Class group of the Segre Cubic
Class group of the Segre Cubic
This page displays a subgroup lattice of S6, and the corresponding list of G-invariant Class groups of the Segre cubic X3.
Here is the source code.
Partially ordered set of subgroup classes of S6
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[56] Order 720 Length 1 Maximal Subgroups: 50 51 52 53 54 55
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[55] Order 360 Length 1 Maximal Subgroups: 39 41 47 48 49
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[54] Order 120 Length 6 Nonsolvable Maximal Subgroups: 32 37 42 49
[53] Order 120 Length 6 Nonsolvable Maximal Subgroups: 33 37 44 48
[52] Order 72 Length 10 Maximal Subgroups: 27 45 46 47
[51] Order 48 Length 15 Maximal Subgroups: 32 38 40 41 42
[50] Order 48 Length 15 Maximal Subgroups: 33 38 39 43 44
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[49] Order 60 Length 6 Maximal Subgroups: 18 22 31
[48] Order 60 Length 6 Maximal Subgroups: 17 22 30
[47] Order 36 Length 10 Maximal Subgroups: 11 34
[46] Order 36 Length 10 Maximal Subgroups: 32 34 35
[45] Order 36 Length 10 Maximal Subgroups: 33 34 36
[44] Order 24 Length 15 Maximal Subgroups: 16 28 30
[43] Order 24 Length 15 Maximal Subgroups: 19 24 30
[42] Order 24 Length 15 Maximal Subgroups: 15 29 31
[41] Order 24 Length 15 Maximal Subgroups: 18 26 31
[40] Order 24 Length 15 Maximal Subgroups: 20 23 31
[39] Order 24 Length 15 Maximal Subgroups: 17 26 30
[38] Order 16 Length 45 Maximal Subgroups: 23 24 25 26 27 28 29
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[37] Order 20 Length 36 Maximal Subgroups: 13 22
[36] Order 18 Length 20 Maximal Subgroups: 16 19 21
[35] Order 18 Length 20 Maximal Subgroups: 15 20 21
[34] Order 18 Length 10 Maximal Subgroups: 17 18 21
[33] Order 12 Length 60 Maximal Subgroups: 14 16 17 19
[32] Order 12 Length 60 Maximal Subgroups: 12 15 18 20
[31] Order 12 Length 15 Maximal Subgroups: 5 8
[30] Order 12 Length 15 Maximal Subgroups: 6 9
[29] Order 8 Length 45 Maximal Subgroups: 8 12 13
[28] Order 8 Length 45 Maximal Subgroups: 9 13 14
[27] Order 8 Length 45 Maximal Subgroups: 11 12 14
[26] Order 8 Length 45 Maximal Subgroups: 8 9 11
[25] Order 8 Length 45 Maximal Subgroups: 10 11 13
[24] Order 8 Length 15 Maximal Subgroups: 9 10 12
[23] Order 8 Length 15 Maximal Subgroups: 8 10 14
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[22] Order 10 Length 36 Maximal Subgroups: 4 7
[21] Order 9 Length 10 Maximal Subgroups: 5 6
[20] Order 6 Length 60 Maximal Subgroups: 2 5
[19] Order 6 Length 60 Maximal Subgroups: 3 6
[18] Order 6 Length 60 Maximal Subgroups: 4 5
[17] Order 6 Length 60 Maximal Subgroups: 4 6
[16] Order 6 Length 20 Maximal Subgroups: 3 6
[15] Order 6 Length 20 Maximal Subgroups: 2 5
[14] Order 4 Length 45 Maximal Subgroups: 3 4
[13] Order 4 Length 45 Maximal Subgroups: 4
[12] Order 4 Length 45 Maximal Subgroups: 2 4
[11] Order 4 Length 45 Maximal Subgroups: 4
[10] Order 4 Length 45 Maximal Subgroups: 2 3 4
[ 9] Order 4 Length 15 Maximal Subgroups: 4
[ 8] Order 4 Length 15 Maximal Subgroups: 4
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[ 7] Order 5 Length 36 Maximal Subgroups: 1
[ 6] Order 3 Length 20 Maximal Subgroups: 1
[ 5] Order 3 Length 20 Maximal Subgroups: 1
[ 4] Order 2 Length 45 Maximal Subgroups: 1
[ 3] Order 2 Length 15 Maximal Subgroups: 1
[ 2] Order 2 Length 15 Maximal Subgroups: 1
Rank of the G-invariant class group (the numbering corresponds to the subgroup lattice):
2. Rank of Cl(X3)G where G= C2 is 5
3. Rank of Cl(X3)G where G= C2 is 3
4. Rank of Cl(X3)G where G= C2 is 4
5. Rank of Cl(X3)G where G= C3 is 4
6. Rank of Cl(X3)G where G= C3 is 2
7. Rank of Cl(X3)G where G= C5 is 2
8. Rank of Cl(X3)G where G= C2^2 is 3
9. Rank of Cl(X3)G where G= C2^2 is 3
10. Rank of Cl(X3)G where G= C2^2 is 3
11. Rank of Cl(X3)G where G= C4 is 2
12. Rank of Cl(X3)G where G= C2^2 is 4
13. Rank of Cl(X3)G where G= C4 is 3
14. Rank of Cl(X3)G where G= C2^2 is 2
15. Rank of Cl(X3)G where G= S3 is 4
16. Rank of Cl(X3)G where G= S3 is 1
17. Rank of Cl(X3)G where G= S3 is 2
18. Rank of Cl(X3)G where G= S3 is 3
19. Rank of Cl(X3)G where G= C6 is 1
20. Rank of Cl(X3)G where G= C6 is 3
21. Rank of Cl(X3)G where G= C3^2 is 2
22. Rank of Cl(X3)G where G= D5 is 2
23. Rank of Cl(X3)G where G= C2^3 is 2
24. Rank of Cl(X3)G where G= C2^3 is 3
25. Rank of Cl(X3)G where G= C2*C4 is 2
26. Rank of Cl(X3)G where G= D4 is 2
27. Rank of Cl(X3)G where G= D4 is 2
28. Rank of Cl(X3)G where G= D4 is 2
29. Rank of Cl(X3)G where G= D4 is 3
30. Rank of Cl(X3)G where G= A4 is 1
31. Rank of Cl(X3)G where G= A4 is 3
32. Rank of Cl(X3)G where G= D6 is 3
33. Rank of Cl(X3)G where G= D6 is 1
34. Rank of Cl(X3)G where G= C3:S3 is 2
35. Rank of Cl(X3)G where G= C3*S3 is 2
36. Rank of Cl(X3)G where G= C3*S3 is 1
37. Rank of Cl(X3)G where G= F5 is 2
38. Rank of Cl(X3)G where G= C2*D4 is 2
39. Rank of Cl(X3)G where G= S4 is 1
40. Rank of Cl(X3)G where G= C2*A4 is 2
41. Rank of Cl(X3)G where G= S4 is 2
42. Rank of Cl(X3)G where G= S4 is 3
43. Rank of Cl(X3)G where G= C2*A4 is 1
44. Rank of Cl(X3)G where G= S4 is 1
45. Rank of Cl(X3)G where G= S3^2 is 1
46. Rank of Cl(X3)G where G= S3^2 is 2
47. Rank of Cl(X3)G where G= C3:S3.C2 is 1
48. Rank of Cl(X3)G where G= A5 is 1
49. Rank of Cl(X3)G where G= A5 is 2
50. Rank of Cl(X3)G where G= C2*S4 is 1
51. Rank of Cl(X3)G where G= C2*S4 is 2
52. Rank of Cl(X3)G where G= S3wrC2 is 1
53. Rank of Cl(X3)G where G= S5 is 1
54. Rank of Cl(X3)G where G= S5 is 2
55. Rank of Cl(X3)G where G= A6 is 1
56. Rank of Cl(X3)G where G= S6 is 1