The series
The group
is the group generated by a rotation of angle 2pi/m around a fixed axis D and m rotation of order 2 around m axis Dm orthogonal to D.
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
m elements: m composition of plane symmetry by P and rotation of angle (1+2k)pi/m
m elements: plane symmetries by planes Pk containing D and going between Dk and Dk+1.
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 C2h
Examples
First Fulleren of symmetry D2d:
First Fulleren of symmetry D3d:
First Fulleren of symmetry D5d:
First Fulleren of symmetry D6d:
First 4n of symmetry D2d:
First 4n of symmetry D3d: