# Hexicated 7-simplexes

| 7-simplex | Hexicated 7-simplex | Hexitruncated 7-simplex | Hexicantellated 7-simplex |
| Hexiruncinated 7-simplex | Hexicantitruncated 7-simplex | Hexiruncitruncated 7-simplex | Hexiruncicantellated 7-simplex |
| Hexisteritruncated 7-simplex | Hexistericantellated 7-simplex | Hexipentitruncated 7-simplex | Hexiruncicantitruncated 7-simplex |
| Hexistericantitruncated 7-simplex | Hexisteriruncitruncated 7-simplex | Hexisteriruncicantellated 7-simplex | Hexipenticantitruncated 7-simplex |
| Hexipentiruncitruncated 7-simplex | Hexisteriruncicantitruncated 7-simplex | Hexipentiruncicantitruncated 7-simplex | Hexipentistericantitruncated 7-simplex |
| Hexipentisteriruncicantitruncated 7-simplex (Omnitruncated 7-simplex) ||||
| Orthogonal projections in A$_7$ Coxeter plane ||||
In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-simplex.
There are 20 unique hexications for the 7-simplex, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations.
The simple hexicated 7-simplex is also called an expanded 7-simplex, with only the first and last nodes ringed, is constructed by an expansion operation applied to the regular 7-simplex. The highest form, the hexipentisteriruncicantitruncated 7-simplex is more simply called a omnitruncated 7-simplex with all of the nodes ringed.

## Hexicated 7-simplex

Hexicated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,6}${3$^6$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | 254: 8+8 {3$^5$} 28+28x{3$^4$} 56+56 {3}x{3,3,3} 70 {3,3}x{3,3} |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 336 |
| Vertices | 56 |
| Vertex figure | 5-simplex antiprism |
| Coxeter group | A$_7$×2, [[3$^6$]], order 80640 |
| Properties | convex |
In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, a hexication (6th order truncation) of the regular 7-simplex, or alternately can be seen as an expansion operation.

### Root vectors

Its 56 vertices represent the root vectors of the simple Lie group A$_7$.

### Alternate names

* Expanded 7-simplex
* Small petated hexadecaexon (acronym: suph) (Jonathan Bowers)

### Coordinates

The vertices of the hexicated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,1,2). This construction is based on facets of the hexicated 8-orthoplex, .
A second construction in 8-space, from the center of a rectified 8-orthoplex is given by coordinate permutations of:
    (1,-1,0,0,0,0,0,0)

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [[7]] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [[5]] | [4] | [[3]] |

## Hexitruncated 7-simplex

hexitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,6}${3$^6$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 1848 |
| Vertices | 336 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |

### Alternate names

* Petitruncated octaexon (acronym: puto) (Jonathan Bowers)

### Coordinates

The vertices of the hexitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,1,2,3). This construction is based on facets of the hexitruncated 8-orthoplex, .

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Hexicantellated 7-simplex

Hexicantellated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,2,6}${3$^6$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 5880 |
| Vertices | 840 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |

### Alternate names

* Petirhombated octaexon (acronym: puro) (Jonathan Bowers)

### Coordinates

The vertices of the hexicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,2,3). This construction is based on facets of the hexicantellated 8-orthoplex, .

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Hexiruncinated 7-simplex

Hexiruncinated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,3,6}${3$^6$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 8400 |
| Vertices | 1120 |
| Vertex figure | |
| Coxeter group | A$_7$×2, [[3$^6$]], order 80640 |
| Properties | convex |

### Alternate names

* Petiprismated hexadecaexon (acronym: puph) (Jonathan Bowers)

### Coordinates

The vertices of the hexiruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,2,2,3). This construction is based on facets of the hexiruncinated 8-orthoplex, .

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [[7]] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [[5]] | [4] | [[3]] |

## Hexicantitruncated 7-simplex

Hexicantitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,6}${3$^6$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 8400 |
| Vertices | 1680 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |

### Alternate names

* Petigreatorhombated octaexon (acronym: pugro) (Jonathan Bowers)

### Coordinates

The vertices of the hexicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,1,2,3,4). This construction is based on facets of the hexicantitruncated 8-orthoplex, .

