David J. Green

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Cohomology of group number 266 of order 128

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

General information on the group

• The group has 3 minimal generators and exponent 8.
• It is non-abelian.
• It has p-Rank 3.
• Its center has rank 2.
• It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 3.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 3 and depth 2.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  ( − 1) · (t7  +  t5  +  t4  +  t2  +  1)

  (t  −  1)3 · (t2  +  1)2 · (t4  +  1)

• The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Benson test.

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

Ring generators

The cohomology ring has 18 minimal generators of maximal degree 8:

1. a_1_0, a nilpotent element of degree 1
2. a_1_1, a nilpotent element of degree 1
3. b_1_2, an element of degree 1
4. a_2_4, a nilpotent element of degree 2
5. a_3_5, a nilpotent element of degree 3
6. a_3_6, a nilpotent element of degree 3
7. a_4_7, a nilpotent element of degree 4
8. a_4_8, a nilpotent element of degree 4
9. c_4_9, a Duflot regular element of degree 4
10. a_5_11, a nilpotent element of degree 5
11. a_5_13, a nilpotent element of degree 5
12. a_5_14, a nilpotent element of degree 5
13. a_6_19, a nilpotent element of degree 6
14. a_7_22, a nilpotent element of degree 7
15. a_7_17, a nilpotent element of degree 7
16. a_8_25, a nilpotent element of degree 8
17. a_8_26, a nilpotent element of degree 8
18. c_8_29, a Duflot regular element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

Ring relations

There are 119 minimal relations of maximal degree 16:

