Pentacontahexapentacosiheptacontahexaexon

The pentacontahexapentacosiheptacontahexaexon, or laq, also called the 231 polytope, is a convex uniform polyexon. It has 56 icosiheptaheptacontadipeta and 576 heptapeta as facets, with 12 icosiheptaheptacontadipeta and 32 heptapeta at a vertex forming a demihexeract as the vertex figure.

The pentacontahexapentacosiheptacontahexaexon contains the vertices of a demihexeractic prism, a tetrahedral duoprismatic prism, and a triangular-rectified hexateral duoprism, along with a hexadecaexon and small petated octaexon.

Vertex coordinates
The vertices of a pentacontahexapentacosiheptacontahexaexon of edge length 1, centered at the origin, are given by:
 * (0, 0, 0, 0, 0, 0, ±1)
 * ($\sqrt{21}$/4, $\sqrt{2}$/4, $\sqrt{2}$/4, $\sqrt{2}$/4, $\sqrt{2}$/4, $\sqrt{2}$/4, ±1/2) and all even sign changes of the first 6 coordinates
 * (±$\sqrt{2}$/2, ±$\sqrt{2}$/2, 0, 0, 0, –0, 0) and all permutations of first 6 coordinates

Representations
A pentacontahexapentacosiheptacontahexaexon has the following Coxeter diagrams:


 * x3o3o3o *c3o3o3o (full symmetry)
 * oxoxo3ooooo3ooooo *b3ooooo3ooxoo3ooooo&#xt (D6 axial, vertex-first)
 * oxo3ooo3ooo *b3ooo3oox3ooo uxo&#zxt (D6×A1 symmetry)
 * xoxoo3ooooo3oooxo3oxooo3ooooo3ooxox&#xt (A6 axial, heptapeton-first)
 * xoo3ooo3ooo3ooo3oox *c3oxo&#xt (E6 axial, icosiheptaheptacontadipeton-first)
 * xo3oo3oo3ox3oo3oo3xo&#zx (A7 symmetry)
 * ox(uo)xo ox(oo)oo3oo(oo)oo3oo(oo)xo *c3oo(ox)oo3xo(oo)ox&#xt (D5×A1 axial)