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 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.


E.g. the 3 subgroups with 24 elements correspond to a nonuniform snub cube with chiral octahedral symmetry, a nonuniform truncated octahedron with full tetrahedral symmetry and a nonuniform rhombicuboctahedron with pyritohedral symmetry (the cantic snub octahedron).


The unique subgroup with 12 elements is the alternating group A₄. It corresponds to a nonuniform icosahedron with chiral tetrahedral symmetry.
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 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.

E.g. the 3 subgroups with 24 elements correspond to a nonuniform snub cube with chiral octahedral symmetry, a nonuniform truncated octahedron with full tetrahedral symmetry and a nonuniform rhombicuboctahedron with pyritohedral symmetry (the cantic snub octahedron).

The unique subgroup with 12 elements is the alternating group A₄. It corresponds to a nonuniform icosahedron with chiral tetrahedral symmetry.
Truncated cuboctahedron