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Top of Page Molecular Structures of Organic Compounds - Three-dimensional Stereographs
Three-dimensional Stereographs and Symmetry Elements:

The table below provides an overview on the three-dimensional stereographic representations of point groups (including the 32 'Crystallographic Point Groups'). Please note, that although any positive integral value of n is allowed for the Cn, Cnv, Cnh, Dn, Dnh, Dnd, and Sn point groups of molecules, only a limited number is listed here. Additionally, the S1 point group is equivalent to Cs and S2 corresponds to Ci, any Sn with odd values of n is categorized under Cnh and thus not considered here.

Chiral point groups are marked by bold-face point group symbols, all other point groups are achiral. The order of each point group is indicated in parenthesis, it is equivalent to the number of symmetry related positions in the stereographic projections. The point group Kh of a sphere was included for completeness, no molecule belongs to this group.

Symmetry and Point Groups
Symmetry Elements of a 3D-Object
Top of Page Three-dimensional Stereographic Representations of Point Groups
General*   Point Groups and Three-dimensional Stereographic Projections*
   
n = 1
n = 2
n = 3
n = 4
n = 5
n = 6
n = ∞
Cn  
Stereographic Projection of Point Group C1
Stereographic Projection of Point Group C2
Stereographic Projection of Point Group C3
Stereographic Projection of Point Group C4
Stereographic Projection of Point Group C5
Stereographic Projection of Point Group C6
-
   
C1 (1)
C2 (2)
C3 (3)
C4 (4)
C5 (5)
C6 (6)
 
Cnv  
-
Stereographic Projection of Point Group C2v
Stereographic Projection of Point Group C3v
Stereographic Projection of Point Group C4v
Stereographic Projection of Point Group C5v
Stereographic Projection of Point Group C6v
Stereographic Projection of Point Group Cinfv
     
C2v (4)
C3v (6)
C4v (8)
C5v (10)
C6v (12)
C∞v (∞)
Cnh  
Stereographic Projection of Point Group Cs
Stereographic Projection of Point Group C2h
Stereographic Projection of Point Group C3h
Stereographic Projection of Point Group C4h
Stereographic Projection of Point Group C5h
Stereographic Projection of Point Group C6h
-
   
(C1h = ) Cs (2)
C2h (4)
C3h (6)
C4h (8)
C5h (10)
C6h (12)
 
Dn  
-
Stereographic Projection of Point Group D2
Stereographic Projection of Point Group D3
Stereographic Projection of Point Group D4
Stereographic Projection of Point Group D5
Stereographic Projection of Point Group D6
-
     
D2 (4)
D3 (6)
D4 (8)
D5 (10)
D6 (12)
 
Dnh  
-
Stereographic Projection of Point Group D2h
Stereographic Projection of Point Group D3h
Stereographic Projection of Point Group D4h
Stereographic Projection of Point Group D5h
Stereographic Projection of Point Group D6h
Stereographic Projection of Point Group Dinfh
     
D2h (8)
D3h (12)
D4h (16)
D5h (20)
D6h (24)
D∞h (∞)
Dnd  
-
Stereographic Projection of Point Group D2d
Stereographic Projection of Point Group D3d
-
-
-
-
     
D2d (8)
D3d (12)
       
Sn  
-
Stereographic Projection of Point Group Ci
-
Stereographic Projection of Point Group S4
-
Stereographic Projection of Point Group S6
-
   
(S1 = Cs)
(S2 = ) Ci (2)
(S3 = C3h)
S4 (4)
(S5 = C5h)
S6 (6)
 
T  
Stereographic Projection of Point Group T
Stereographic Projection of Point Group Th
Stereographic Projection of Point Group Td
       
   
T (12)
Th (24)
Td (24)
       
O  
Stereographic Projection of Point Group O
Stereographic Projection of Point Group Oh
         
   
O (24)
Oh (48)
         
I  
Stereographic Projection of Point Group I
Stereographic Projection of Point Group Ih
         
   
I (60)
Ih (120)
         
K  
Stereographic Projection of Point Group Kh
           
   
Kh (∞)
           
[*] The stereographic projections are illustrations of the set of symmetry operations of an object (i.e. a molecular geometry). The equator plane of all objects is marked by a pale yellow circular plane, all mirror planes are designated by transparent orange planes, axes of rotation and rotary-reflection are indicated by solid orange lines. All symmetry elements intersect in the geometrical center of each object. The graphics provide two views for each point group (front-view and top-view).