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Hexiruncitruncated 7-simplex

Hexiruncitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,3,6}${3$^6$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 20160 |
| Vertices | 3360 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |

### Alternate names

* Petiprismatotruncated octaexon (acronym: pupato) (Jonathan Bowers)

### Coordinates

The vertices of the hexiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,2,3,4). This construction is based on facets of the hexiruncitruncated 8-orthoplex, .

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Hexiruncicantellated 7-simplex

Hexiruncicantellated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,2,3,6}${3$^6$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 16800 |
| Vertices | 3360 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |
In seven-dimensional geometry, a hexiruncicantellated 7-simplex is a uniform 7-polytope.

### Alternate names

* Petiprismatorhombated octaexon (acronym: pupro) (Jonathan Bowers)

### Coordinates

The vertices of the hexiruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,1,2,3,3,4). This construction is based on facets of the hexiruncicantellated 8-orthoplex, .

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Hexisteritruncated 7-simplex

hexisteritruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,4,6}${3$^6$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 20160 |
| Vertices | 3360 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |

### Alternate names

* Peticellitruncated octaexon (acronym: pucto) (Jonathan Bowers)

### Coordinates

The vertices of the hexisteritruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,2,3,4). This construction is based on facets of the hexisteritruncated 8-orthoplex, .

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Hexistericantellated 7-simplex

hexistericantellated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,2,4,6}${3$^6$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | t$_{0,2,4}${3,3,3,3,3}xt$_{0,2,4}${3,3,3,3} {3}xt$_{0,2}${3,3,3} t$_{0,2}${3,3}xt$_{0,2}${3,3} |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 30240 |
| Vertices | 5040 |
| Vertex figure | |
| Coxeter group | A$_7$×2, [[3$^6$]], order 80640 |
| Properties | convex |

### Alternate names

* Peticellirhombihexadecaexon (acronym: pucroh) (Jonathan Bowers)

### Coordinates

The vertices of the hexistericantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,3,4). This construction is based on facets of the hexistericantellated 8-orthoplex, .

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [[7]] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [[5]] | [4] | [[3]] |

## Hexipentitruncated 7-simplex

Hexipentitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,5,6}${3$^6$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 8400 |
| Vertices | 1680 |
| Vertex figure | |
| Coxeter group | A$_7$×2, [[3$^6$]], order 80640 |
| Properties | convex |

### Alternate names

* Petiteritruncated hexadecaexon (acronym: putath) (Jonathan Bowers)

### Coordinates

The vertices of the hexipentitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,2,2,3,4). This construction is based on facets of the hexipentitruncated 8-orthoplex, .

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [[7]] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [[5]] | [4] | [[3]] |

## Hexiruncicantitruncated 7-simplex

Hexiruncicantitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,3,6}${3$^6$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 30240 |
| Vertices | 6720 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |

### Alternate names

* Petigreatoprismated octaexon (acronym: pugopo) (Jonathan Bowers)

### Coordinates

The vertices of the hexiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,4,5). This construction is based on facets of the hexiruncicantitruncated 8-orthoplex, .

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [[7]] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [[5]] | [4] | [[3]] |

## Hexistericantitruncated 7-simplex

Hexistericantitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,4,6}${3$^6$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 50400 |
| Vertices | 10080 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |

### Alternate names

* Peticelligreatorhombated octaexon (acronym: pucagro) (Jonathan Bowers)

### Coordinates

The vertices of the hexistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,2,3,4,5). This construction is based on facets of the hexistericantitruncated 8-orthoplex, .

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [[7]] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [[5]] | [4] | [[3]] |

## Hexisteriruncitruncated 7-simplex

Hexisteriruncitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,3,4,6}${3$^6$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 45360 |
| Vertices | 10080 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |

### Alternate names

* Peticelliprismatotruncated octaexon (acronym: pucpato) (Jonathan Bowers)

### Coordinates

The vertices of the hexisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,3,4,5). This construction is based on facets of the hexisteriruncitruncated 8-orthoplex, .

