# 1 42 polytope

| 4$_{21}$ | 1$_{42}$ | 2$_{41}$ |
| Rectified 4$_{21}$ | Rectified 1$_{42}$ | Rectified 2$_{41}$ |
| Birectified 4$_{21}$ | Trirectified 4$_{21}$ |
| Orthogonal projections in E$_6$ Coxeter plane |||
In 8-dimensional geometry, the 1$_{42}$ is a uniform 8-polytope, constructed within the symmetry of the E$_8$ group.
Its Coxeter symbol is 1$_{42}$, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequences.
The rectified 1$_{42}$ is constructed by points at the mid-edges of the 1$_{42}$ and is the same as the birectified 2$_{41}$, and the quadrirectified 4$_{21}$.
These polytopes are part of a family of 255 (2$^8$ − 1) convex uniform polytopes in 8 dimensions, made of uniform polytope facets and vertex figures, defined by all non-empty combinations of rings in this Coxeter-Dynkin diagram: .

## 1$_{42}$ polytope

1$_{42}$ :
| Type | Uniform 8-polytope |
| Family | 1$_{k2}$ polytope |
| Schläfli symbol | {3,3$^{4,2}$} |
| Coxeter symbol | 1$_{42}$ |
| Coxeter diagrams | |
| 7-faces | 2400: 240 1$_{32}$ 2160 1$_{41}$ |
| 6-faces | 106080: 6720 1$_{22}$ 30240 1$_{31}$ 69120 {3$^5$} |
| 5-faces | 725760: 60480 1$_{12}$ 181440 1$_{21}$ 483840 {3$^4$} |
| 4-faces | 2298240: 241920 1$_{02}$ 604800 1$_{11}$ 1451520 {3$^3$} |
| Cells | 3628800: 1209600 1$_{01}$ 2419200 {3$^2$} |
| Faces | 2419200 {3} |
| Edges | 483840 |
| Vertices | 17280 |
| Vertex figure | t$_2${3$^6$} |
| Petrie polygon | 30-gon |
| Coxeter group | E$_8$, [3$^{4,2,1}$] |
| Properties | convex |
The 1$_{42}$ is composed of 2400 facets: 240 1$_{32}$ polytopes, and 2160 7-demicubes (1$_{41}$). Its vertex figure is a birectified 7-simplex.
This polytope, along with the demiocteract, can tessellate 8-dimensional space, represented by the symbol 1$_{52}$, and Coxeter-Dynkin diagram: .

### Alternate names

* E. L. Elte (1912) excluded this polytope from his listing of semiregular polytopes, because it has more than two types of 6-faces, but under his naming scheme it would be called V$_{17280}$ for its 17280 vertices.
* Coxeter named it 1$_{42}$ for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch.
* Diacositetracont-dischiliahectohexaconta-zetton (acronym bif) - 240-2160 facetted polyzetton (Jonathan Bowers)

### Coordinates

The 17280 vertices can be defined as sign and location permutations of:
All sign combinations (32): (280×32=8960 vertices)
    (4, 2, 2, 2, 2, 0, 0, 0)
Half of the sign combinations (128): ((1+8+56)×128=8320 vertices)
    (2, 2, 2, 2, 2, 2, 2, 2)
    (5, 1, 1, 1, 1, 1, 1, 1)
    (3, 3, 3, 1, 1, 1, 1, 1)
The edge length is 2√2 in this coordinate set, and the polytope radius is 4√2.

