title: 1 42 polytope
text: # 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 |
  | $A_4A_2A_1$ | { } | $f_1$ | 2 | 483840 | 15 | 15 | 30 | 5 | 30 | 30 | 10 | 30 | 15 | 10 | 15 | 3 | 5 | 3 | {3}x{3,3,3} | $E_8/A_4A_2A_1$ = 192*10!/5!/2/2 = 483840 |
  | $A_3A_2A_1$ | {3} | $f_2$ | 3 | 3 | 2419200 | 2 | 4 | 1 | 8 | 6 | 4 | 12 | 4 | 6 | 8 | 1 | 4 | 2 | {3.3}v{ } | $E_8/A_3A_2A_1$ = 192*10!/4!/3!/2 = 2419200 |
  | $A_3A_3$ | $1_{10}$ | $f_3$ | 4 | 6 | 4 | 1209600 | * | 1 | 4 | 0 | 4 | 6 | 0 | 6 | 4 | 0 | 4 | 1 | {3,3}v( ) | $E_8/A_3A_3$ = 192*10!/4!/4! = 1209600 |
  | $A_3A_2A_1$ | $1_{10}$ | $f_3$ | 4 | 6 | 4 | * | 2419200 | 0 | 2 | 3 | 1 | 6 | 3 | 3 | 6 | 1 | 3 | 2 | {3}v{ } | $E_8/A_3A_2A_1$ = 192*10!/4!/3!/2 = 2419200 |
  | $A_4A_3$ | $1_{20}$ | $f_4$ | 5 | 10 | 10 | 5 | 0 | 241920 | * | * | 4 | 0 | 0 | 6 | 0 | 0 | 4 | 0 | {3,3} | $E_8/A_4A_3$ = 192*10!/4!/4! = 241920 |
  | $D_4A_2$ | $1_{11}$ | $f_4$ | 8 | 24 | 32 | 8 | 8 | * | 604800 | * | 1 | 3 | 0 | 3 | 3 | 0 | 3 | 1 | {3}v( ) | $E_8/D_4A_2$ = 192*10!/8/4!/3! = 604800 |
  | $A_4A_1A_1$ | $1_{20}$ | $f_4$ | 5 | 10 | 10 | 0 | 5 | * | * | 1451520 | 0 | 2 | 2 | 1 | 4 | 1 | 2 | 2 | { }v{ } | $E_8/A_4A_1A_1$ = 192*10!/5!/2/2 = 1451520 |
  | $D_5A_2$ | $1_{21}$ | $f_5$ | 16 | 80 | 160 | 80 | 40 | 16 | 10 | 0 | 60480 | * | * | 3 | 0 | 0 | 3 | 0 | {3} | $E_8/D_5A_2$ = 192*10!/16/5!/3! = 40480 |
  | $D_5A_1$ | $1_{21}$ | $f_5$ | 16 | 80 | 160 | 40 | 80 | 0 | 10 | 16 | * | 181440 | * | 1 | 2 | 0 | 2 | 1 | { }v( ) | $E_8/D_5A_1$ = 192*10!/16/5!/2 = 181440 |
  | $A_5A_1$ | $1_{30}$ | $f_5$ | 6 | 15 | 20 | 0 | 15 | 0 | 0 | 6 | * | * | 483840 | 0 | 2 | 1 | 1 | 2 | { }v( ) | $E_8/A_5A_1$ = 192*10!/6!/2 = 483840 |
  | $E_6A_1$ | $1_{22}$ | $f_6$ | 72 | 720 | 2160 | 1080 | 1080 | 216 | 270 | 216 | 27 | 27 | 0 | 6720 | * | * | 2 | 0 | { } | $E_8/E_6A_1$ = 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 |
  | $A_6A_1$ | $1_{40}$ | $f_6$ | 7 | 21 | 35 | 0 | 35 | 0 | 0 | 21 | 0 | 0 | 7 | * | * | 69120 | 0 | 2 | { } | $E_8/A_6A_1$ = 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 | $E_3=A_2A_1$ | $E_4=A_4$ | $E_5=D_5$ | $E_6$ | $E_7$ | $E_8$ | $E_9$ = ${\tilde {E}}_{8}$ = $E_8^+$ | $E_{10}$ = ${\bar {T}}_{8}$ = $E_8^{++}$ |
  | 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 |
  | $A_4A_2A_1$ | ( ) | $f_0$ | 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 |
  | $A_3A_1A_1$ | { } | $f_1$ | 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 | |
  | $A_3A_2$ | {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 | |
  | $A_3A_2A_1$ | {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 | |
  | $A_2A_2A_1$ | {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 | |
  | $A_3A_3$ | $0_{200}$ | $f_3$ | 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 | |
  | $A_3A_3$ | $0_{110}$ | $f_3$ | 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 | |
  | $A_3A_2$ | $0_{110}$ | $f_3$ | 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 | |
  | $A_3A_2A_1$ | $0_{110}$ | $f_3$ | 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 | |
  | $A_3A_1A_1$ | $0_{200}$ | $f_3$ | 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 | |
  | $A_4A_3$ | $0_{210}$ | $f_4$ | 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 | |
  | $A_4A_2$ | $0_{210}$ | $f_4$ | 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 | |
  | $D_4A_2$ | $0_{111}$ | $f_4$ | 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 | |
  | $A_4A_1$ | $0_{210}$ | $f_4$ | 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 | |
  | $A_4A_1A_1$ | $0_{210}$ | $f_4$ | 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 | |
  | $A_4A_1$ | $0_{300}$ | $f_4$ | 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 | |
  | $D_5A_2$ | $0_{211}$ | $f_5$ | 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} |
  | $A_5A_1$ | $0_{220}$ | $f_5$ | 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 |
  | $D_5A_1$ | $0_{211}$ | $f_5$ | 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 |
  | $A_5A_1$ | $0_{310}$ | $f_5$ | 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 |
  | $A_5A_1$ | $0_{400}$ | $f_5$ | 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 |
  | $E_6A_1$ | $0_{221}$ | $f_6$ | 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 | { } |
  | $A_6A_1$ | $0_{410}$ | $f_6$ | 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] |
  