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Hexisteriruncicantellated 7-simplex

Hexisteriruncicantellated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,2,3,4,6}${3$^6$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 45360 |
| Vertices | 10080 |
| Vertex figure | |
| Coxeter group | A$_7$×2, [[3$^6$]], order 80640 |
| Properties | convex |

### Alternate names

* Peticelliprismatorhombihexadecaexon (acronym: pucproh) (Jonathan Bowers)

### Coordinates

The vertices of the hexisteriruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,4,4,5). This construction is based on facets of the hexisteriruncitruncated 8-orthoplex, .

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [[7]] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [[5]] | [4] | [[3]] |

## Hexipenticantitruncated 7-simplex

hexipenticantitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,5,6}${3$^6$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 30240 |
| Vertices | 6720 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |

### Alternate names

* Petiterigreatorhombated octaexon (acronym: putagro) (Jonathan Bowers)

### Coordinates

The vertices of the hexipenticantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,2,3,4,5). This construction is based on facets of the hexipenticantitruncated 8-orthoplex, .

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [7] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [5] | [4] | [3] |

## Hexipentiruncitruncated 7-simplex

Hexipentiruncitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,3,5,6}${3$^6$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | 10080 |
| Vertex figure | |
| Coxeter group | A$_7$×2, [[3$^6$]], order 80640 |
| Properties | convex |

### Alternate names

* Petiteriprismatotruncated hexadecaexon (acronym: putpath) (Jonathan Bowers)

### Coordinates

The vertices of the hexipentiruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,3,4,4,5). This construction is based on facets of the hexipentiruncitruncated 8-orthoplex, .

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [[7]] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [[5]] | [4] | [[3]] |

## Hexisteriruncicantitruncated 7-simplex

Hexisteriruncicantitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,3,4,6}${3$^6$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 80640 |
| Vertices | 20160 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |

### Alternate names

* Petigreatocellated octaexon (acronym: pugaco) (Jonathan Bowers)

### Coordinates

The vertices of the hexisteriruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,1,2,3,4,5,6). This construction is based on facets of the hexisteriruncicantitruncated 8-orthoplex, .

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [[7]] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [[5]] | [4] | [[3]] |

## Hexipentiruncicantitruncated 7-simplex

Hexipentiruncicantitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,3,5,6}${3$^6$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 80640 |
| Vertices | 20160 |
| Vertex figure | |
| Coxeter group | A$_7$, [3$^6$], order 40320 |
| Properties | convex |

### Alternate names

* Petiterigreatoprismated octaexon (acronym: putgapo) (Jonathan Bowers)

### Coordinates

The vertices of the hexipentiruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,2,3,4,5,6). This construction is based on facets of the hexipentiruncicantitruncated 8-orthoplex, .

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [[7]] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [[5]] | [4] | [[3]] |

## Hexipentistericantitruncated 7-simplex

Hexipentistericantitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,4,5,6}${3$^6$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 80640 |
| Vertices | 20160 |
| Vertex figure | |
| Coxeter group | A$_7$×2, [[3$^6$]], order 80640 |
| Properties | convex |

### Alternate names

* Petitericelligreatorhombihexadecaexon (acronym: putcagroh) (Jonathan Bowers)

### Coordinates

The vertices of the hexipentistericantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,3,3,4,5,6). This construction is based on facets of the hexipentistericantitruncated 8-orthoplex, .

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [[7]] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [[5]] | [4] | [[3]] |

## Omnitruncated 7-simplex

Omnitruncated 7-simplex :
| Type | uniform 7-polytope |
| Schläfli symbol | t$_{0,1,2,3,4,5,6}${3$^6$} |
| Coxeter-Dynkin diagrams | |
| 6-faces | 254 |
| 5-faces | 5796 |
| 4-faces | 40824 |
| Cells | 126000 |
| Faces | 191520 |
| Edges | 141120 |
| Vertices | 40320 |
| Vertex figure | Irr. 6-simplex |
| Coxeter group | A$_7$×2, [[3$^6$]], order 80640 |
| Properties | convex |
The omnitruncated 7-simplex is composed of 40320 (8 factorial) vertices and is the largest uniform 7-polytope in the A$_7$ symmetry of the regular 7-simplex. It can also be called the hexipentisteriruncicantitruncated 7-simplex which is the long name for the omnitruncation for 7 dimensions, with all reflective mirrors active.