### Construction

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram: .
Removing the node on the end of the 2-length branch leaves the 7-demicube, 1$_{41}$, .
Removing the node on the end of the 4-length branch leaves the 1$_{32}$, .
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 7-simplex, 0$_{42}$, .
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.
| | Configuration matrix |
| E$_8$ | k-face | f$_k$ | f$_0$ | f$_1$ | f$_2$ | f$_3$ | f$_3$ | f$_4$ | f$_4$ | f$_4$ | f$_5$ | f$_5$ | f$_5$ | f$_6$ | f$_6$ | f$_6$ | f$_7$ | f$_7$ | k-figure | notes |
| A$_7$ | ( ) | f$_0$ | 17280 | 56 | 420 | 280 | 560 | 70 | 280 | 420 | 56 | 168 | 168 | 28 | 56 | 28 | 8 | 8 | 2r{3$^6$} | E$_8$/A$_7$ = 192*10!/8! = 17280 |
| A$_4$A$_2$A$_1$ | { } | f$_1$ | 2 | 483840 | 15 | 15 | 30 | 5 | 30 | 30 | 10 | 30 | 15 | 10 | 15 | 3 | 5 | 3 | {3}x{3,3,3} | E$_8$/A$_4$A$_2$A$_1$ = 192*10!/5!/2/2 = 483840 |
| A$_3$A$_2$A$_1$ | {3} | f$_2$ | 3 | 3 | 2419200 | 2 | 4 | 1 | 8 | 6 | 4 | 12 | 4 | 6 | 8 | 1 | 4 | 2 | {3.3}v{ } | E$_8$/A$_3$A$_2$A$_1$ = 192*10!/4!/3!/2 = 2419200 |
| A$_3$A$_3$ | 1$_{10}$ | f$_3$ | 4 | 6 | 4 | 1209600 | * | 1 | 4 | 0 | 4 | 6 | 0 | 6 | 4 | 0 | 4 | 1 | {3,3}v( ) | E$_8$/A$_3$A$_3$ = 192*10!/4!/4! = 1209600 |
| A$_3$A$_2$A$_1$ | 1$_{10}$ | f$_3$ | 4 | 6 | 4 | * | 2419200 | 0 | 2 | 3 | 1 | 6 | 3 | 3 | 6 | 1 | 3 | 2 | {3}v{ } | E$_8$/A$_3$A$_2$A$_1$ = 192*10!/4!/3!/2 = 2419200 |
| A$_4$A$_3$ | 1$_{20}$ | f$_4$ | 5 | 10 | 10 | 5 | 0 | 241920 | * | * | 4 | 0 | 0 | 6 | 0 | 0 | 4 | 0 | {3,3} | E$_8$/A$_4$A$_3$ = 192*10!/4!/4! = 241920 |
| D$_4$A$_2$ | 1$_{11}$ | f$_4$ | 8 | 24 | 32 | 8 | 8 | * | 604800 | * | 1 | 3 | 0 | 3 | 3 | 0 | 3 | 1 | {3}v( ) | E$_8$/D$_4$A$_2$ = 192*10!/8/4!/3! = 604800 |
| A$_4$A$_1$A$_1$ | 1$_{20}$ | f$_4$ | 5 | 10 | 10 | 0 | 5 | * | * | 1451520 | 0 | 2 | 2 | 1 | 4 | 1 | 2 | 2 | { }v{ } | E$_8$/A$_4$A$_1$A$_1$ = 192*10!/5!/2/2 = 1451520 |
| D$_5$A$_2$ | 1$_{21}$ | f$_5$ | 16 | 80 | 160 | 80 | 40 | 16 | 10 | 0 | 60480 | * | * | 3 | 0 | 0 | 3 | 0 | {3} | E$_8$/D$_5$A$_2$ = 192*10!/16/5!/3! = 40480 |
| D$_5$A$_1$ | 1$_{21}$ | f$_5$ | 16 | 80 | 160 | 40 | 80 | 0 | 10 | 16 | * | 181440 | * | 1 | 2 | 0 | 2 | 1 | { }v( ) | E$_8$/D$_5$A$_1$ = 192*10!/16/5!/2 = 181440 |
| A$_5$A$_1$ | 1$_{30}$ | f$_5$ | 6 | 15 | 20 | 0 | 15 | 0 | 0 | 6 | * | * | 483840 | 0 | 2 | 1 | 1 | 2 | { }v( ) | E$_8$/A$_5$A$_1$ = 192*10!/6!/2 = 483840 |
| E$_6$A$_1$ | 1$_{22}$ | f$_6$ | 72 | 720 | 2160 | 1080 | 1080 | 216 | 270 | 216 | 27 | 27 | 0 | 6720 | * | * | 2 | 0 | { } | E$_8$/E$_6$A$_1$ = 192*10!/72/6!/2 = 6720 |
| D$_6$ | 1$_{31}$ | f$_6$ | 32 | 240 | 640 | 160 | 480 | 0 | 60 | 192 | 0 | 12 | 32 | * | 30240 | * | 1 | 1 | { } | E$_8$/D$_6$ = 192*10!/32/6! = 30240 |
| A$_6$A$_1$ | 1$_{40}$ | f$_6$ | 7 | 21 | 35 | 0 | 35 | 0 | 0 | 21 | 0 | 0 | 7 | * | * | 69120 | 0 | 2 | { } | E$_8$/A$_6$A$_1$ = 192*10!/7!/2 = 69120 |
| E$_7$ | 1$_{32}$ | f$_7$ | 576 | 10080 | 40320 | 20160 | 30240 | 4032 | 7560 | 12096 | 756 | 1512 | 2016 | 56 | 126 | 0 | 240 | * | ( ) | E$_8$/E$_7$ = 192*10!/72/8! = 240 |
| D$_7$ | 1$_{41}$ | f$_7$ | 64 | 672 | 2240 | 560 | 2240 | 0 | 280 | 1344 | 0 | 84 | 448 | 0 | 14 | 64 | * | 2160 | ( ) | E$_8$/D$_7$ = 192*10!/64/7! = 2160 |

