> (girlin)||||||||||||||241920
|- align=center
|69||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea_1}}||celligreatorhombated 2<sub>31</sub> (cagrilq)||||||||||||||362880
|- align=center
|70||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||cellidemitruncated 2<sub>31</sub> (chotalq)||||||||||||||241920
|- align=center
|71||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea|3a|nodea_1}}||prismatotruncated 1<sub>32</sub> (patlin)||||||||||||||362880
|- align=center
|72||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea}}||biprismatorhombated 3<sub>21</sub> (bipirnaq)||||||||||||||362880
|- align=center
|73||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea_1}}||tritruncated 1<sub>32</sub> (tatlin)||||||||||||||241920
|- align=center
|74||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea_1}}||cellidemiprismatorhombated 2<sub>31</sub> (chopralq)||||||||||||||362880
|- align=center
|75||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea}}||great demibiprismated 3<sub>21</sub> (ghobipnaq)||||||||||||||362880
|- align=center
|76||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea_1}}||celliprismated 2<sub>31</sub> (caplaq)||||||||||||||241920
|- align=center
|77||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea}}||biprismatotruncated 3<sub>21</sub> (boptanq)||||||||||||||362880
|- align=center
|78||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea}}||great trirhombated 2<sub>31</sub> (gatralaq)||||||||||||||241920
|- align=center
|79||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea_1}}||terigreatorhombated 2<sub>31</sub> (togrilq)||||||||||||||241920
|- align=center
|80||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||teridemitruncated 2<sub>31</sub> (thotalq)||||||||||||||120960
|- align=center
|81||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_11|3a|nodea|3a|nodea_1}}||teridemirhombated 2<sub>31</sub> (thorlaq)||||||||||||||241920
|- align=center
|82||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea}}||celliprismated 3<sub>21</sub> (capnaq)||||||||||||||241920
|- align=center
|83||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea_1}}||teridemiprismatotruncated 2<sub>31</sub> (thoptalq)||||||||||||||241920
|- align=center
|84||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea_1}}||teriprismatorhombated 3<sub>21</sub> (tapronaq)||||||||||||||362880
|- align=center
|85||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea}}||demicelliprismatorhombated 3<sub>21</sub> (hacpranq)||||||||||||||362880
|- align=center
|86||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea_1}}||teriprismated 2<sub>31</sub> (toplaq)||||||||||||||241920
|- align=center
|87||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea}}||cellirhombated 3<sub>21</sub> (cranq)||||||||||||||362880
|- align=center
|88||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea}}||demiprismatorhombated 3<sub>21</sub> (hapranq)||||||||||||||241920
|- align=center
|89||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch|3a|nodea_1|3a|nodea_1}}||tericellitruncated 2<sub>31</sub> (tectalq)||||||||||||||120960
|- align=center
|90||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea|3a|nodea_1}}||teriprismatotruncated 3<sub>21</sub> (toptanq)||||||||||||||362880
|- align=center
|91||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea}}||demicelliprismatotruncated 3<sub>21</sub> (hecpotanq)||||||||||||||362880
|- align=center
|92||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea_1}}||teridemitruncated 3<sub>21</sub> (thotanq)||||||||||||||120960
|- align=center
|93||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea}}||cellitruncated 3<sub>21</sub> (catnaq)||||||||||||||241920
|- align=center
|94||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}||demiprismatotruncated 3<sub>21</sub> (hiptanq)||||||||||||||241920
|- align=center
|95||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea|3a|nodea_1}}||terigreatorhombated 3<sub>21</sub> (tagranq)||||||||||||||120960
|- align=center
|96||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea}}||demicelligreatorhombated 3<sub>21</sub> (hicgarnq)||||||||||||||241920
|- align=center
|97||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea}}||great prismated 3<sub>21</sub> (gopanq)||||||||||||||241920
|- align=center
|98||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea}}||great demirhombated 3<sub>21</sub> (gahranq)||||||||||||||120960
|- align=center
|99||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea_1}}||great prismated 2<sub>31</sub> (gopalq)||||||||||||||483840
|- align=center
|100||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea_1}}||great cellidemified 2<sub>31</sub> (gechalq)||||||||||||||725760
|- align=center
|101||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea_1}}||great birhombated 1<sub>32</sub> (gebrolin)||||||||||||||725760
|- align=center
|102||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||prismatorhombated 1<sub>32</sub> (prolin)||||||||||||||725760
|- align=center
|103||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea_1}}||celliprismatorhombated 2<sub>31</sub> (caprolaq)||||||||||||||725760
|- align=center
|104||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea}}||great biprismated 2<sub>31</sub> (gobpalq)||||||||||||||725760
|- align=center
|105||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea_1}}||tericelliprismated 3<sub>21</sub> (ticpanq)||||||||||||||483840
|- align=center
|106||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea_1}}||teridemigreatoprismated 2<sub>31</sub> (thegpalq)||||||||||||||725760
|- align=center
|107||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||teriprismatotruncated 2<sub>31</sub> (teptalq)||||||||||||||725760
|- align=center
|108||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea_1}}||teriprismatorhombated 2<sub>31</sub> (topralq)||||||||||||||725760
|- align=center
|109||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea}}||cellipriemsatorhombated 3<sub>21</sub> (copranq)||||||||||||||725760
|- align=center
|110||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea_1}}||tericelligreatorhombated 2<sub>31</sub> (tecgrolaq)||||||||||||||725760
|- align=center
|111||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||tericellitruncated 3<sub>21</sub> (tectanq)||||||||||||||483840
|- align=center
|112||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea|3a|nodea_1}}||teridemiprismatotruncated 3<sub>21</sub> (thoptanq)||||||||||||||725760
|- align=center
|113||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea}}||celliprismatotruncated 3<sub>21</sub> (coptanq)||||||||||||||725760
|- align=center
|114||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea_1}}||teridemicelligreatorhombated 3<sub>21</sub> (thocgranq)||||||||||||||483840
|- align=center
|115||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea_1}}||terigreatoprismated 3<sub>21</sub> (tagpanq)||||||||||||||725760
|- align=center
|116||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea}}||great demicellated 3<sub>21</sub> (gahcnaq)||||||||||||||725760
|- align=center
|117||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea_1}}||tericelliprismated laq  (tecpalq)||||||||||||||483840
|- align=center
|118||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea}}||celligreatorhombated 3<sub>21</sub> (cogranq)||||||||||||||725760
|- align=center
|119||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea}}||great demified 3<sub>21</sub> (gahnq)||||||||||||||483840
|- align=center
|120||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea_1}}||great cellated 2<sub>31</sub> (gocalq)||||||||||||||1451520
|- align=center
|121||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea_1}}||terigreatoprismated 2<sub>31</sub> (tegpalq)||||||||||||||1451520
|- align=center
|122||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea_1}}||tericelliprismatotruncated 3<sub>21</sub> (tecpotniq)||||||||||||||1451520
|- align=center
|123||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea_1}}||tericellidemigreatoprismated 2<sub>31</sub> (techogaplaq)||||||||||||||1451520
|- align=center
|124||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||tericelligreatorhombated 3<sub>21</sub> (tacgarnq)||||||||||||||1451520
|- align=center
|125||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea_1}}||tericelliprismatorhombated 2<sub>31</sub> (tecprolaq)||||||||||||||1451520
|- align=center
|126||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea}}||great cellated 3<sub>21</sub> (gocanq)||||||||||||||1451520
|- align=center
|127||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea_1}}||great terated 3<sub>21</sub> (gotanq)||||||||||||||2903040
|}
 