Top of Page Symmetry Elements of Point Groups
This table lists the symmetry elements of point groups including the order of the group (i.e. the number of symmetry related positions in the stereographic projections). See also the 'Hierarchy of Point Groups - Symmetry Correlations'.

Schönflies
Symbol
International
Symbol
Group
Order
Crystal Class Symmetry Operations
x i
C1
1
1
triclinic
E
Ci
C2
2
2
monoclinic
E, C2
C2h
C3
3
3
trigonal
E, C3
S6
C4
4
4
tetragonal
E, C4, C2
C4h
C5
5
5
non-crystallographic
E, C5
S10
C6
6
6
hexagonal
E, C6, C3, C2
C6h
C2v
mm2
4
orthorhombic
E, C2, 2σv
D2h
C3v
3m
6
trigonal
E, C3, 3σv
D3d
C4v
4mm
8
tetragonal
E, C4, C2, 2σv, 2σd
D4h
C5v
-
10
non-crystallographic
E, C5, 5σv
D5d
C6v
6mm
12
hexagonal
E, C6, C3, C2, 3σv, 3σd
D6h
Cs (= C1h)
m
2
monoclinic
E, σh
C2h
C2h
2/m
4
monoclinic
E, C2, i, σh
C2h
C3h (= S3)
6-bar
6
hexagonal
E, C3, S3, σh
C6h
C4h
4/m
8
tetragonal
E, C4, C2, S4, i, σh
C4h
C5h
-
10
non-crystallographic
E, C5, S5, σh
C10h
C6h
6/m
12
hexagonal
E, C6, C3, C2, S6, S3, i, σh
C6h
D2
222
4
orthorhombic
E, 3C2
D2h
D3
32
6
trigonal
E, C3, 3C2
D3d
D4
422
8
tetragonal
E, C4, 5C2
D4h
D5
-
10
non-crystallographic
E, C5, 5C2
D5d
D6
622
12
hexagonal
E, C6, C3, 7C2
D6h
D2h
mmm
8
orthorhombic
E, 3C2, i, 3σ
D2h
D3h
6-barm2
12
hexagonal
E, C3, 3C2, S3, σh, 3σv
D6h
D4h
4/mmm
16
tetragonal
E, C4, 5C2, S4, i, σh, 2σv, 2σd
D4h
D5h
-
20
non-crystallographic
E, C5, 5C2, S5, σh, 5σd
D10h
D6h
6/mmm
24
hexagonal
E, C6, C3, 7C2, S6, S3, i, σh, 3σv, 3σd
D6h
D2d
4-bar2m
8
tetragonal
E, 3C2, S4, 2σd
D4h
D3d
3-barm
12
trigonal
E, C3, 3C2, S6, i, 3σd
D3d
D4d
-
16
non-crystallographic
E, C4, 5C2, S8, 4σd
D8h
D5d
-
20
non-crystallographic
E, C5, 5C2, S10, i, 5σd
D5d
D6d
-
24
non-crystallographic
E, C6, C3, 7C2, S12, S4, 6σd
D12h
Ci (= S2)
1-bar
2
triclinic
E, i
Ci
S4
4-bar
4
tetragonal
E, C2, S4
C4h
S6
3-bar
6
trigonal
E, C3, S6, i
S6
S8
-
8
non-crystallographic
E, C4, C2, S8
C8h
T
23
12
cubic
E, 4C3, 3C2
Th
Th
m3
24
cubic
E, 4C3, 3C2, 4S6, i, 3σv
Th
Td
4-bar3m
24
cubic
E, 4C3, 3C2, 3S4, 6σd
Oh
O
432
24
cubic
E, 3C4, 4C3, 9C2
Oh
Oh
m3m
48
cubic
E, 3C4, 4C3, 9C2, 4S6, 3S4, i, 3σh, 6σd
Oh
I
-
60
non-crystallographic
E, 6C5, 10C3, 15C2
Ih
Ih
-
120
non-crystallographic
E, 6C5, 10C3, 15C2, 6S10, 10S6, i, 15σ
Ih
C∞v
m
non-crystallographic
E, C, ∞σv
D∞h
D∞h
∞/mm
non-crystallographic
E, C, ∞C2, S, i, ∞σv
D∞h
Kh
-
non-crystallographic
E, ∞C, ∞S, i, ∞σv
Kh

For more information on other research topics, please refer to the complete list of publications and to the gallery of graphics and animations.

© Copyright PD Dr. S. Immel

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