### Projections

| E8 [30] | E7 [18] | E6 [12] |
| (1) | (1,3,6) | (8,16,24,32,48,64,96) |
| [20] | [24] | [6] |
| | | (1,2,3,4,5,6,7,8,10,11,12,14,16,18,19,20) |
Orthographic projections are shown for the sub-symmetries of E$_8$: E$_7$, E$_6$, B$_8$, B$_7$, B$_6$, B$_5$, B$_4$, B$_3$, B$_2$, A$_7$, and A$_5$ Coxeter planes, as well as two more symmetry planes of order 20 and 24. Vertices are shown as circles, colored by their order of overlap in each projective plane.
| D3 / B2 / A3 [4] | D4 / B3 / A2 [6] | D5 / B4 [8] |
| (32,160,192,240,480,512,832,960) | (72,216,432,720,864,1080) | (8,16,24,32,48,64,96) |
| D6 / B5 / A4 [10] | D7 / B6 [12] | D8 / B7 / A6 [14] |
| B8 [16/2] | A5 [6] | A7 [8] |

### Related polytopes and honeycombs

1$_{k2}$ figures in n dimensions :
| Space | Finite | Finite | Finite | Finite | Finite | Finite | Euclidean | Hyperbolic |
| n | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Coxeter group | E$_3$=A$_2$A$_1$ | E$_4$=A$_4$ | E$_5$=D$_5$ | E$_6$ | E$_7$ | E$_8$ | E$_9$ = ${\tilde {E}}_{8}$ = E$_8$$^+$ | E$_{10}$ = ${\bar {T}}_{8}$ = E$_8$$^{++}$ |
| Coxeter diagram | | | | | | | | |
| Symmetry (order) | [3$^{−1,2,1}$] | [3$^{0,2,1}$] | [3$^{1,2,1}$] | [[3$^{2,2,1}$]] | [3$^{3,2,1}$] | [3$^{4,2,1}$] | [3$^{5,2,1}$] | [3$^{6,2,1}$] |
| Order | 12 | 120 | 1,920 | 103,680 | 2,903,040 | 696,729,600 | ∞ | ∞ |
| Graph | | | | | | | - | - |
| Name | 1$_{−1,2}$ | 1$_{02}$ | 1$_{12}$ | 1$_{22}$ | 1$_{32}$ | 1$_{42}$ | 1$_{52}$ | 1$_{62}$ |

## Rectified 1$_{42}$ polytope

Rectified 1$_{42}$ :
| Type | Uniform 8-polytope |
| Schläfli symbol | t$_1${3,3$^{4,2}$} |
| Coxeter symbol | 0$_{421}$ |
| Coxeter diagrams | |
| 7-faces | 19680 |
| 6-faces | 382560 |
| 5-faces | 2661120 |
| 4-faces | 9072000 |
| Cells | 16934400 |
| Faces | 16934400 |
| Edges | 7257600 |
| Vertices | 483840 |
| Vertex figure | {3,3,3}×{3}× |
| Coxeter group | E$_8$, [3$^{4,2,1}$] |
| Properties | convex |
The rectified 1$_{42}$ is named from being a rectification of the 1$_{42}$ polytope, with vertices positioned at the mid-edges of the 1$_{42}$. It can also be called a 0$_{421}$ polytope with the ring at the center of 3 branches of length 4, 2, and 1.

### Alternate names

* 0$_{421}$ polytope
* Birectified 2$_{41}$ polytope
* Quadrirectified 4$_{21}$ polytope
* Rectified diacositetracont-dischiliahectohexaconta-zetton as a rectified 240-2160 facetted polyzetton (acronym buffy) (Jonathan Bowers)

### Construction

It is created by a Wythoff construction upon a set of 8 hyperplane mirrors in 8-dimensional space.
The facet information can be extracted from its Coxeter-Dynkin diagram: .
Removing the node on the end of the 1-length branch leaves the birectified 7-simplex,
Removing the node on the end of the 2-length branch leaves the birectified 7-cube, .
Removing the node on the end of the 3-length branch leaves the rectified 1$_{32}$, .
The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 5-cell-triangle duoprism prism, .
Seen in a configuration matrix, the element counts can be derived by mirror removal and ratios of Coxeter group orders.
| | Configuration matrix |
| E$_8$ | k-face | f$_k$ | f$_0$ | f$_1$ | f$_2$ | f$_2$ | f$_2$ | f$_3$ | f$_3$ | f$_3$ | f$_3$ | f$_3$ | f$_4$ | f$_4$ | f$_4$ | f$_4$ | f$_4$ | f$_4$ | f$_5$ | f$_5$ | f$_5$ | f$_5$ | f$_5$ | f$_5$ | f$_6$ | f$_6$ | f$_6$ | f$_6$ | f$_6$ | f$_7$ | f$_7$ | f$_7$ | k-figure |
| A$_4$A$_2$A$_1$ | ( ) | f$_0$ | 483840 | 30 | 30 | 15 | 60 | 10 | 15 | 60 | 30 | 60 | 5 | 20 | 30 | 60 | 30 | 30 | 10 | 20 | 30 | 30 | 15 | 6 | 10 | 10 | 15 | 6 | 3 | 5 | 2 | 3 | {3,3,3}x{3,3}x |
| A$_3$A$_1$A$_1$ | { } | f$_1$ | 2 | 7257600 | 2 | 1 | 4 | 1 | 2 | 8 | 4 | 6 | 1 | 4 | 8 | 12 | 6 | 4 | 4 | 6 | 12 | 8 | 4 | 1 | 6 | 4 | 8 | 2 | 1 | 4 | 1 | 2 | |
| A$_3$A$_2$ | {3} | f$_2$ | 3 | 3 | 4838400 | * | * | 1 | 1 | 4 | 0 | 0 | 1 | 4 | 4 | 6 | 0 | 0 | 4 | 6 | 6 | 4 | 0 | 0 | 6 | 4 | 4 | 1 | 0 | 4 | 1 | 1 | |
| A$_3$A$_2$A$_1$ | {3} | f$_2$ | 3 | 3 | * | 2419200 | * | 0 | 2 | 0 | 4 | 0 | 1 | 0 | 8 | 0 | 6 | 0 | 4 | 0 | 12 | 0 | 4 | 0 | 6 | 0 | 8 | 0 | 1 | 4 | 0 | 2 | |
| A$_2$A$_2$A$_1$ | {3} | f$_2$ | 3 | 3 | * | * | 9676800 | 0 | 0 | 2 | 1 | 3 | 0 | 1 | 2 | 6 | 3 | 3 | 1 | 3 | 6 | 6 | 3 | 1 | 3 | 3 | 6 | 2 | 1 | 3 | 1 | 2 | |
| A$_3$A$_3$ | 0$_{200}$ | f$_3$ | 4 | 6 | 4 | 0 | 0 | 1209600 | * | * | * | * | 1 | 4 | 0 | 0 | 0 | 0 | 4 | 6 | 0 | 0 | 0 | 0 | 6 | 4 | 0 | 0 | 0 | 4 | 1 | 0 | |
| A$_3$A$_3$ | 0$_{110}$ | f$_3$ | 6 | 12 | 4 | 4 | 0 | * | 1209600 | * | * | * | 1 | 0 | 4 | 0 | 0 | 0 | 4 | 0 | 6 | 0 | 0 | 0 | 6 | 0 | 4 | 0 | 0 | 4 | 0 | 1 | |