== Regular and uniform honeycombs ==
[[File:Coxeter diagram affine rank7 correspondence.png|518px|thumb|Coxeter-Dynkin diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence.]]
There are five fundamental affine [[Coxeter groups]] and sixteen prismatic groups that generate regular and uniform tessellations in 6-space:
{| class="wikitable"
|-
!#
!colspan=2|[[Coxeter group]]
![[Coxeter diagram]]
!Forms
|- align=center
|1||<math>{\tilde{A}}_6</math>||[3<sup>[7]</sup>]||{{CDD|branch|3ab|nodes|3ab|nodes|split2|node}}||17
|- align=center
|2||<math>{\tilde{C}}_6</math>||[4,3<sup>4</sup>,4]||{{CDD|node|4|node|3|node|3|node|3|node|3|node|4|node}}||71
|- align=center
|3||<math>{\tilde{B}}_6</math>||h[4,3<sup>4</sup>,4]<br />[4,3<sup>3</sup>,3<sup>1,1</sup>]||{{CDD|nodes|split2|node|3|node|3|node|3|node|4|node}}||95 (32 new)
|- align=center
|4||<math>{\tilde{D}}_6</math>||q[4,3<sup>4</sup>,4]<br />[3<sup>1,1</sup>,3<sup>2</sup>,3<sup>1,1</sup>]||{{CDD|nodes|split2|node|3|node|3|node|split1|nodes}}|| 41 (6 new)
|- align=center
|5||<math>{\tilde{E}}_6</math>||[3<sup>2,2,2</sup>]||{{CDD|nodes|3ab|nodes|split2|node|3|node|3|node}}||39
|}
 
