Prismatic uniform 4-polytope

In four-dimensional geometry, a prismatic uniform 4-polytope is a uniform 4-polytope with a nonconnected Coxeter diagram symmetry group. These figures are analogous to the set of prisms and antiprism uniform polyhedra, but add a third category called duoprisms, constructed as a product of two regular polygons.
The prismatic uniform 4-polytopes consist of two infinite families:

Polyhedral prisms: products of a line segment and a uniform polyhedron. This family is infinite because it includes prisms built on 3-dimensional prisms and antiprisms.
Duoprisms: product of two regular polygons.

Convex polyhedral prisms

The most obvious family of prismatic 4-polytopes is the polyhedral prisms, i.e. products of a polyhedron with a line segment. The cells of such a 4-polytope are two identical uniform polyhedra lying in parallel hyperplanes and a layer of prisms joining them. This family includes prisms for the 75 nonprismatic uniform polyhedra.
There are 18 convex polyhedral prisms created from 5 Platonic solids and 13 Archimedean solids as well as for the infinite families of three-dimensional prisms and antiprisms. The symmetry number of a polyhedral prism is twice that of the base polyhedron.

Tetrahedral prisms: A₃ × A₁

Octahedral prisms: BC₃ × A₁

Icosahedral prisms: H₃ × A₁

Duoprisms: p × q

The second is the infinite family of uniform duoprisms, products of two regular polygons.
Their Coxeter diagram is of the form
This family overlaps with the first: when one of the two "factor" polygons is a square, the product is equivalent to a hyperprism whose base is a three-dimensional prism. The symmetry number of a duoprism whose factors are a p-gon and a q-gon is 4pq if p≠''q; if the factors are both p''-gons, the symmetry number is 8p². The tesseract can also be considered a 4,4-duoprism.
The elements of a p,q-duoprism are:

Cells: p q-gonal prisms, q p-gonal prisms
Faces: pq squares, p q-gons, q p-gons
Edges: 2pq
Vertices: pq

There is no uniform analogue in four dimensions to the infinite family of three-dimensional antiprisms with the exception of the great duoantiprism.
Infinite set of p-q duoprism - - p q-gonal prisms, q p-gonal prisms:3-3 duoprism - - 6 triangular prisms3-4 duoprism - - 3 cubes, 4 triangular prisms4-4 duoprism - - 8 cubes 3-5 duoprism - - 3 pentagonal prisms, 5 triangular prisms4-5 duoprism - - 4 pentagonal prisms, 5 cubes5-5 duoprism - - 10 pentagonal prisms3-6 duoprism - - 3 hexagonal prisms, 6 triangular prisms4-6 duoprism - - 4 hexagonal prisms, 6 cubes5-6 duoprism - - 5 hexagonal prisms, 6 pentagonal prisms6-6 duoprism - - 12 hexagonal prisms

Polygonal prismatic prisms

The infinite set of uniform prismatic prisms overlaps with the 4-p duoprisms: - - p cubes and 4 p-gonal prisms - Triangular prismatic prism - - 3 cubes and 4 triangular prisms - Square prismatic prism - - 4 cubes and 4 cubes - Pentagonal prismatic prism - - 5 cubes and 4 pentagonal prisms - Hexagonal prismatic prism - - 6 cubes and 4 hexagonal prisms - Heptagonal prismatic prism - - 7 cubes and 4 heptagonal prisms - Octagonal prismatic prism - - 8 cubes and 4 octagonal prisms -

Uniform antiprismatic prism

The infinite sets of uniform antiprismatic prisms or antiduoprisms are constructed from two parallel uniform antiprisms: - - 2 p-gonal antiprisms, connected by 2 p-gonal prisms and 2p triangular prisms.
A p-gonal antiprismatic prism has 4p triangle, 4p square and 4 p-gon faces. It has 10p edges, and 4p vertices.