1. a_1_0²
2. a_1_0·a_1_1
3. a_1_1³
4. a_1_0·b_1_2² + a_2_4·a_1_1
5. a_2_4·a_1_0
6. a_2_4²
7. a_1_1·a_3_5
8. a_1_0·a_3_5
9. a_2_4·b_1_2² + a_1_1·a_3_6 + a_2_4·a_1_1·b_1_2
10. a_1_0·a_3_6
11. a_2_4·a_3_5
12. a_1_1²·a_3_6
13. a_2_4·a_3_6
14. b_1_2²·a_3_5 + a_4_7·a_1_1
15. a_4_7·a_1_0
16. a_4_8·a_1_0
17. a_3_5²
18. a_3_6² + a_1_1·b_1_2²·a_3_6 + a_1_1²·b_1_2⁴
19. a_3_5·a_3_6 + a_2_4·a_4_7
20. a_3_5·a_3_6 + a_4_8·a_1_1²
21. a_3_5·a_3_6 + a_2_4·a_4_8
22. a_1_1·a_5_11
23. a_1_0·a_5_11
24. a_1_0·a_5_13
25. a_4_7·b_1_2² + a_1_1·a_5_14 + a_1_1·a_5_13 + a_4_7·a_1_1·b_1_2 + c_4_9·a_1_1²
26. a_3_5·a_3_6 + a_1_0·a_5_14
27. a_4_7·a_3_5
28. a_4_8·a_1_1·b_1_2² + a_4_7·a_3_6 + a_4_8·a_1_1²·b_1_2
29. a_4_8·a_3_5
30. a_2_4·a_5_11
31. a_4_8·a_3_6 + a_4_7·a_3_6 + a_1_1²·a_5_13
32. b_1_2²·a_5_11 + a_1_1·b_1_2³·a_3_6 + a_4_8·a_3_6 + a_4_7·a_3_6 + a_2_4·a_5_13
33. a_1_1²·a_5_14
34. b_1_2²·a_5_11 + a_1_1·b_1_2³·a_3_6 + a_4_8·a_3_6 + a_2_4·a_5_14 + a_2_4·c_4_9·a_1_1
35. b_1_2²·a_5_11 + a_1_1·b_1_2·a_5_14 + a_1_1·b_1_2·a_5_13 + a_1_1·b_1_2³·a_3_6
       + a_6_19·a_1_1 + a_4_7·a_3_6 + c_4_9·a_1_1²·b_1_2
36. a_6_19·a_1_0 + a_4_8·a_1_1·b_1_2² + a_4_7·a_3_6
37. a_4_7²
38. a_4_8²
39. a_3_5·a_5_11 + a_4_7·a_4_8
40. a_3_6·a_5_11
41. a_3_5·a_5_13 + a_4_7·a_4_8
42. a_3_5·a_5_14 + a_4_7·a_4_8
43. a_4_7·a_4_8 + a_2_4·a_1_1·a_5_14
44. a_4_8·b_1_2⁴ + a_3_6·a_5_14 + a_3_6·a_5_13 + a_1_1·b_1_2⁴·a_3_6 + c_4_9·a_1_1·a_3_6
45. b_1_2³·a_5_14 + b_1_2³·a_5_13 + a_6_19·b_1_2² + a_4_8·b_1_2⁴ + a_3_6·a_5_13
       + a_1_1·b_1_2²·a_5_13 + a_1_1·b_1_2⁴·a_3_6 + a_2_4·b_1_2·a_5_14 + a_6_19·a_1_1²
       + c_4_9·a_1_1·b_1_2³
46. a_4_7·a_4_8 + a_2_4·b_1_2·a_5_14 + a_2_4·b_1_2·a_5_13 + a_2_4·a_6_19
       + a_2_4·c_4_9·a_1_1·b_1_2
47. a_1_1·a_7_22 + a_1_1·b_1_2²·a_5_13 + a_1_1·b_1_2⁴·a_3_6 + a_6_19·a_1_1·b_1_2
       + a_4_7·a_4_8 + c_4_9·a_1_1·a_3_6
48. a_1_0·a_7_22
49. b_1_2³·a_5_14 + b_1_2³·a_5_13 + a_6_19·b_1_2² + a_1_1·a_7_17 + a_1_1·b_1_2²·a_5_13
       + a_1_1·b_1_2⁴·a_3_6 + a_6_19·a_1_1·b_1_2 + a_4_7·a_4_8 + a_2_4·b_1_2·a_5_13
       + c_4_9·a_1_1·b_1_2³ + c_4_9·a_1_1·a_3_6 + c_4_9·a_1_1²·b_1_2²
50. a_1_0·a_7_17