Regular and uniform tessellations include:
* <math>{\tilde{A}}_6</math>, 17 forms
** Uniform [[6-simplex honeycomb]]: {3<sup>[7]</sup>} {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch}}
* <math>{\tilde{C}}_6</math>, [4,3<sup>4</sup>,4], 71 forms
** Regular [[6-cube honeycomb]], represented by symbols {4,3<sup>4</sup>,4}, {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|4|node}}
* <math>{\tilde{B}}_6</math>, [3<sup>1,1</sup>,3<sup>3</sup>,4], 95 forms, 64 shared with <math>{\tilde{C}}_6</math>, 32 new
** Uniform [[6-demicube honeycomb]], represented by symbols h{4,3<sup>4</sup>,4} = {3<sup>1,1</sup>,3<sup>3</sup>,4}, {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|4|node}} = {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|4|node}}
* <math>{\tilde{D}}_6</math>, [3<sup>1,1</sup>,3<sup>2</sup>,3<sup>1,1</sup>], 41 unique ringed permuations, most shared with <math>{\tilde{B}}_6</math> and <math>{\tilde{C}}_6</math>, and 6 are new. Coxeter calls the first one a [[quarter 6-cubic honeycomb]].
** {{CDD|nodes_10ru|split2|node|3|node|3|node|split1|nodes_10lu}}, {{CDD|nodes_10ru|split2|node_1|3|node|3|node|split1|nodes_10lu}}, {{CDD|nodes_10ru|split2|node|3|node_1|3|node|split1|nodes_10lu}}, {{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|split1|nodes_10lu}}, {{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|split1|nodes_10lu}}, {{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|split1|nodes_10lu}}
* <math>{\tilde{E}}_6</math>: [3<sup>2,2,2</sup>], 39 forms
** Uniform [[Gosset 2 22 honeycomb|2<sub>22</sub> honeycomb]]: represented by symbols {3<sup>2,2,2</sup>}, {{CDD|nodes|3ab|nodes|split2|node|3|node|3|node_1}}
 
{| class=wikitable
|+ Prismatic groups
|-
!#
!colspan=2|[[Coxeter group]]
![[Coxeter-Dynkin diagram]]
|-
|1||<math>{\tilde{A}}_5</math>x<math>{\tilde{I}}_1</math>||[3<sup>[6]</sup>,2,∞]||{{CDD|node|split1|nodes|3ab|nodes|split2|node|2|node|infin|node}}
|-
|2||<math>{\tilde{B}}_5</math>x<math>{\tilde{I}}_1</math>||[4,3,3<sup>1,1</sup>,2,∞]||{{CDD|node|4|node|3|node|3|node|3|node|4|node|2|node|infin|node}}
|-
|3||<math>{\tilde{C}}_5</math>x<math>{\tilde{I}}_1</math>||[4,3<sup>3</sup>,4,2,∞]||{{CDD|nodes|split2|node|3|node|3|node|4|node|2|node|infin|node}}
|-
|4||<math>{\tilde{D}}_5</math>x<math>{\tilde{I}}_1</math>||[3<sup>1,1</sup>,3,3<sup>1,1</sup>,2,∞]||{{CDD|nodes|split2|node|3|node|split1|nodes|2|node|infin|node}}
|-
|5||<math>{\tilde{A}}_4</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[3<sup>[5]</sup>,2,∞,2,∞,2,∞]||{{CDD|branch|3ab|nodes|split2|node|2|node|infin|node|2|node|infin|node}}
|-
|6||<math>{\tilde{B}}_4</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[4,3,3<sup>1,1</sup>,2,∞,2,∞]||{{CDD|nodes|split2|node|3|node|4|node|2|node|infin|node|2|node|infin|node}}
|-
|7||<math>{\tilde{C}}_4</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[4,3,3,4,2,∞,2,∞]||{{CDD|node|4|node|3|node|3|node|4|node|2|node|infin|node|2|node|infin|node}}
|-
|8||<math>{\tilde{D}}_4</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[3<sup>1,1,1,1</sup>,2,∞,2,∞]||{{CDD|nodes|split2|node|split1|nodes|2|node|infin|node|2|node|infin|node}}
|-
|9||<math>{\tilde{F}}_4</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[3,4,3,3,2,∞,2,∞]||{{CDD|node|3|node|4|node|3|node|3|node|2|node|infin|node|2|node|infin|node}}
|-
|10||<math>{\tilde{C}}_3</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[4,3,4,2,∞,2,∞,2,∞]||{{CDD|node|4|node|3|node|4|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}}
|-
|11||<math>{\tilde{B}}_3</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[4,3<sup>1,1</sup>,2,∞,2,∞,2,∞]||{{CDD|nodes|split2|node|4|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}}
|-
|12||<math>{\tilde{A}}_3</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[3<sup>[4]</sup>,2,∞,2,∞,2,∞]||{{CDD|branch|3ab|branch|2|node|infin|node|2|node|infin|node|2|node|infin|node}}
|-
|13||<math>{\tilde{C}}_2</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[4,4,2,∞,2,∞,2,∞,2,∞]||{{CDD|node|4|node|4|node|2|node|infin|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}}
|-
|14||<math>{\tilde{H}}_2</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[6,3,2,∞,2,∞,2,∞,2,∞]||{{CDD|node|6|node|3|node|2|node|infin|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}}
|-
|15||<math>{\tilde{A}}_2</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[3<sup>[3]</sup>,2,∞,2,∞,2,∞,2,∞]||{{CDD|node|split1|branch|2|node|infin|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}}
|-
|16||<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[∞,2,∞,2,∞,2,∞,2,∞]||{{CDD|node|infin|node|2|node|infin|node|2|node|infin|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}}
|}
 
=== 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 [[Coxeter-Dynkin_diagram#Rank_4_to_10|3 noncompact hyperbolic Coxeter groups]] of rank 7, each generating uniform honeycombs in 6-space as permutations of rings of the Coxeter diagrams.  
{| class=wikitable
|align=right|<math>{\bar{P}}_6</math> = [3,3<sup>[6]</sup>]:<BR>{{CDD|node|split1|nodes|3ab|nodes|split2|node|3|node}}
|align=right|<math>{\bar{Q}}_6</math> = [3<sup>1,1</sup>,3,3<sup>2,1</sup>]:<BR>{{CDD|nodea|3a|branch|3a|branch|3a|nodea|3a|nodea}}
 
|align=right|<math>{\bar{S}}_6</math> = [4,3,3,3<sup>2,1</sup>]:<BR>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|4a|nodea}}
|}
 
== Notes on the Wythoff construction for the uniform 7-polytopes ==
 
The reflective 7-dimensional [[uniform polytope]]s 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 polytope]]s 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.
 
