# Generalized hypercube

γ p
n

Rankn
Dimensionn
TypeRegular
SpaceComplex
Notation
Coxeter diagram...
Schläfli symbolp{4}2{3}2...2{3}2{3}2
Elements
Facetspn  γ p
n-1

Edges2p n-1  p -edges
Verticesp n
Vertex figure(n -1)-simplex
Related polytopes
DualGeneralized orthoplex
Abstract & topological properties
Flag count${\displaystyle n!p^{n}}$
Properties
Symmetryp[4]2[3]2...2[3]2[3]2, order ${\displaystyle n!p^{n}}$

The generalized hypercubes, γ p
n

, are a family of regular complex polytopes that generalize the hypercubes.

The generalized hypercube γ p
n

can be formed as the prism product of n  identical p -edges. This means that the real analogs of generalized hypercubes are multiprisms.

The generalized hypercubes of the form γ 2
n

have real valued coordinates and are exactly the hypercubes.

## Vertex coordinates

Vertex coordinates for the generalized hypercube γ p
n

can be given as:

• ${\displaystyle \left(e^{x_{1}2\pi i/p},e^{x_{2}2\pi i/p},e^{x_{3}2\pi i/p},\dots ,e^{x_{n}2\pi i/p}\right)}$,

where each x k  is an integer ranging between 1 and p  inclusive.