There are two Coxeter groups associated with the 10-orthoplex, one regular, dual of the 10-cube with the C10 or 4,38 symmetry group, and a lower symmetry with two copies of 9-simplex facets, alternating, with the D10 or 37,1,1 symmetry group.
There are three Coxeter groups associated with the 5-orthoplex, one regular, dual of the penteract with the C5 or 4,3,3,3 Coxeter group, and a lower symmetry with two copies of 5-cell facets, alternating, with the D5 or 32,1,1 Coxeter group, and the final one as a dual 5-orthotope, called a 5-fusil which can have a variety of subsymmetries.
There are three Coxeter groups associated with the 6-orthoplex, one regular, dual of the hexeract with the C6 or 4,3,3,3,3 Coxeter group, and a half symmetry with two copies of 5-simplex facets, alternating, with the D6 or 33,1,1 Coxeter group.
There are two Coxeter groups associated with the 7-orthoplex, one regular, dual of the hepteract with the C7 or 4,3,3,3,3,3 symmetry group, and a half symmetry with two copies of 6-simplex facets, alternating, with the D7 or 34,1,1 symmetry group.
There are two Coxeter groups associated with the 8-cube, one regular, dual of the octeract with the C8 or 4,3,3,3,3,3,3 symmetry group, and a half symmetry with two copies of 7-simplex facets, alternating, with the D8 or 35,1,1 symmetry group.A lowest symmetry construction is based on a dual of a 8-orthotope, called a 8-fusil.
There are two Coxeter groups associated with the 9-orthoplex, one regular, dual of the enneract with the C9 or 4,37 symmetry group, and a lower symmetry with two copies of 8-simplex facets, alternating, with the D9 or 36,1,1 symmetry group.
Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes.
Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes (5-cells and 16-cells in the case of the rectified 5-cell).
There are two Coxeter groups associated with the bicantitruncated 6-orthoplex, one with the BC6 or 4,3,3,3,3 Coxeter group, and a lower symmetry with the D6 or 33,1,1 Coxeter group.
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There are two Coxeter groups associated with the cantitruncated 6-orthoplex, one with the BC6 or 4,3,3,3,3 Coxeter group, and a lower symmetry with the D6 or 33,1,1 Coxeter group.
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There are two Coxeter groups associated with the cantellated 6-orthoplex, one with the BC6 or 4,3,3,3,3 Coxeter group, and a lower symmetry with the D6 or 33,1,1 Coxeter group.
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There are two Coxeter groups associated with the bicantellated 6-orthoplex, one with the BC6 or 4,3,3,3,3 Coxeter group, and a lower symmetry with the D6 or 33,1,1 Coxeter group.
There are two Coxeter groups associated with the rectified 10-orthoplex, one with the C10 or 4,38 Coxeter group, and a lower symmetry with two copies of 9-orthoplex facets, alternating, with the D10 or 37,1,1 Coxeter group.
There are two Coxeter groups associated with the rectified 8-orthoplex, one with the C8 or 4,36 Coxeter group, and a lower symmetry with two copies of heptcross facets, alternating, with the D8 or 35,1,1 Coxeter group.
There are two Coxeter groups associated with the rectified 9-orthoplex, one with the C9 or 4,37 Coxeter group, and a lower symmetry with two copies of 8-orthoplex facets, alternating, with the D9 or 36,1,1 Coxeter group.
There are two Coxeter groups associated with the truncated 7-orthoplex, one with the C7 or 4,35 Coxeter group, and a lower symmetry with the D7 or 34,1,1 Coxeter group.
There are two Coxeter groups associated with the truncated 8-orthoplex, one with the C8 or 4,3,3,3,3,3,3 Coxeter group, and a lower symmetry with the D8 or 35,1,1 Coxeter group.
The orthoplex faces are constructed from the Coxeter group Dn−1 and have a Schlafli symbol of {31,n−1,1} rather than the regular {3n−2,4}.