Class group of the Burkhardt quartic
Class group of the Burkhardt quartic
This page displays a subgroup lattice of PSp4(F3), and the corresponding list of G-invariant Class groups of the Burkhardt quartic X4.
Here is the source code.
Partially ordered set of subgroup classes PSp4(F3)
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[116] Order 25920 Length 1 Maximal Subgroups: 111 112 113 114 115
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[115] Order 960 Length 27 Maximal Subgroups: 77 102 103 109
[114] Order 576 Length 45 Maximal Subgroups: 73 108 109 110
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[113] Order 720 Length 36 Maximal Subgroups: 86 87 88 95 96 107
[112] Order 648 Length 40 Maximal Subgroups: 68 97 105 106
[111] Order 648 Length 40 Maximal Subgroups: 89 97 104
[110] Order 288 Length 45 Maximal Subgroups: 89 99 100 101
[109] Order 192 Length 135 Maximal Subgroups: 84 98 99
[108] Order 192 Length 45 Maximal Subgroups: 87 98 100
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[107] Order 360 Length 36 Maximal Subgroups: 64 67 72 77 78
[106] Order 324 Length 40 Maximal Subgroups: 36 79 91
[105] Order 216 Length 120 Maximal Subgroups: 69 88 91 93 94
[104] Order 216 Length 40 Maximal Subgroups: 63 92
[103] Order 160 Length 162 Maximal Subgroups: 25 82 90
[102] Order 96 Length 270 Maximal Subgroups: 69 82 83 84 86
[101] Order 96 Length 90 Maximal Subgroups: 63 80 85
[100] Order 96 Length 45 Maximal Subgroups: 61 66 80
[ 99] Order 96 Length 45 Maximal Subgroups: 62 65 80
[ 98] Order 64 Length 135 Maximal Subgroups: 80 81 82
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[ 97] Order 162 Length 160 Maximal Subgroups: 74 76 79
[ 96] Order 120 Length 216 Maximal Subgroups: 41 49 67 77
[ 95] Order 120 Length 216 Maximal Subgroups: 37 49 68 78
[ 94] Order 108 Length 120 Maximal Subgroups: 39 72 75
[ 93] Order 108 Length 120 Maximal Subgroups: 70 73 75 76
[ 92] Order 108 Length 120 Maximal Subgroups: 40 74
[ 91] Order 108 Length 40 Maximal Subgroups: 71 75
[ 90] Order 80 Length 162 Maximal Subgroups: 7 53
[ 89] Order 72 Length 360 Maximal Subgroups: 48 59 61 62 63
[ 88] Order 72 Length 360 Maximal Subgroups: 34 70 71 72
[ 87] Order 48 Length 540 Maximal Subgroups: 37 57 64 66 68
[ 86] Order 48 Length 270 Maximal Subgroups: 41 57 60 67
[ 85] Order 48 Length 270 Maximal Subgroups: 40 54 59
[ 84] Order 48 Length 270 Maximal Subgroups: 38 53 60 65
[ 83] Order 48 Length 270 Maximal Subgroups: 39 58 60
[ 82] Order 32 Length 405 Maximal Subgroups: 53 55 56 57 58
[ 81] Order 32 Length 405 Maximal Subgroups: 55 56
[ 80] Order 32 Length 45 Maximal Subgroups: 54 55
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[ 79] Order 81 Length 160 Maximal Subgroups: 50 51 52
[ 78] Order 60 Length 216 Maximal Subgroups: 17 25 36
[ 77] Order 60 Length 216 Maximal Subgroups: 18 25 35
[ 76] Order 54 Length 240 Maximal Subgroups: 43 44 45 48 50
[ 75] Order 54 Length 120 Maximal Subgroups: 42 46 47 50
[ 74] Order 54 Length 40 Maximal Subgroups: 45 51
[ 73] Order 36 Length 720 Maximal Subgroups: 37 38 44 47 48
[ 72] Order 36 Length 360 Maximal Subgroups: 13 42
[ 71] Order 36 Length 360 Maximal Subgroups: 41 42 46
[ 70] Order 36 Length 120 Maximal Subgroups: 37 42 43
[ 69] Order 24 Length 1080 Maximal Subgroups: 34 38 39 41
[ 68] Order 24 Length 540 Maximal Subgroups: 14 34 36
[ 67] Order 24 Length 540 Maximal Subgroups: 18 33 35
[ 66] Order 24 Length 540 Maximal Subgroups: 16 27 36
[ 65] Order 24 Length 540 Maximal Subgroups: 19 28 35
[ 64] Order 24 Length 540 Maximal Subgroups: 17 33 36
[ 63] Order 24 Length 360 Maximal Subgroups: 26 40
[ 62] Order 24 Length 360 Maximal Subgroups: 19 26
[ 61] Order 24 Length 360 Maximal Subgroups: 16 26
[ 60] Order 24 Length 270 Maximal Subgroups: 20 29 35
[ 59] Order 24 Length 90 Maximal Subgroups: 15 26
[ 58] Order 16 Length 810 Maximal Subgroups: 29 30 32
[ 57] Order 16 Length 810 Maximal Subgroups: 27 29 32 33 34
[ 56] Order 16 Length 405 Maximal Subgroups: 28 32
[ 55] Order 16 Length 405 Maximal Subgroups: 27 28 30 31
[ 54] Order 16 Length 270 Maximal Subgroups: 26 30 31
[ 53] Order 16 Length 27 Maximal Subgroups: 28 29
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[ 52] Order 27 Length 320 Maximal Subgroups: 22 24
[ 51] Order 27 Length 40 Maximal Subgroups: 22
[ 50] Order 27 Length 40 Maximal Subgroups: 21 22 23
[ 49] Order 20 Length 1296 Maximal Subgroups: 13 25
[ 48] Order 18 Length 720 Maximal Subgroups: 15 16 19 23
[ 47] Order 18 Length 720 Maximal Subgroups: 17 20 23
[ 46] Order 18 Length 720 Maximal Subgroups: 18 20 21
[ 45] Order 18 Length 480 Maximal Subgroups: 14 15 22
[ 44] Order 18 Length 240 Maximal Subgroups: 14 19 23
[ 43] Order 18 Length 240 Maximal Subgroups: 14 16 21
[ 42] Order 18 Length 120 Maximal Subgroups: 17 18 21
[ 41] Order 12 Length 1080 Maximal Subgroups: 12 18 20
[ 40] Order 12 Length 1080 Maximal Subgroups: 8 15
[ 39] Order 12 Length 1080 Maximal Subgroups: 13 20
[ 38] Order 12 Length 1080 Maximal Subgroups: 9 19 20
[ 37] Order 12 Length 720 Maximal Subgroups: 9 14 16 17
[ 36] Order 12 Length 540 Maximal Subgroups: 6 12
[ 35] Order 12 Length 270 Maximal Subgroups: 5 10
[ 34] Order 8 Length 1620 Maximal Subgroups: 9 12 13
[ 33] Order 8 Length 1620 Maximal Subgroups: 10 12 13
[ 32] Order 8 Length 810 Maximal Subgroups: 11 13
[ 31] Order 8 Length 810 Maximal Subgroups: 8 11
[ 30] Order 8 Length 405 Maximal Subgroups: 8 11
[ 29] Order 8 Length 270 Maximal Subgroups: 9 10 11
[ 28] Order 8 Length 135 Maximal Subgroups: 10 11
[ 27] Order 8 Length 135 Maximal Subgroups: 11 12
[ 26] Order 8 Length 90 Maximal Subgroups: 8
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[ 25] Order 10 Length 1296 Maximal Subgroups: 3 7
[ 24] Order 9 Length 960 Maximal Subgroups: 4
[ 23] Order 9 Length 240 Maximal Subgroups: 4 5 6
[ 22] Order 9 Length 160 Maximal Subgroups: 4 6
[ 21] Order 9 Length 120 Maximal Subgroups: 5 6
[ 20] Order 6 Length 1080 Maximal Subgroups: 3 5
[ 19] Order 6 Length 720 Maximal Subgroups: 2 5
[ 18] Order 6 Length 720 Maximal Subgroups: 3 5
[ 17] Order 6 Length 720 Maximal Subgroups: 3 6
[ 16] Order 6 Length 720 Maximal Subgroups: 2 6
[ 15] Order 6 Length 360 Maximal Subgroups: 2 4
[ 14] Order 6 Length 240 Maximal Subgroups: 2 6
[ 13] Order 4 Length 1620 Maximal Subgroups: 3
[ 12] Order 4 Length 540 Maximal Subgroups: 3
[ 11] Order 4 Length 405 Maximal Subgroups: 2 3
[ 10] Order 4 Length 270 Maximal Subgroups: 3
[ 9] Order 4 Length 270 Maximal Subgroups: 2 3
[ 8] Order 4 Length 270 Maximal Subgroups: 2
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[ 7] Order 5 Length 1296 Maximal Subgroups: 1
[ 6] Order 3 Length 240 Maximal Subgroups: 1
[ 5] Order 3 Length 120 Maximal Subgroups: 1
[ 4] Order 3 Length 40 Maximal Subgroups: 1
[ 3] Order 2 Length 270 Maximal Subgroups: 1
[ 2] Order 2 Length 45 Maximal Subgroups: 1
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Rank of the G-invariant class group (the numbering corresponds to the subgroup lattice):
2. Rank of Cl(X4)G where G= C2 is 8
3. Rank of Cl(X4)G where G= C2 is 8
4. Rank of Cl(X4)G where G= C3 is 10
5. Rank of Cl(X4)G where G= C3 is 8
6. Rank of Cl(X4)G where G= C3 is 6
7. Rank of Cl(X4)G where G= C5 is 4
8. Rank of Cl(X4)G where G= C4 is 6
9. Rank of Cl(X4)G where G= C2^2 is 4
10. Rank of Cl(X4)G where G= C2^2 is 4
11. Rank of Cl(X4)G where G= C2^2 is 4
12. Rank of Cl(X4)G where G= C2^2 is 4
13. Rank of Cl(X4)G where G= C4 is 4
14. Rank of Cl(X4)G where G= S3 is 3
15. Rank of Cl(X4)G where G= C6 is 6
16. Rank of Cl(X4)G where G= C6 is 4
17. Rank of Cl(X4)G where G= S3 is 3
18. Rank of Cl(X4)G where G= S3 is 4
19. Rank of Cl(X4)G where G= C6 is 4
20. Rank of Cl(X4)G where G= C6 is 4
21. Rank of Cl(X4)G where G= C3^2 is 4
22. Rank of Cl(X4)G where G= C3^2 is 4
23. Rank of Cl(X4)G where G= C3^2 is 6
24. Rank of Cl(X4)G where G= C9 is 4
25. Rank of Cl(X4)G where G= D5 is 2
26. Rank of Cl(X4)G where G= Q8 is 5
27. Rank of Cl(X4)G where G= C2^3 is 2
28. Rank of Cl(X4)G where G= C2^3 is 2
29. Rank of Cl(X4)G where G= C2^3 is 2
30. Rank of Cl(X4)G where G= C2*C4 is 4
31. Rank of Cl(X4)G where G= D4 is 3
32. Rank of Cl(X4)G where G= C2*C4 is 2
33. Rank of Cl(X4)G where G= D4 is 2
34. Rank of Cl(X4)G where G= D4 is 2
35. Rank of Cl(X4)G where G= A4 is 4
36. Rank of Cl(X4)G where G= A4 is 2
37. Rank of Cl(X4)G where G= D6 is 2
38. Rank of Cl(X4)G where G= C2*C6 is 2
39. Rank of Cl(X4)G where G= C3:C4 is 2
40. Rank of Cl(X4)G where G= C12 is 4
41. Rank of Cl(X4)G where G= D6 is 2
42. Rank of Cl(X4)G where G= C3:S3 is 2
43. Rank of Cl(X4)G where G= C3*S3 is 3
44. Rank of Cl(X4)G where G= C3*S3 is 3
45. Rank of Cl(X4)G where G= C3*S3 is 3
46. Rank of Cl(X4)G where G= C3*S3 is 2
47. Rank of Cl(X4)G where G= C3*S3 is 3
48. Rank of Cl(X4)G where G= C3*C6 is 4
49. Rank of Cl(X4)G where G= F5 is 1
50. Rank of Cl(X4)G where G= C3^3 is 4
51. Rank of Cl(X4)G where G= He3 is 2
52. Rank of Cl(X4)G where G= C9:C3 is 2
53. Rank of Cl(X4)G where G= C2^4 is 1
54. Rank of Cl(X4)G where G= D4:C2 is 3
55. Rank of Cl(X4)G where G= C2*D4 is 2
56. Rank of Cl(X4)G where G= C2^2:C4 is 1
57. Rank of Cl(X4)G where G= C2*D4 is 1
58. Rank of Cl(X4)G where G= C2^2:C4 is 2
59. Rank of Cl(X4)G where G= SL(2,3) is 5
60. Rank of Cl(X4)G where G= C2*A4 is 2
61. Rank of Cl(X4)G where G= SL(2,3) is 3
62. Rank of Cl(X4)G where G= SL(2,3) is 3
63. Rank of Cl(X4)G where G= C3*Q8 is 3
64. Rank of Cl(X4)G where G= S4 is 1
65. Rank of Cl(X4)G where G= C2*A4 is 2
66. Rank of Cl(X4)G where G= C2*A4 is 2
67. Rank of Cl(X4)G where G= S4 is 2
68. Rank of Cl(X4)G where G= S4 is 1
69. Rank of Cl(X4)G where G= C3:D4 is 1
70. Rank of Cl(X4)G where G= S3^2 is 2
71. Rank of Cl(X4)G where G= S3^2 is 1
72. Rank of Cl(X4)G where G= C3:S3.C2 is 1
73. Rank of Cl(X4)G where G= C6*S3 is 2
74. Rank of Cl(X4)G where G= C3^2:S3 is 2
75. Rank of Cl(X4)G where G= C3*C3:S3 is 2
76. Rank of Cl(X4)G where G= C3^2*S3 is 3
77. Rank of Cl(X4)G where G= A5 is 2
78. Rank of Cl(X4)G where G= A5 is 1
79. Rank of Cl(X4)G where G= C3wrC3 is 2
80. Rank of Cl(X4)G where G= Q8:C2^2 is 2
81. Rank of Cl(X4)G where G= C2^2.D4 is 1
82. Rank of Cl(X4)G where G= C2^2wrC2 is 1
83. Rank of Cl(X4)G where G= A4:C4 is 2
84. Rank of Cl(X4)G where G= C2^2*A4 is 1
85. Rank of Cl(X4)G where G= SL(2,3):C2 is 3
86. Rank of Cl(X4)G where G= C2*S4 is 1
87. Rank of Cl(X4)G where G= C2*S4 is 1
88. Rank of Cl(X4)G where G= S3wrC2 is 1
89. Rank of Cl(X4)G where G= C3*SL(2,3) is 3
90. Rank of Cl(X4)G where G= C2^4:C5 is 1
91. Rank of Cl(X4)G where G= C3^3:C2^2 is 1
92. Rank of Cl(X4)G where G= C3^2:S3.C2 is 2
93. Rank of Cl(X4)G where G= C3*S3^2 is 2
94. Rank of Cl(X4)G where G= C3^2:(C3:C4) is 1
95. Rank of Cl(X4)G where G= S5 is 1
96. Rank of Cl(X4)G where G= S5 is 1
97. Rank of Cl(X4)G where G= C3wrS3 is 2
98. Rank of Cl(X4)G where G= C2wrC2^2 is 1
99. Rank of Cl(X4)G where G= C2^3:A4 is 2
100. Rank of Cl(X4)G where G= C2^3:A4 is 2
101. Rank of Cl(X4)G where G= Q8.A4 is 2
102. Rank of Cl(X4)G where G= GL(2,Z/4) is 1
103. Rank of Cl(X4)G where G= C2^4:D5 is 1
104. Rank of Cl(X4)G where G= SU(3,2) is 2
105. Rank of Cl(X4)G where G= S3^2:S3 is 1
106. Rank of Cl(X4)G where G= C3^3:C2^2:C3 is 1
107. Rank of Cl(X4)G where G= A6 is 1
108. Rank of Cl(X4)G where G= C2^3:S4 is 1
109. Rank of Cl(X4)G where G= C2wrA4 is 1
110. Rank of Cl(X4)G where G= SL(2,3):A4 is 2
111. Rank of Cl(X4)G where G= SU(3,2).C3 is 2
112. Rank of Cl(X4)G where G= C3^3.S4 is 1
113. Rank of Cl(X4)G where G= S6 is 1
114. Rank of Cl(X4)G where G= C2.A4wrC2 is 1
115. Rank of Cl(X4)G where G= C2^4.A5 is 1
116. Rank of Cl(X4)G where G= C(2,3) is 1