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Table. 2.

Orbit numbers of finite group actions on S2.

Number G Orbit Numbers of non-regular points LCM |G|/LCM OC Satisfied?
1 Trivial 1 1 1 all b
2 n 1, 1 n 1 all b
3 Dih(2n) 2, 2n, 2n 2n 2 b even
4 Dih(2n+1) 2, 2n + 1, 2n + 1 4n + 2 1 all b
5 A4 4, 4, 6 6 2 b even
6 S4 6, 8, 12 12 2 b even
7 A5 6, 10, 15 30 1 all b
8 2 1 2 1 all b
9 2n×2 2, n 2n 2 b even
10 4n+2 2, 2n + 1 4n + 2 1 all b
11 Dih(n) 1, 1 2n 1 all b
12 Dih(2n)2 2, 2n, 2n 4n 2 b even
13 Dih(2n+1)2 2, 2n + 1, 2n + 1 8n + 4 1 all b
14 A42 4, 4, 6 12 2 b even
15 S4×2 6, 8, 12 24 2 b even
16 A5×2 6, 10, 15 60 2 b even
17 Dih(2n)2 2, 4n 4n 2 b even
18 Dih(2n+1)2 2, 4n + 2 4n + 2 2 b even
19 A4×2 6, 8 12 2 b even
20 2 1 2 b even
21 2n 2 n 2 b even
Kyungpook Mathematical Journal 2022;62:363~388 https://doi.org/10.5666/KMJ.2022.62.2.363
© KYUNGPOOK Math. J.