Editing Truncated cuboctahedron (section)

==Full octahedral group==
[[File:Full octahedral group elements in truncated cuboctahedron; JF.png|thumb|right]]
Like many other solids the truncated octahedron has full [[octahedral symmetry]] - but its relationship with the full octahedral group is closer than that: Its 48 vertices correspond to the elements of the group, and each face of [[Disdyakis dodecahedron|its dual]] is a [[fundamental domain]] of the group.

The image on the right shows the 48 permutations in the group applied to an example object (namely the light JF compound on the left). The 24 light elements are rotations, and the dark ones are their reflections.

The edges of the solid correspond to the 9 reflections in the group:
* Those between octagons and squares correspond to the 3 reflections between opposite octagons.
* Hexagon edges correspond to the 6 reflections between opposite squares.
* (There are no reflections between opposite hexagons.)

The subgroups correspond to solids that share the respective vertices of the truncated octahedron.<br>
E.g. the 3 subgroups with 24 elements correspond to a nonuniform [[snub cube]] with chiral octahedral symmetry, a nonuniform [[rhombicuboctahedron]] with [[pyritohedral symmetry]] (the [[cantic snub octahedron]]) and a nonuniform [[truncated octahedron]] with [[full tetrahedral symmetry]]. The unique subgroup with 12 elements is the [[alternating group]] A<sub>4</sub>. It corresponds to a nonuniform [[icosahedron]] with [[chiral tetrahedral symmetry]].

{| class="wikitable collapsible" style="text-align: center;"
!colspan="5"| Subgroups and corresponding solids
|- valign=top
!Truncated cuboctahedron<BR>{{CDD|node_1|4|node_1|3|node_1}}<BR>tr{4,3}
![[Snub cube]]<BR>{{CDD|node_h|4|node_h|3|node_h}}<BR>sr{4,3}
![[Rhombicuboctahedron#Pyritohedral_symmetry|Rhombicuboctahedron]]<BR>{{CDD|node_1|4|node_h|3|node_h}}<BR>s<sub>2</sub>{3,4}
![[Truncated octahedron]]<BR>{{CDD|node_h|4|node_1|3|node_1}}<BR>h<sub>1,2</sub>{4,3}
![[Regular_icosahedron#Symmetries|Icosahedron]]<BR>{{CDD|node_h|2|4|2|node_h|3|node_h}}
|-
![4,3]<BR>[[Octahedral_symmetry#Subgroups_of_full_octahedral_symmetry|Full octahedral]]
![4,3]<sup>+</sup><BR>Chiral octahedral
![4,3<sup>+</sup>]<BR>[[Tetrahedral_symmetry#Subgroups_of_pyritohedral_symmetry|Pyritohedral]]
![1<sup>+</sup>,4,3] = [3,3]<BR>[[Tetrahedral_symmetry#Subgroups_of_achiral_tetrahedral_symmetry|Full tetrahedral]]
![1<sup>+</sup>,4,3<sup>+</sup>] = [3,3]<sup>+</sup><BR>Chiral tetrahedral
|-
| [[File:Polyhedron great rhombi 6-8 max.png|150px]]
| [[File:Polyhedron great rhombi 6-8 subsolid snub right maxmatch.png|150px]]
| [[File:Polyhedron great rhombi 6-8 subsolid pyritohedral maxmatch.png|150px]]
| [[File:Polyhedron great rhombi 6-8 subsolid tetrahedral maxmatch.png|150px]]
| [[File:Polyhedron great rhombi 6-8 subsolid 20 maxmatch.png|150px]]
|-
| all 48 vertices
|colspan="3"| 24 vertices
| 12 vertices
|}