Symmetric cubic surfaces

Type	Group name	GAP id	Normalizer id	Normalizer solvable?	Centralizer id	Centralizer solvable?	Normalizer mod group (id)	Normalizer mod group solvable?	Conjugacy classes (Atlas)	Conjugacy classes (Carter)
1	\(C_3^{\times 3} \rtimes S_4\)	[648, 704]	[1296, 3490]	true	[ 1, 1 ]	true	[2, 1]	true	[‘1A’, ‘3A’, ‘3D’, ‘6A’, ‘2A’, ‘3C’, ‘9A’, ‘6C’, ‘6E’, ‘2B’, ‘6F’, ‘4B’]	[‘emptyset’, ‘3A2’, ‘2A2’, ‘E6(a2)’, ‘4A1’, ‘A2’, ‘E6(a1)’, ‘D4’, ‘A1+A5’, ‘2A1’, ‘2A1+A2’, ‘A1+A3’]
2	\(S_5\)	[120, 34]	[240, 189]	false	[ 2, 1 ]	true	[2, 1]	true	[‘1A’, ‘3D’, ‘6E’, ‘4B’, ‘2B’, ‘2A’, ‘5A’]	[‘emptyset’, ‘2A2’, ‘A1+A5’, ‘A1+A3’, ‘2A1’, ‘4A1’, ‘A4’]
3	\(H_3(3) \rtimes C_4\)	[108, 15]	[432, 520]	true	[ 3, 1 ]	true	[4, 2]	true	[‘1A’, ‘4A’, ‘6A’, ‘2A’, ‘3A’, ‘3D’, ‘12A’]	[‘emptyset’, ‘D4(a1)’, ‘E6(a2)’, ‘4A1’, ‘3A2’, ‘2A2’, ‘E6’]
4	\(H_3(3)\rtimes C_2\)	[54, 8]	[1296, 2891]	true	[ 3, 1 ]	true	[24, 12]	true	[‘1A’, ‘3A’, ‘3D’, ‘2A’, ‘6A’]	[‘emptyset’, ‘3A2’, ‘2A2’, ‘4A1’, ‘E6(a2)’]
5	\(S_4\)	[24, 12]	[96, 226]	true	[ 4, 2 ]	true	[4, 2]	true	[‘1A’, ‘4B’, ‘2A’, ‘2B’, ‘3D’]	[‘emptyset’, ‘A1+A3’, ‘4A1’, ‘2A1’, ‘2A2’]
6	\(S_3 \times C_2\)	[12, 4]	[72, 46]	true	[ 12, 4 ]	true	[6, 1]	true	[‘1A’, ‘6E’, ‘2A’, ‘3D’, ‘2B’]	[‘emptyset’, ‘A1+A5’, ‘4A1’, ‘2A2’, ‘2A1’]
7	\(C_8\)	[8, 1]	[32, 43]	true	[ 8, 1 ]	true	[4, 2]	true	[‘1A’, ‘8A’, ‘4A’, ‘2A’]	[‘emptyset’, ‘D5’, ‘D4(a1)’, ‘4A1’]
8	\(S_3\)	[6, 1]	[216, 162]	true	[ 36, 10 ]	true	[36, 10]	true	[‘1A’, ‘3D’, ‘2A’]	[‘emptyset’, ‘2A2’, ‘4A1’]
9	\(C_4\)	[4, 1]	[192, 988]	true	[ 96, 67 ]	true	[48, 48]	true	[‘1A’, ‘4A’, ‘2A’]	[‘emptyset’, ‘D4(a1)’, ‘4A1’]
10	\(C_2 \times C_2\)	[4, 2]	[192, 1472]	true	[ 96, 226 ]	true	[48, 48]	true	[‘1A’, ‘2A’, ‘2B’]	[‘emptyset’, ‘4A1’, ‘2A1’]
11	\(C_2\)	[2, 1]	[1152, 157478]	true	[ 1152, 157478 ]	true	[576, 8654]	true	[‘1A’, ‘2A’]	[‘emptyset’, ‘4A1’]

Subgroup lattice

Here is the subgroup lattice, where an edge indicates subconjugacy in \(W(E_6)\).