7cube Hepteract  

Orthogonal projection inside Petrie polygon The central orange vertex is doubled  
Type  Regular 7polytope 
Family  hypercube 
Schläfli symbol  {4,3^{5}} 
CoxeterDynkin diagrams 

6faces  14 {4,3^{4}} 
5faces  84 {4,3^{3}} 
4faces  280 {4,3,3} 
Cells  560 {4,3} 
Faces  672 {4} 
Edges  448 
Vertices  128 
Vertex figure  6simplex 
Petrie polygon  tetradecagon 
Coxeter group  C_{7}, [3^{5},4] 
Dual  7orthoplex 
Properties  convex, Hanner polytope 
In geometry, a 7cube is a sevendimensional hypercube with 128 vertices, 448 edges, 672 square faces, 560 cubic cells, 280 tesseract 4faces, 84 penteract 5faces, and 14 hexeract 6faces.
It can be named by its Schläfli symbol {4,3^{5}}, being composed of 3 6cubes around each 5face. It can be called a hepteract, a portmanteau of tesseract (the 4cube) and hepta for seven (dimensions) in Greek. It can also be called a regular tetradeca7tope or tetradecaexon, being a 7 dimensional polytope constructed from 14 regular facets.
The 7cube is 7th in a series of hypercube:
Line segment  Square  Cube  4cube  5cube  6cube  7cube  8cube  9cube  10cube 
The dual of a 7cube is called a 7orthoplex, and is a part of the infinite family of crosspolytopes.
Applying an alternation operation, deleting alternating vertices of the hepteract, creates another uniform polytope, called a demihepteract, (part of an infinite family called demihypercubes), which has 14 demihexeractic and 64 6simplex 6faces.
This configuration matrix represents the 7cube. The rows and columns correspond to vertices, edges, faces, cells, 4faces, 5faces and 6faces. The diagonal numbers say how many of each element occur in the whole 7cube. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^{[1]}^{[2]}
Cartesian coordinates for the vertices of a hepteract centered at the origin and edge length 2 are
while the interior of the same consists of all points (x_{0}, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}) with 1 < x_{i} < 1.
This hypercube graph is an orthogonal projection. This orientation shows columns of vertices positioned a vertexedgevertex distance from one vertex on the left to one vertex on the right, and edges attaching adjacent columns of vertices. The number of vertices in each column represents rows in Pascal's triangle, being 1:7:21:35:35:21:7:1. 
Coxeter plane  B_{7} / A_{6}  B_{6} / D_{7}  B_{5} / D_{6} / A_{4} 

Graph  
Dihedral symmetry  [14]  [12]  [10] 
Coxeter plane  B_{4} / D_{5}  B_{3} / D_{4} / A_{2}  B_{2} / D_{3} 
Graph  
Dihedral symmetry  [8]  [6]  [4] 
Coxeter plane  A_{5}  A_{3}  
Graph  
Dihedral symmetry  [6]  [4] 