In 7-dimensional geometry, **1**_{32} is a uniform polytope, constructed from the E7 group.

Its Coxeter symbol is **1**_{32}, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences.

The **rectified 1**_{32} is constructed by points at the mid-edges of the **1**_{32}.

These polytopes are part of a family of 127 (2^{7}-1) convex uniform polytopes in 7-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .

This polytope can tessellate 7-dimensional space, with symbol **1**_{33}, and Coxeter-Dynkin diagram, . It is the Voronoi cell of the dual E_{7}^{*} lattice.

Emanuel Lodewijk Elte named it V_{576} (for its 576 vertices) in his 1912 listing of semiregular polytopes.
Coxeter called it **1**_{32} for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node branch.
*Pentacontihexa-hecatonicosihexa-exon* (Acronym lin) - 56-126 facetted polyexon (Jonathan Bowers)
It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-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 6-demicube, 1_{31},

Removing the node on the end of the 3-length branch leaves the 1_{22},

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the birectified 6-simplex, 0_{32},

The 1_{32} is third in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 1_{3k} series. The next figure is the Euclidean honeycomb 1_{33} and the final is a noncompact hyperbolic honeycomb, 1_{34}.

The **rectified 1**_{32} (also called **0**_{321}) is a rectification of the 1_{32} polytope, creating new vertices on the center of edge of the 1_{32}. Its vertex figure is a duoprism prism, the product of a regular tetrahedra and triangle, doubled into a prism: {3,3}×{3}×{}.

Rectified pentacontihexa-hecatonicosihexa-exon for rectified 56-126 facetted polyexon (acronym rolin) (Jonathan Bowers)
It is created by a Wythoff construction upon a set of 7 hyperplane mirrors in 7-dimensional space. These mirrors are represented by its Coxeter-Dynkin diagram, , and the ring represents the position of the active mirror(s).

Removing the node on the end of the 3-length branch leaves the rectified 1_{22} polytope,

Removing the node on the end of the 2-length branch leaves the demihexeract, 1_{31},

Removing the node on the end of the 1-length branch leaves the birectified 6-simplex,

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the tetrahedron-triangle duoprism prism, {3,3}×{3}×{},