In 7-dimensional geometry, **2**_{31} is a uniform polytope, constructed from the E7 group.

Its Coxeter symbol is **2**_{31}, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node branch.

The **rectified 2**_{31} is constructed by points at the mid-edges of the **2**_{31}.

These polytopes are part of a family of 127 (or 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: .

The **2**_{31} is composed of 126 vertices, 2016 edges, 10080 faces (Triangles), 20160 cells (tetrahedra), 16128 4-faces (3-simplexes), 4788 5-faces (756 pentacrosses, and 4032 5-simplexes), 632 6-faces (576 6-simplexes and 56 **2**_{21}). Its vertex figure is a 6-demicube. Its 126 vertices represent the root vectors of the simple Lie group E_{7}.

This polytope is the vertex figure for a uniform tessellation of 7-dimensional space, **3**_{31}.

E. L. Elte named it V_{126} (for its 126 vertices) in his 1912 listing of semiregular polytopes.
It was called **2**_{31} by Coxeter for its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence.
*Pentacontihexa-pentacosiheptacontihexa-exon* (Acronym laq) - 56-576 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 short branch leaves the 6-simplex. There are 576 of these facets. These facets are centered on the locations of the vertices of the 3_{21} polytope, .

Removing the node on the end of the 3-length branch leaves the 2_{21}. There are 56 of these facets. These facets are centered on the locations of the vertices of the 1_{32} polytope, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node. This makes the 6-demicube, 1_{31}, .

The **rectified 2**_{31} is a rectification of the 2_{31} polytope, creating new vertices on the center of edge of the 2_{31}.

Rectified pentacontihexa-pentacosiheptacontihexa-exon - as a rectified 56-576 facetted polyexon (acronym rolaq) (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 short branch leaves the rectified 6-simplex, .

Removing the node on the end of the 2-length branch leaves the, 6-demicube, .

Removing the node on the end of the 3-length branch leaves the rectified 2_{21}, .

The vertex figure is determined by removing the ringed node and ringing the neighboring node.