| A$_3$A$_2$ | 0$_{110}$ | f$_3$ | 6 | 12 | 4 | 0 | 4 | * | * | 4838400 | * | * | 0 | 1 | 1 | 3 | 0 | 0 | 1 | 3 | 3 | 3 | 0 | 0 | 3 | 3 | 3 | 1 | 0 | 3 | 1 | 1 | |
| A$_3$A$_2$A$_1$ | 0$_{110}$ | f$_3$ | 6 | 12 | 0 | 4 | 4 | * | * | * | 2419200 | * | 0 | 0 | 2 | 0 | 3 | 0 | 1 | 0 | 6 | 0 | 3 | 0 | 3 | 0 | 6 | 0 | 1 | 3 | 0 | 2 | |
| A$_3$A$_1$A$_1$ | 0$_{200}$ | f$_3$ | 4 | 6 | 0 | 0 | 4 | * | * | * | * | 7257600 | 0 | 0 | 0 | 2 | 1 | 2 | 0 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 4 | 2 | 1 | 2 | 1 | 2 | |
| A$_4$A$_3$ | 0$_{210}$ | f$_4$ | 10 | 30 | 20 | 10 | 0 | 5 | 5 | 0 | 0 | 0 | 241920 | * | * | * | * | * | 4 | 0 | 0 | 0 | 0 | 0 | 6 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | |
| A$_4$A$_2$ | 0$_{210}$ | f$_4$ | 10 | 30 | 20 | 0 | 10 | 5 | 0 | 5 | 0 | 0 | * | 967680 | * | * | * | * | 1 | 3 | 0 | 0 | 0 | 0 | 3 | 3 | 0 | 0 | 0 | 3 | 1 | 0 | |
| D$_4$A$_2$ | 0$_{111}$ | f$_4$ | 24 | 96 | 32 | 32 | 32 | 0 | 8 | 8 | 8 | 0 | * | * | 604800 | * | * | * | 1 | 0 | 3 | 0 | 0 | 0 | 3 | 0 | 3 | 0 | 0 | 3 | 0 | 1 | |
| A$_4$A$_1$ | 0$_{210}$ | f$_4$ | 10 | 30 | 10 | 0 | 20 | 0 | 0 | 5 | 0 | 5 | * | * | * | 2903040 | * | * | 0 | 1 | 1 | 2 | 0 | 0 | 1 | 2 | 2 | 1 | 0 | 2 | 1 | 1 | |
| A$_4$A$_1$A$_1$ | 0$_{210}$ | f$_4$ | 10 | 30 | 0 | 10 | 20 | 0 | 0 | 0 | 5 | 5 | * | * | * | * | 1451520 | * | 0 | 0 | 2 | 0 | 2 | 0 | 1 | 0 | 4 | 0 | 1 | 2 | 0 | 2 | |
| A$_4$A$_1$ | 0$_{300}$ | f$_4$ | 5 | 10 | 0 | 0 | 10 | 0 | 0 | 0 | 0 | 5 | * | * | * | * | * | 2903040 | 0 | 0 | 0 | 2 | 1 | 1 | 0 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | |
| D$_5$A$_2$ | 0$_{211}$ | f$_5$ | 80 | 480 | 320 | 160 | 160 | 80 | 80 | 80 | 40 | 0 | 16 | 16 | 10 | 0 | 0 | 0 | 60480 | * | * | * | * | * | 3 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | {3} |
| A$_5$A$_1$ | 0$_{220}$ | f$_5$ | 20 | 90 | 60 | 0 | 60 | 15 | 0 | 30 | 0 | 15 | 0 | 6 | 0 | 6 | 0 | 0 | * | 483840 | * | * | * | * | 1 | 2 | 0 | 0 | 0 | 2 | 1 | 0 | { }v |
| D$_5$A$_1$ | 0$_{211}$ | f$_5$ | 80 | 480 | 160 | 160 | 320 | 0 | 40 | 80 | 80 | 80 | 0 | 0 | 10 | 16 | 16 | 0 | * | * | 181440 | * | * | * | 1 | 0 | 2 | 0 | 0 | 2 | 0 | 1 | { }v |
| A$_5$ | 0$_{310}$ | f$_5$ | 15 | 60 | 20 | 0 | 60 | 0 | 0 | 15 | 0 | 30 | 0 | 0 | 0 | 6 | 0 | 6 | * | * | * | 967680 | * | * | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | ( )v( )v |
