Symmetries of Canonical Self-Dual Polyhedra

To produce the table below, all 3-connected self-dual planar simple graphs with up to 16 vertices were generated using the plantri software developed by Gunnar Brinkmann and Brendan McKay. From each graph, a canonical polyhedron was produced and its symmetry determined. A self-dual polyhedron has the same number of faces as vertices, so the table shows the counts of all canonical self-dual polyhedra with up to 16 faces, cross tabulated by symmetry type and number of faces. Many of the counts can be clicked to show representative polyhedra.

Symmetries of All Canonical Self-Dual Polyhedra with up to 16 Faces

Symmetry 4 faces 5 faces 6 faces 7 faces 8 faces 9 faces 10 faces 11 faces 12 faces 13 faces 14 faces 15 faces 16 faces TOTAL
C1 ...... ...... ...... 2 10 39 137 514 1817 6490 23199 83406 300928 416542
Ci = S2 ...... ...... ...... ...... ...... ...... 1 ...... 2 ...... 8 ...... 29 40
Cs=C1h=C1v ...... ...... ...... ...... 1 4 7 16 41 76 180 346 746 1417
C2 ...... ...... 1 1 3 4 10 18 37 68 155 275 587 1159
C3 ...... ...... ...... ...... ...... ...... 3 ...... ...... 12 ...... ...... 52 67
C4 ...... ...... ...... ...... ...... ...... ...... ...... ...... 3 ...... ...... ...... 3
C5 ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... 3 3
C2h ...... ...... ...... ...... ...... ...... ...... ...... 2 ...... 3 ...... 5 10
C2v ...... ...... ...... ...... ...... ...... 2 3 5 4 7 18 18 57
C3v ...... ...... ...... 2 ...... ...... 3 ...... ...... 8 ...... ...... 23 36
C4v ...... 1 ...... ...... ...... 2 ...... ...... ...... 3 ...... ...... ...... 6
C5v ...... ...... 1 ...... ...... ...... ...... 2 ...... ...... ...... ...... 3 6
C6v ...... ...... ...... 1 ...... ...... ...... ...... ...... 2 ...... ...... ...... 3
C7v ...... ...... ...... ...... 1 ...... ...... ...... ...... ...... ...... 2 ...... 3
C8v ...... ...... ...... ...... ...... 1 ...... ...... ...... ...... ...... ...... ...... 1
C9v ...... ...... ...... ...... ...... ...... 1 ...... ...... ...... ...... ...... ...... 1
C10v ...... ...... ...... ...... ...... ...... ...... 1 ...... ...... ...... ...... ...... 1
C11v ...... ...... ...... ...... ...... ...... ...... ...... 1 ...... ...... ...... ...... 1
C12v ...... ...... ...... ...... ...... ...... ...... ...... ...... 1 ...... ...... ...... 1
C13v ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... 1 ...... ...... 1
C14v ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... 1 ...... 1
C15v ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... 1 1
S4 ...... ...... ...... ...... ...... ...... ...... ...... 1 ...... ...... ...... 1 2
D2 ...... ...... ...... ...... ...... ...... 1 ...... 2 ...... 3 ...... 6 12
D2h ...... ...... ...... ...... 1 ...... ...... ...... ...... ...... ...... ...... ...... 1
D2v ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... 1 1
T ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... 1 1
Td 1 ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... 1
Th ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ......
O ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ......
Oh ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ......
I ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ......
Ih ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ......
TOTAL 1 1 2 6 16 50 165 554 1908 6667 23556 84048 302404 419378
non-C1 1 1 2 4 6 11 28 40 91 177 357 642 1476 2836

The following table shows the counts of all canonical self-dual polyhedra with up to 16 faces that have symmetry type other than C1, cross tabulated by number of different edge lengths and number of faces.

Edge Length Counts of Symmetric Canonical Self-Dual Polyhedra with up to 16 Faces

Edge lengths 4 faces 5 faces 6 faces 7 faces 8 faces 9 faces 10 faces 11 faces 12 faces 13 faces 14 faces 15 faces 16 faces TOTAL
1 length 1 ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... 1
2 lengths ...... 1 1 1 1 1 1 1 2 1 1 1 2 14
3 lengths ...... ...... ...... 1 1 1 ...... 1 ...... 1 ...... 1 ...... 6
4 lengths ...... ...... ...... 1 ...... 1 2 1 ...... 2 ...... 1 1 9
5 lengths ...... ...... 1 1 ...... ...... 4 ...... 2 6 ...... ...... 7 21
6 lengths ...... ...... ...... ...... 1 1 5 4 5 8 4 3 16 47
7 lengths ...... ...... ...... ...... 2 3 ...... 1 ...... 11 5 2 11 35
8 lengths ...... ...... ...... ...... 1 2 3 4 7 9 9 15 24 74
9 lengths ...... ...... ...... ...... ...... 1 7 10 4 3 3 8 39 75
10 lengths ...... ...... ...... ...... ...... 1 5 6 10 11 5 9 37 84
11 lengths ...... ...... ...... ...... ...... ...... 1 3 24 44 12 14 14 112
12 lengths ...... ...... ...... ...... ...... ...... ...... 8 28 22 35 42 32 167
13 lengths ...... ...... ...... ...... ...... ...... ...... 1 7 12 97 157 33 307
14 lengths ...... ...... ...... ...... ...... ...... ...... ...... 2 42 122 110 116 392
15 lengths ...... ...... ...... ...... ...... ...... ...... ...... ...... 5 31 50 356 442
16 lengths ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... 31 191 454 676
17 lengths ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... 2 32 113 147
18 lengths ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... 6 206 212
19 lengths ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... 15 15
TOTAL 1 1 2 4 6 11 28 40 91 177 357 642 1476 2836

References:[1]Canonical Polyhedron (Wolfram MathWorld)
[2]Gunnar Brinkmann and Brendan D. McKay, Fast generation of planar graphs, MATCH-Communications in Mathematical and in Computer Chemistry 58(2) (2007), 323-357.