51. a_4_8·a_5_11 + a_4_7·a_5_11
52. a_4_7·a_5_14 + a_4_7·a_5_13 + a_6_19·a_1_1²·b_1_2 + a_4_7·c_4_9·a_1_1
53. a_6_19·a_3_5
54. a_4_8·a_5_11 + a_2_4·a_6_19·a_1_1
55. b_1_2²·a_7_22 + b_1_2⁴·a_5_13 + b_1_2⁶·a_3_6 + a_6_19·b_1_2³ + a_1_1·b_1_2³·a_5_13
       + a_6_19·a_1_1·b_1_2² + a_4_7·a_5_14 + a_1_1·a_3_6·a_5_13 + c_4_9·b_1_2²·a_3_6
       + c_4_9·a_1_1²·b_1_2³
56. a_4_8·a_5_14 + a_4_8·a_5_13 + a_2_4·a_7_22 + a_4_8·c_4_9·a_1_1
57. b_1_2·a_3_6·a_5_13 + a_1_1·b_1_2·a_7_17 + a_1_1·b_1_2⁵·a_3_6 + a_6_19·a_3_6
       + a_6_19·a_1_1·b_1_2² + a_4_8·a_5_11 + a_4_7·a_5_14 + a_4_7·a_5_13
       + c_4_9·a_1_1·b_1_2·a_3_6 + c_4_9·a_1_1²·b_1_2³ + a_4_7·c_4_9·a_1_1
58. a_4_8·a_5_14 + a_4_8·a_5_13 + a_4_7·a_5_14 + a_4_7·a_5_13 + a_1_1²·a_7_17
       + a_4_8·c_4_9·a_1_1 + a_4_7·c_4_9·a_1_1
59. a_4_8·a_5_11 + a_2_4·a_7_17 + a_1_1·a_3_6·a_5_13
60. a_1_1·b_1_2⁵·a_3_6 + a_8_25·a_1_1 + a_6_19·a_1_1·b_1_2² + a_4_8·a_5_14 + a_4_8·a_5_13
       + a_4_8·a_5_11 + a_4_7·a_5_13 + a_4_8·c_4_9·a_1_1 + a_4_7·c_4_9·a_1_1
61. a_8_25·a_1_0 + a_4_8·a_5_11
62. a_8_26·a_1_1 + a_6_19·a_1_1·b_1_2² + a_4_8·a_5_14 + a_4_7·a_5_14 + a_1_1·a_3_6·a_5_13
       + c_4_9·a_1_1·b_1_2·a_3_6 + a_4_7·c_4_9·a_1_1
63. a_8_26·a_1_0
64. a_5_11²
65. a_5_11·a_5_14 + a_4_8·a_6_19 + a_4_8·a_1_1·a_5_13
66. a_5_11·a_5_14 + a_5_11·a_5_13 + a_6_19·a_1_1·a_3_6 + a_4_8·a_1_1·a_5_13
67. a_4_7·a_6_19 + a_6_19·a_1_1²·b_1_2² + a_4_8·a_1_1·a_5_13
68. a_3_5·a_7_22 + a_4_8·c_4_9·a_1_1²
69. a_5_11·a_5_14 + a_3_6·a_7_22 + a_1_1·b_1_2²·a_7_17 + a_1_1²·b_1_2⁸
       + a_6_19·a_1_1·b_1_2³ + a_6_19·a_1_1²·b_1_2² + a_4_8·c_4_9·a_1_1²
70. a_5_11·a_5_14 + a_1_1²·b_1_2·a_7_17 + a_6_19·a_1_1²·b_1_2² + a_4_8·a_1_1·a_5_13
71. a_3_5·a_7_17 + a_4_8·a_1_1·a_5_13 + a_4_8·c_4_9·a_1_1²
72. a_5_11·a_5_14 + a_5_11·a_5_13 + a_3_6·a_7_17 + a_3_6·a_7_22 + a_1_1·b_1_2⁴·a_5_13
       + a_6_19·b_1_2·a_3_6 + a_6_19·a_1_1·b_1_2³ + a_4_8·a_1_1·a_5_13
       + c_4_9·a_1_1·b_1_2²·a_3_6 + a_4_8·c_4_9·a_1_1²
73. a_5_14² + a_6_19·a_1_1²·b_1_2² + c_8_29·a_1_1² + c_4_9·a_1_1·b_1_2²·a_3_6
       + c_4_9·a_1_1²·b_1_2⁴ + a_4_8·c_4_9·a_1_1² + c_4_9²·a_1_1²
74. b_1_2⁷·a_3_6 + a_8_25·b_1_2² + a_6_19·b_1_2⁴ + a_5_13·a_5_14 + a_5_13²
       + a_3_6·a_7_22 + a_1_1²·b_1_2⁸ + a_4_7·b_1_2·a_5_13 + a_6_19·a_1_1²·b_1_2²
       + c_4_9·a_1_1·a_5_14 + c_4_9·a_1_1²·b_1_2⁴ + a_4_8·c_4_9·a_1_1² + c_4_9²·a_1_1²
75. a_5_14² + a_5_13² + a_5_11·a_5_14 + a_1_1·b_1_2⁴·a_5_13 + a_8_25·a_1_1·b_1_2
       + a_6_19·a_1_1·b_1_2³ + a_4_7·b_1_2·a_5_13 + a_4_8·a_1_1·a_5_13
       + c_4_9·a_1_1·b_1_2²·a_3_6 + a_4_7·c_4_9·a_1_1·b_1_2 + c_4_9²·a_1_1²
76. a_5_11·a_5_14 + a_5_11·a_5_13 + a_2_4·a_8_25 + a_4_8·c_4_9·a_1_1²
77. a_5_11·a_5_14 + a_5_11·a_5_13 + a_2_4·a_8_26 + a_4_8·a_1_1·a_5_13 + a_4_8·c_4_9·a_1_1²
78. a_6_19·a_5_11 + a_6_19·a_1_1·b_1_2·a_3_6 + a_4_8·a_1_1·b_1_2·a_5_13
79. a_4_7·a_7_22 + a_1_1·a_5_13·a_5_14 + a_6_19·a_1_1·b_1_2·a_3_6
       + a_4_8·a_1_1·b_1_2·a_5_13 + a_2_4·c_4_9·a_5_14 + a_2_4·c_4_9·a_5_13
       + c_4_9·a_1_1²·a_5_13 + a_2_4·c_4_9²·a_1_1
80. a_4_8·a_7_17 + a_4_8·b_1_2²·a_5_13 + a_1_1·a_5_13·a_5_14 + a_6_19·a_1_1·b_1_2·a_3_6
       + a_6_19·a_1_1²·b_1_2³ + a_4_8·c_4_9·a_1_1²·b_1_2
81. a_4_8·a_7_22 + a_4_8·b_1_2²·a_5_13 + a_1_1²·b_1_2²·a_7_17
       + a_6_19·a_1_1·b_1_2·a_3_6 + a_4_8·a_1_1·b_1_2·a_5_13 + a_2_4·c_4_9·a_5_14
       + a_2_4·c_4_9·a_5_13 + c_4_9·a_1_1²·a_5_13 + a_4_8·c_4_9·a_1_1²·b_1_2
       + a_2_4·c_4_9²·a_1_1
82. a_4_8·b_1_2²·a_5_13 + a_4_7·a_7_17 + a_6_19·a_1_1·b_1_2·a_3_6
       + a_6_19·a_1_1²·b_1_2³ + a_2_4·c_4_9·a_5_14 + a_2_4·c_4_9·a_5_13
       + c_4_9·a_1_1²·a_5_13 + a_4_8·c_4_9·a_1_1²·b_1_2 + a_2_4·c_4_9²·a_1_1
83. b_1_2·a_5_13·a_5_14 + a_6_19·a_5_14 + a_1_1·a_5_13·a_5_14 + a_6_19·a_1_1²·b_1_2³
       + c_8_29·a_1_1²·b_1_2 + c_4_9·a_1_1·b_1_2·a_5_13 + c_4_9·a_1_1·b_1_2³·a_3_6
       + c_4_9·a_1_1²·b_1_2⁵ + c_4_9·a_6_19·a_1_1 + a_2_4·c_8_29·a_1_1
       + c_4_9·a_1_1²·a_5_13
84. a_1_1·b_1_2⁵·a_5_13 + a_8_25·a_1_1·b_1_2² + a_6_19·a_5_14 + a_6_19·a_5_13
       + a_6_19·a_1_1·b_1_2⁴ + a_4_8·a_7_22 + a_6_19·a_1_1·b_1_2·a_3_6
       + a_6_19·a_1_1²·b_1_2³ + c_4_9·a_1_1·b_1_2³·a_3_6 + c_4_9·a_6_19·a_1_1
       + a_2_4·c_4_9·a_5_14 + a_2_4·c_4_9·a_5_13 + a_2_4·c_4_9²·a_1_1
85. a_8_25·a_3_5
86. a_1_1·b_1_2⁵·a_5_13 + a_1_1²·b_1_2⁹ + a_8_25·a_3_6 + a_6_19·a_5_14 + a_6_19·a_5_13
       + a_6_19·b_1_2²·a_3_6 + a_1_1·a_5_13·a_5_14 + a_6_19·a_1_1·b_1_2·a_3_6
       + a_6_19·a_1_1²·b_1_2³ + c_4_9·a_1_1·b_1_2³·a_3_6 + c_4_9·a_6_19·a_1_1
       + a_2_4·c_4_9·a_5_14 + a_2_4·c_4_9·a_5_13 + c_4_9·a_1_1²·a_5_13 + a_2_4·c_4_9²·a_1_1
87. a_8_26·a_3_5 + a_4_8·c_4_9·a_1_1²·b_1_2
88. a_8_26·a_3_6 + a_6_19·b_1_2²·a_3_6 + a_1_1·a_5_13·a_5_14 + a_6_19·a_1_1·b_1_2·a_3_6
       + a_6_19·a_1_1²·b_1_2³ + a_4_8·a_1_1·b_1_2·a_5_13 + c_4_9·a_1_1·b_1_2³·a_3_6
       + c_4_9·a_1_1²·b_1_2⁵ + a_2_4·c_4_9·a_5_14 + a_2_4·c_4_9·a_5_13
       + a_2_4·c_4_9²·a_1_1
89. a_5_11·a_7_22 + a_1_1²·b_1_2³·a_7_17 + a_6_19·a_1_1²·b_1_2⁴ + a_2_4·a_5_13·a_5_14
       + a_2_4·c_4_9·a_1_1·a_5_14
90. a_6_19·b_1_2·a_5_14 + a_6_19·b_1_2·a_5_13 + a_6_19² + a_4_7·b_1_2·a_7_17
       + a_6_19·a_1_1·a_5_13 + a_6_19·a_1_1·b_1_2²·a_3_6 + a_6_19·a_1_1²·b_1_2⁴
       + c_4_9·a_6_19·a_1_1·b_1_2 + a_2_4·c_4_9·a_6_19 + c_4_9·a_6_19·a_1_1²
91. a_5_11·a_7_17 + a_5_11·a_7_22 + a_6_19·a_1_1·b_1_2²·a_3_6 + a_2_4·c_4_9·a_1_1·a_5_14
92. a_5_14·a_7_17 + a_5_13·a_7_22 + a_5_11·a_7_22 + a_6_19·b_1_2³·a_3_6
       + a_6_19·a_1_1·a_5_13 + a_6_19·a_1_1²·b_1_2⁴ + a_2_4·a_5_13·a_5_14
       + c_8_29·a_1_1·a_3_6 + c_8_29·a_1_1²·b_1_2² + c_4_9·a_3_6·a_5_13
       + c_4_9·a_1_1·b_1_2²·a_5_13 + a_2_4·c_4_9·a_6_19 + c_4_9·a_6_19·a_1_1²
       + c_4_9²·a_1_1·a_3_6 + c_4_9²·a_1_1²·b_1_2² + a_2_4·c_4_9²·a_1_1·b_1_2
93. a_5_14·a_7_22 + a_5_13·a_7_22 + a_6_19·b_1_2·a_5_13 + a_6_19·b_1_2³·a_3_6 + a_6_19²
       + a_6_19·a_1_1·b_1_2²·a_3_6 + c_4_9·a_3_6·a_5_13 + c_4_9·a_1_1·a_7_17
       + c_4_9·a_1_1·b_1_2²·a_5_13 + a_2_4·c_8_29·a_1_1·b_1_2 + a_2_4·c_4_9·a_6_19
       + c_4_9²·a_1_1²·b_1_2² + a_2_4·c_4_9²·a_1_1·b_1_2
94. a_5_11·a_7_22 + a_4_8·a_8_25 + a_6_19·a_1_1·b_1_2²·a_3_6 + a_6_19·a_1_1²·b_1_2⁴
       + a_2_4·a_5_13·a_5_14 + a_2_4·c_4_9·a_1_1·a_5_14
95. a_5_13·a_7_22 + a_5_11·a_7_22 + a_1_1·b_1_2⁴·a_7_17 + a_8_25·a_1_1·b_1_2³
       + a_6_19·b_1_2·a_5_13 + a_6_19·b_1_2³·a_3_6 + a_6_19² + a_6_19·a_1_1²·b_1_2⁴
       + c_8_29·a_1_1²·b_1_2² + c_4_9·a_3_6·a_5_13 + a_2_4·c_4_9·a_1_1·a_5_14
