Table. 1.
Orbit numbers of finite group actions on S2
| Number |
Underlying Space |
G |
Data |
Generators |
| 1 |
S2 |
Trivial |
() |
id |
| 2 |
ℤn |
(n,n) |
rotnz |
| 3 |
Dih(ℤ2n) |
(2,2,2n) |
rot2nz,rot2y |
| 4 |
Dih(ℤ2n+1) |
(2,2,2n+1) |
rot2n+1z,rot2y |
| 5 |
A4 |
(2,3,3) |
rot2z,rot3L1 |
| 6 |
S4 |
(2,3,4) |
rot2z,rot3L2 |
| 7 |
A5 |
(2,3,5) |
rot2z,rot3L3 |
| 8 |
D |
ℤ2 |
(;) |
refxy |
| 9 |
ℤ2n×ℤ2 |
(2n;) |
rot2nz,refxy |
| 10 |
ℤ4n+2 |
(2n+1;) |
rot2n+1z∘refxy |
| 11 |
Dih(ℤn) |
(;n,n) |
rotnz,refyz |
| 12 |
Dih(ℤ2n)∘−ℤ2 |
(;2,2,2n) |
rot2nz,rot2y,refyz |
| 13 |
Dih(ℤ2n+1)∘−ℤ2 |
(;2,2,2n+1) |
rot2n+1z,rot2y,refyz |
| 14 |
A4°−ℤ2 |
(;2,3,3) |
rot2z,rot3L1,refyz |
| 15 |
S4×ℤ2 |
(;2,3,4) |
rot2z,rot3L2,refxy |
| 16 |
A5×ℤ2 |
(;2,3,5) |
rot2z,rot3L3,refxy |
| 17 |
Dih(ℤ2n)∘−ℤ2 |
(2;2n) |
rot2nz,rot2y,refxz |
| 18 |
Dih(ℤ2n+1)∘−ℤ2 |
(2;2n+1) |
rot2n+1z,rot2y,refxz |
| 19 |
A4×ℤ2 |
(3;2) |
rot2z,rot3L1,refxy |
| 20 |
ℙ2 |
ℤ2 |
() |
rot2z∘refxy |
| 21 |
ℤ2n |
(n) |
rot2nz∘refxy |
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