Symmetric cubic surfaces
| Type | Group name | GAP id | Normalizer | Centralizer | Normalizer mod group | Conjugacy classes in the centralizer |
|---|---|---|---|---|---|---|
1 | \(C_3^{\times 3} \rtimes S_4\) | [648, 704] | [1296, 3490] | [ 1, 1 ] | [2, 1] | [‘emptyset’] |
2 | \(S_5\) | [120, 34] | [240, 189] | [ 2, 1 ] | [2, 1] | [‘emptyset’, ‘A1’] |
3 | \(H_3(3) \rtimes C_4\) | [108, 15] | [432, 520] | [ 3, 1 ] | [4, 2] | [‘emptyset’, ‘3A2’] |
4 | \(H_3(3)\rtimes C_2\) | [54, 8] | [1296, 2891] | [ 3, 1 ] | [24, 12] | [‘emptyset’, ‘3A2’] |
5 | \(S_4\) | [24, 12] | [96, 226] | [ 4, 2 ] | [4, 2] | [‘emptyset’, ‘4A1’, ‘3A1’, ‘A1’] |
6 | \(S_3 \times C_2\) | [12, 4] | [72, 46] | [ 12, 4 ] | [6, 1] | [‘emptyset’, ‘A2’, ‘4A1’, ‘D4’, ‘3A1’, ‘A1’] |
7 | \(C_8\) | [8, 1] | [32, 43] | [ 8, 1 ] | [4, 2] | [‘emptyset’, ‘4A1’, ‘D4(a1)’, ‘D5’] |
8 | \(S_3\) | [6, 1] | [216, 162] | [ 36, 10 ] | [36, 10] | [‘emptyset’, ‘2A2’, ‘A2’, ‘3A2’, ‘4A1’, ‘D4’, ‘A1’, ‘A1+A5’, ‘3A1’] |
9 | \(C_4\) | [4, 1] | [192, 988] | [ 96, 67 ] | [48, 48] | [‘emptyset’, ‘4A1’, ‘D4(a1)’, ‘3A2’, ‘E6(a2)’, ‘2A1’, ‘E6’, ‘A3’, ‘2A1+A3’, ‘D5’] |
10 | \(C_2 \times C_2\) | [4, 2] | [192, 1472] | [ 96, 226 ] | [48, 48] | [‘emptyset’, ‘4A1’, ‘2A1+A3’, ‘A2’, ‘A3’, ‘D4’, ‘2A1’, ‘A1’, ‘2A1+A2’, ‘3A1’] |
11 | \(C_2\) | [2, 1] | [1152, 157478] | [ 1152, 157478 ] | [576, 8654] | [‘emptyset’, ‘4A1’, ‘2A1’, ‘2A2’, ‘A1+A5’, ‘A1+A3’, ‘A2’, ‘D4’, ‘2A1+A3’, ‘A3’, ‘D4(a1)’, ‘3A1’, ‘2A1+A2’, ‘E6’, ‘A5’, ‘3A2’, ‘E6(a2)’, ‘D5’, ‘A1’] |
Some facts about this (data omitted from the table above):
- For each of the 11 groups $G$ except $S_5$, we have that $N_{W(E_6)}(G)$ is solvable
- For each of the 11 groups $G$, we have that $C_{W(E_6)}(G)$ is solvable.
- For each of the 11 groups $G$, we have that $N_{W(E_6)}(G)/G$ is solvable.
Subgroup lattice
Here is the subgroup lattice, where an edge indicates subconjugacy in \(W(E_6)\).