The series

Features

• Algebraic structure: group of order 4m containing as normal subgroup.
• Kind of elements
  • 1 element: identity,
  • m-1 elements: rotation of angle k2pi/m with 1<=k<=m-1 around D,
  • m elements: m rotation of order 2 around axis Dk, 1<=k<=m with angle between Dk and Dk+1 equal to pi/m
  • 1 element: symmetry by the plane P,
  • m-1 elements: composition of symmetry by the plane P and rotation of angle k2pi/m,
  • m elements: plane symmetry by the plane containing Dk and D.
• The group contains an inversion if and only if m is even
• A simple way to distinguish between serie and serie :
  • In case, the two-fold axis belong to symmetry plane
  • In case, the two-fold axis do not belong to symmetry planes
• Particular cases
  • if m=1, then the group is C2v
  • if m=2, the group is Z2xZ2xZ2 and is so commutative.
• Examples
  • First Fulleren of symmetry D2h:
    
  • First Fulleren of symmetry D3h:
    
  • First Fulleren of symmetry D5h:
    
  • First Fulleren of symmetry D6h:
    
  • First 4n of symmetry D2h:
    
  • First 4n of symmetry D3h:
    
  • First 4n of symmetry D6h: