Picard group of <span style="text-decoration: overline;"> M</span><sub>0,6</sub>
Picard group of M0,6
This page displays a subgroup lattice of S6, and the corresponding list of G-invariant Picard groups of M0,6, a smooth model of the Segre cubic.
Notice that H1(G,Pic( M0,6))=0 for all subgroups G of S6. But there are three conjugacy classes starred in the subgroup lattice, #9, 30 and 48, having nontrivial first cohomology of the dual of the Picard module. This is specified in the list below.
Here is the source code.
Partially ordered set of subgroup classes
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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 42 47 48 49
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[54] Order 120 Length 6 Maximal Subgroups: 32 37 40 49
[53] Order 120 Length 6 Maximal Subgroups: 33 37 44 48
[52] Order 72 Length 10 Maximal Subgroups: 29 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: 17 22 31
*[48] Order 60 Length 6 Maximal Subgroups: 19 22 30
[47] Order 36 Length 10 Maximal Subgroups: 10 34
[46] Order 36 Length 10 Maximal Subgroups: 33 34 35
[45] Order 36 Length 10 Maximal Subgroups: 32 34 36
[44] Order 24 Length 15 Maximal Subgroups: 16 28 30
[43] Order 24 Length 15 Maximal Subgroups: 18 24 30
[42] Order 24 Length 15 Maximal Subgroups: 17 27 31
[41] Order 24 Length 15 Maximal Subgroups: 20 23 31
[40] Order 24 Length 15 Maximal Subgroups: 15 26 31
[39] Order 24 Length 15 Maximal Subgroups: 19 27 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: 15 20 21
[35] Order 18 Length 20 Maximal Subgroups: 16 18 21
[34] Order 18 Length 10 Maximal Subgroups: 17 19 21
[33] Order 12 Length 60 Maximal Subgroups: 12 16 18 19
[32] Order 12 Length 60 Maximal Subgroups: 11 15 17 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: 10 11 12
[28] Order 8 Length 45 Maximal Subgroups: 9 12 13
[27] Order 8 Length 45 Maximal Subgroups: 8 9 10
[26] Order 8 Length 45 Maximal Subgroups: 8 11 13
[25] Order 8 Length 45 Maximal Subgroups: 10 13 14
[24] Order 8 Length 15 Maximal Subgroups: 9 11 14
[23] Order 8 Length 15 Maximal Subgroups: 8 12 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: 4 6
[18] Order 6 Length 60 Maximal Subgroups: 3 6
[17] Order 6 Length 60 Maximal Subgroups: 4 5
[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: 2 3 4
[13] Order 4 Length 45 Maximal Subgroups: 4
[12] Order 4 Length 45 Maximal Subgroups: 3 4
[11] Order 4 Length 45 Maximal Subgroups: 2 4
[10] Order 4 Length 45 Maximal Subgroups: 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
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Rank of the G-invariant Picard group (the numbering corresponds to the subgroup lattice):
2. Rank of Pic( M0,6)G where G= C2 is 12
3. Rank of Pic( M0,6)G where G= C2 is 10
4. Rank of Pic( M0,6)G where G= C2 is 10
5. Rank of Pic( M0,6)G where G= C3 is 8
6. Rank of Pic( M0,6)G where G= C3 is 6
7. Rank of Pic( M0,6)G where G= C5 is 4
8. Rank of Pic( M0,6)G where G= C2^2 is 7
9. Rank of Pic( M0,6)G where G= C2^2 is 7
H^1(9.C2^2, dual of Picard( M0,6))=Z/2.
10. Rank of Pic( M0,6)G where G= C4 is 6
11. Rank of Pic( M0,6)G where G= C2^2 is 9
12. Rank of Pic( M0,6)G where G= C2^2 is 7
13. Rank of Pic( M0,6)G where G= C4 is 6
14. Rank of Pic( M0,6)G where G= C2^2 is 8
15. Rank of Pic( M0,6)G where G= S3 is 8
16. Rank of Pic( M0,6)G where G= S3 is 5
17. Rank of Pic( M0,6)G where G= S3 is 6
18. Rank of Pic( M0,6)G where G= C6 is 4
19. Rank of Pic( M0,6)G where G= S3 is 5
20. Rank of Pic( M0,6)G where G= C6 is 6
21. Rank of Pic( M0,6)G where G= C3^2 is 4
22. Rank of Pic( M0,6)G where G= D5 is 4
23. Rank of Pic( M0,6)G where G= C2^3 is 6
24. Rank of Pic( M0,6)G where G= C2^3 is 7
25. Rank of Pic( M0,6)G where G= C2*C4 is 5
26. Rank of Pic( M0,6)G where G= D4 is 6
27. Rank of Pic( M0,6)G where G= D4 is 5
28. Rank of Pic( M0,6)G where G= D4 is 5
29. Rank of Pic( M0,6)G where G= D4 is 6
30. Rank of Pic( M0,6)G where G= A4 is 3
H^1(30.A4, dual of Picard( M0,6))=Z/2.
31. Rank of Pic( M0,6)G where G= A4 is 5
32. Rank of Pic( M0,6)G where G= D6 is 6
33. Rank of Pic( M0,6)G where G= D6 is 4
34. Rank of Pic( M0,6)G where G= C3:S3 is 4
35. Rank of Pic( M0,6)G where G= C3*S3 is 3
36. Rank of Pic( M0,6)G where G= C3*S3 is 4
37. Rank of Pic( M0,6)G where G= F5 is 3
38. Rank of Pic( M0,6)G where G= C2*D4 is 5
39. Rank of Pic( M0,6)G where G= S4 is 3
40. Rank of Pic( M0,6)G where G= S4 is 5
41. Rank of Pic( M0,6)G where G= C2*A4 is 4
42. Rank of Pic( M0,6)G where G= S4 is 4
43. Rank of Pic( M0,6)G where G= C2*A4 is 3
44. Rank of Pic( M0,6)G where G= S4 is 3
45. Rank of Pic( M0,6)G where G= S3^2 is 4
46. Rank of Pic( M0,6)G where G= S3^2 is 3
47. Rank of Pic( M0,6)G where G= C3:S3.C2 is 3
48. Rank of Pic( M0,6)G where G= A5 is 2
H^1(48.A5, dual of Picard( M0,6))=Z/2.
49. Rank of Pic( M0,6)G where G= A5 is 3
50. Rank of Pic( M0,6)G where G= C2*S4 is 3
51. Rank of Pic( M0,6)G where G= C2*S4 is 4
52. Rank of Pic( M0,6)G where G= S3wrC2 is 3
53. Rank of Pic( M0,6)G where G= S5 is 2
54. Rank of Pic( M0,6)G where G= S5 is 3
55. Rank of Pic( M0,6)G where G= A6 is 2
56. Rank of Pic( M0,6)G where G= S6 is 2