Octahedral symmetry

In mathematics, the octahedral group is depending on the Convention

• The symmetry group of an octahedron, so the amount of the pictures back on itself an octahedron, ie Represent corners on corners, edges to edges, etc.;
• Or the rotation group of an octahedron, a subgroup of the symmetry group, are not permitted in the glare and reflections rotation. The full symmetry group is then called to distinguish complete, binary or extended (F. Klein) octahedral group.

Together, the two groups are the following illustrations as elements:

• 90 °, 180 °, 270 ° - rotation about the three four-fold axes of rotation ( through opposite corners)
• 120 °, 240 ° - rotation about the three-fold rotation axes 4 ( through opposite centroids )
• 180 ° rotation about the twofold axes of rotation 6 ( through opposite edge center points)
• The identity

This results in elements of the rotation group, combined with reflections arise resulting elements of the symmetry group.

The groups for the octahedron and cube are isomorphic as dual body have the same symmetry type. Therefore, one can call good and dice the octahedral group as well. The rotation group of the cube is canonically isomorphic to the symmetric group on the set of spatial diagonals, ie the group of permutations of the four space diagonals; Rotation group of cube and octahedron are thus (not canonically ) isomorphic to the symmetric group. The symmetry group is the direct product of the rotation group to the two-element group, which is produced by the reflection at the center point.

In crystallography is called the rotation group of the octahedron and the complete symmetry group.

• The finite set
• Space geometry
• Group Theory