{|class="wikitable"
|-
!Operation
!Extended<br/>[[Schläfli symbol]]
!width=110|[[Coxeter-Dynkin diagram|Coxeter-<br/>Dynkin<br/>diagram]]
!Description
|-
! Parent
|width=70| t<sub>0</sub>{p,q,r,s,t,u}
|{{CDD|node_1|p|node|q|node|r|node|s|node|t|node|u|node}}
| Any regular 7-polytope
|-
! [[Rectification (geometry)|Rectified]]
| t<sub>1</sub>{p,q,r,s,t,u}
|{{CDD|node|p|node_1|q|node|r|node|s|node|t|node|u|node}}
|The edges are fully truncated into single points. The 7-polytope now has the combined faces of the parent and dual.
|-
! Birectified
| t<sub>2</sub>{p,q,r,s,t,u}
|{{CDD|node|p|node|q|node_1|r|node|s|node|t|node|u|node}}
|Birectification reduces [[Cell (geometry)|cells]] to their [[Dual polytope|duals]].
|-
![[Truncation (geometry)|Truncated]]
| t<sub>0,1</sub>{p,q,r,s,t,u}
|{{CDD|node_1|p|node_1|q|node|r|node|s|node|t|node|u|node}}
|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.<br/>[[File:Cube truncation sequence.svg|400px]]
|-
![[Bitruncated]]
| t<sub>1,2</sub>{p,q,r,s,t,u}
|{{CDD|node|p|node_1|q|node_1|r|node|s|node|t|node|u|node}}
|Bitrunction transforms cells to their dual truncation.
|-
!Tritruncated
| t<sub>2,3</sub>{p,q,r,s,t,u}
|{{CDD|node|p|node|q|node_1|r|node_1|s|node|t|node|u|node}}
|Tritruncation transforms 4-faces to their dual truncation.
|-
! [[Cantellation (geometry)|Cantellated]]
| t<sub>0,2</sub>{p,q,r,s,t,u}
|{{CDD|node_1|p|node|q|node_1|r|node|s|node|t|node|u|node}}
|In addition to vertex truncation, each original edge is ''beveled'' with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms.<br/>[[File:Cube cantellation sequence.svg|400px]]
|-
! Bicantellated
| t<sub>1,3</sub>{p,q,r,s,t,u}
|{{CDD|node|p|node_1|q|node|r|node_1|s|node|t|node|u|node}}
|In addition to vertex truncation, each original edge is ''beveled'' with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms.
|-
! [[Runcination (geometry)|Runcinated]]
| t<sub>0,3</sub>{p,q,r,s,t,u}
|{{CDD|node_1|p|node|q|node|r|node_1|s|node|t|node|u|node}}
|Runcination reduces cells and creates new cells at the vertices and edges.
|-
! Biruncinated
| t<sub>1,4</sub>{p,q,r,s,t,u}
|{{CDD|node|p|node_1|q|node|r|node|s|node_1|t|node|u|node}}
|Runcination reduces cells and creates new cells at the vertices and edges.
|-
! [[Sterication|Stericated]]
| t<sub>0,4</sub>{p,q,r,s,t,u}
|{{CDD|node_1|p|node|q|node|r|node|s|node_1|t|node|u|node}}
|Sterication reduces 4-faces and creates new 4-faces at the vertices, edges, and faces in the gaps.
|-
! Pentellated
| t<sub>0,5</sub>{p,q,r,s,t,u}
|{{CDD|node_1|p|node|q|node|r|node|s|node|t|node_1|u|node}}
|Pentellation reduces 5-faces and creates new 5-faces at the vertices, edges, faces, and cells in the gaps.
|-
! Hexicated
| t<sub>0,6</sub>{p,q,r,s,t,u}
|{{CDD|node_1|p|node|q|node|r|node|s|node|t|node|u|node_1}}
|Hexication reduces 6-faces and creates new 6-faces at the vertices, edges, faces, cells, and 4-faces in the gaps. ([[Expansion (geometry)|expansion]] operation for 7-polytopes)
|-
![[Omnitruncation (geometry)|Omnitruncated]]
| t<sub>0,1,2,3,4,5,6</sub>{p,q,r,s,t,u}
|{{CDD|node_1|p|node_1|q|node_1|r|node_1|s|node_1|t|node_1|u|node_1}}
|All six operators, truncation, cantellation, runcination, sterication, pentellation, and hexication are applied.
|}
 
== References ==
 
* [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', [[Messenger of Mathematics]], Macmillan, 1900
* [[Alicia Boole Stott|A. Boole Stott]]: ''Geometrical deduction of semiregular from regular polytopes and space fillings'', Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
* [[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]:
** H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: ''Uniform Polyhedra'', Philosophical Transactions of the Royal Society of London, Londne, 1954
** H.S.M. Coxeter, ''Regular Polytopes'', 3rd Edition, Dover New York, 1973
* '''Kaleidoscopes: Selected Writings of H.S.M. Coxeter''', editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* [[Norman Johnson (mathematician)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
* {{KlitzingPolytopes|polyexa.htm|7D|uniform polytopes (polyexa)}}
 
== External links ==
* [http://www.steelpillow.com/polyhedra/ditela.html Polytope names]
* [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
* {{PolyCell | urlname = glossary.html| title = Glossary for hyperspace}}
{{Polytopes}}
 
[[Category:7-polytopes]]

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