### Permutohedron and related tessellation

The omnitruncated 7-simplex is the permutohedron of order 8. The omnitruncated 7-simplex is a zonotope, the Minkowski sum of eight line segments parallel to the eight lines through the origin and the eight vertices of the 7-simplex.
Like all uniform omnitruncated n-simplices, the omnitruncated 7-simplex can tessellate space by itself, in this case 7-dimensional space with three facets around each ridge. It has Coxeter-Dynkin diagram of .

### Alternate names

* Great petated hexadecaexon (Acronym: guph) (Jonathan Bowers)

### Coordinates

The vertices of the omnitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,1,2,3,4,5,6,7). This construction is based on facets of the hexipentisteriruncicantitruncated 8-orthoplex, t$_{0,1,2,3,4,5,6}${3$^6$,4}, .

### Images

orthographic projections :
| A$_k$ Coxeter plane | A$_7$ | A$_6$ | A$_5$ |
| Graph | | | |
| Dihedral symmetry | [8] | [[7]] | [6] |
| A$_k$ Coxeter plane | A$_4$ | A$_3$ | A$_2$ |
| Graph | | | |
| Dihedral symmetry | [[5]] | [4] | [[3]] |

## Related polytopes

These polytope are a part of 71 uniform 7-polytopes with A$_7$ symmetry.
A7 polytopes :
| t$_0$ | t$_1$ | t$_2$ | t$_3$ | t$_{0,1}$ | t$_{0,2}$ | t$_{1,2}$ | t$_{0,3}$ |
| t$_{1,3}$ | t$_{2,3}$ | t$_{0,4}$ | t$_{1,4}$ | t$_{2,4}$ | t$_{0,5}$ | t$_{1,5}$ | t$_{0,6}$ |
| t$_{0,1,2}$ | t$_{0,1,3}$ | t$_{0,2,3}$ | t$_{1,2,3}$ | t$_{0,1,4}$ | t$_{0,2,4}$ | t$_{1,2,4}$ | t$_{0,3,4}$ |
| t$_{1,3,4}$ | t$_{2,3,4}$ | t$_{0,1,5}$ | t$_{0,2,5}$ | t$_{1,2,5}$ | t$_{0,3,5}$ | t$_{1,3,5}$ | t$_{0,4,5}$ |
| t$_{0,1,6}$ | t$_{0,2,6}$ | t$_{0,3,6}$ | t$_{0,1,2,3}$ | t$_{0,1,2,4}$ | t$_{0,1,3,4}$ | t$_{0,2,3,4}$ | t$_{1,2,3,4}$ |
| t$_{0,1,2,5}$ | t$_{0,1,3,5}$ | t$_{0,2,3,5}$ | t$_{1,2,3,5}$ | t$_{0,1,4,5}$ | t$_{0,2,4,5}$ | t$_{1,2,4,5}$ | t$_{0,3,4,5}$ |
| t$_{0,1,2,6}$ | t$_{0,1,3,6}$ | t$_{0,2,3,6}$ | t$_{0,1,4,6}$ | t$_{0,2,4,6}$ | t$_{0,1,5,6}$ | t$_{0,1,2,3,4}$ | t$_{0,1,2,3,5}$ |
| t$_{0,1,2,4,5}$ | t$_{0,1,3,4,5}$ | t$_{0,2,3,4,5}$ | t$_{1,2,3,4,5}$ | t$_{0,1,2,3,6}$ | t$_{0,1,2,4,6}$ | t$_{0,1,3,4,6}$ | t$_{0,2,3,4,6}$ |
| t$_{0,1,2,5,6}$ | t$_{0,1,3,5,6}$ | t$_{0,1,2,3,4,5}$ | t$_{0,1,2,3,4,6}$ | t$_{0,1,2,3,5,6}$ | t$_{0,1,2,4,5,6}$ | t$_{0,1,2,3,4,5,6}$ |