| A$_5$A$_1$ | 0$_{310}$ | f$_5$ | 15 | 60 | 0 | 20 | 60 | 0 | 0 | 0 | 15 | 30 | 0 | 0 | 0 | 0 | 6 | 6 | * | * | * | * | 483840 | * | 0 | 0 | 2 | 0 | 1 | 1 | 0 | 2 | { }v |
| A$_5$A$_1$ | 0$_{400}$ | f$_5$ | 6 | 15 | 0 | 0 | 20 | 0 | 0 | 0 | 0 | 15 | 0 | 0 | 0 | 0 | 0 | 6 | * | * | * | * | * | 483840 | 0 | 0 | 0 | 2 | 1 | 0 | 1 | 2 | { }v |
| E$_6$A$_1$ | 0$_{221}$ | f$_6$ | 720 | 6480 | 4320 | 2160 | 4320 | 1080 | 1080 | 2160 | 1080 | 1080 | 216 | 432 | 270 | 432 | 216 | 0 | 27 | 72 | 27 | 0 | 0 | 0 | 6720 | * | * | * | * | 2 | 0 | 0 | { } |
| A$_6$ | 0$_{320}$ | f$_6$ | 35 | 210 | 140 | 0 | 210 | 35 | 0 | 105 | 0 | 105 | 0 | 21 | 0 | 42 | 0 | 21 | 0 | 7 | 0 | 7 | 0 | 0 | * | 138240 | * | * | * | 1 | 1 | 0 | { } |
| D$_6$ | 0$_{311}$ | f$_6$ | 240 | 1920 | 640 | 640 | 1920 | 0 | 160 | 480 | 480 | 960 | 0 | 0 | 60 | 192 | 192 | 192 | 0 | 0 | 12 | 32 | 32 | 0 | * | * | 30240 | * | * | 1 | 0 | 1 | { } |
| A$_6$ | 0$_{410}$ | f$_6$ | 21 | 105 | 35 | 0 | 140 | 0 | 0 | 35 | 0 | 105 | 0 | 0 | 0 | 21 | 0 | 42 | 0 | 0 | 0 | 7 | 0 | 7 | * | * | * | 138240 | * | 0 | 1 | 1 | { } |
| A$_6$A$_1$ | 0$_{410}$ | f$_6$ | 21 | 105 | 0 | 35 | 140 | 0 | 0 | 0 | 35 | 105 | 0 | 0 | 0 | 0 | 21 | 42 | 0 | 0 | 0 | 0 | 7 | 7 | * | * | * | * | 69120 | 0 | 0 | 2 | { } |
| E$_7$ | 0$_{321}$ | f$_7$ | 10080 | 120960 | 80640 | 40320 | 120960 | 20160 | 20160 | 60480 | 30240 | 60480 | 4032 | 12096 | 7560 | 24192 | 12096 | 12096 | 756 | 4032 | 1512 | 4032 | 2016 | 0 | 56 | 576 | 126 | 0 | 0 | 240 | * | * | ( ) |
| A$_7$ | 0$_{420}$ | f$_7$ | 56 | 420 | 280 | 0 | 560 | 70 | 0 | 280 | 0 | 420 | 0 | 56 | 0 | 168 | 0 | 168 | 0 | 28 | 0 | 56 | 0 | 28 | 0 | 8 | 0 | 8 | 0 | * | 17280 | * | ( ) |
| D$_7$ | 0$_{411}$ | f$_7$ | 672 | 6720 | 2240 | 2240 | 8960 | 0 | 560 | 2240 | 2240 | 6720 | 0 | 0 | 280 | 1344 | 1344 | 2688 | 0 | 0 | 84 | 448 | 448 | 448 | 0 | 0 | 14 | 64 | 64 | * | * | 2160 | ( ) |

### Projections

Orthographic projections are shown for the sub-symmetries of B$_6$, B$_5$, B$_4$, B$_3$, B$_2$, A$_7$, and A$_5$ Coxeter planes. Vertices are shown as circles, colored by their order of overlap in each projective plane.
(Planes for E$_8$: E$_7$, E$_6$, B$_8$, B$_7$, [24] are not shown for being too large to display.)
| D3 / B2 / A3 [4] | D4 / B3 / A2 [6] | D5 / B4 [8] |
| D6 / B5 / A4 [10] | D7 / B6 [12] | [6] |
| A5 [6] | A7 [8] | [20] |