96. a_4_7·a_8_25 + a_6_19·a_1_1·b_1_2²·a_3_6 + a_6_19·a_1_1²·b_1_2⁴
       + a_2_4·c_4_9·a_1_1·a_5_14
97. a_5_11·a_7_22 + a_4_8·a_8_26 + a_2_4·a_5_13·a_5_14 + a_2_4·c_4_9·a_6_19
       + c_4_9·a_6_19·a_1_1²
98. a_8_26·b_1_2⁴ + a_6_19·b_1_2⁶ + a_5_14·a_7_17 + a_5_14·a_7_22 + a_5_13·a_7_17
       + a_1_1·b_1_2⁴·a_7_17 + a_8_25·b_1_2·a_3_6 + a_6_19·b_1_2·a_5_14
       + a_6_19·b_1_2³·a_3_6 + a_6_19·a_1_1·b_1_2⁵ + a_6_19² + c_4_9·b_1_2⁵·a_3_6
       + c_8_29·a_1_1²·b_1_2² + c_4_9·a_1_1·b_1_2²·a_5_13 + c_4_9·a_1_1·b_1_2⁴·a_3_6
       + a_2_4·c_4_9·b_1_2·a_5_13 + c_4_9·a_6_19·a_1_1² + c_4_9²·a_1_1²·b_1_2²
       + a_2_4·c_4_9²·a_1_1·b_1_2
99. a_4_7·a_8_26 + a_6_19·a_1_1²·b_1_2⁴ + a_2_4·c_4_9·a_6_19
100. a_6_19·a_7_22 + a_6_19·b_1_2²·a_5_13 + a_6_19·b_1_2⁴·a_3_6 + a_6_19²·b_1_2
        + a_6_19·a_1_1·b_1_2·a_5_13 + a_6_19²·a_1_1 + a_2_4·a_6_19·a_5_13 + c_4_9·a_6_19·a_3_6
        + c_4_9·a_6_19·a_1_1²·b_1_2
101. b_1_2·a_5_13·a_7_17 + a_1_1·b_1_2⁵·a_7_17 + a_8_25·a_1_1·b_1_2⁴ + a_6_19·a_7_17
        + a_6_19·b_1_2²·a_5_13 + a_6_19²·b_1_2 + a_6_19·a_1_1·b_1_2³·a_3_6
        + a_2_4·a_6_19·a_5_13 + c_8_29·a_1_1·b_1_2·a_3_6 + c_4_9·a_1_1·b_1_2·a_7_17
        + c_4_9·a_1_1·b_1_2³·a_5_13 + c_4_9·a_1_1²·a_7_17 + c_4_9·a_6_19·a_1_1²·b_1_2
        + a_2_4·c_4_9·a_6_19·a_1_1 + c_4_9²·a_1_1·b_1_2·a_3_6 + c_4_9²·a_1_1²·b_1_2³
102. a_8_25·a_5_11 + a_6_19·a_1_1·b_1_2³·a_3_6
103. b_1_2·a_5_13·a_7_17 + a_8_25·a_5_13 + a_6_19·a_7_17 + a_6_19·b_1_2⁴·a_3_6
        + a_6_19²·b_1_2 + a_1_1·a_5_13·a_7_17 + a_1_1²·b_1_2⁴·a_7_17
        + a_6_19·a_1_1·b_1_2·a_5_13 + c_8_29·a_1_1·b_1_2·a_3_6 + c_4_9·a_1_1·b_1_2·a_7_17
        + c_4_9·a_1_1·b_1_2³·a_5_13 + c_4_9·a_1_1²·b_1_2⁷ + c_4_9·a_8_25·a_1_1
        + c_4_9·a_6_19·a_1_1·b_1_2² + a_4_7·c_8_29·a_1_1 + c_4_9·a_6_19·a_1_1²·b_1_2
        + c_4_9²·a_1_1·b_1_2·a_3_6 + c_4_9²·a_1_1²·b_1_2³ + a_4_7·c_4_9²·a_1_1
104. b_1_2·a_5_13·a_7_17 + a_8_25·a_5_14 + a_6_19·a_7_17 + a_1_1·a_5_13·a_7_17
        + a_6_19·a_1_1·b_1_2³·a_3_6 + a_2_4·a_6_19·a_5_13 + c_8_29·a_1_1·b_1_2·a_3_6
        + c_4_9·a_1_1·b_1_2·a_7_17 + c_4_9·a_1_1·b_1_2³·a_5_13 + c_4_9·a_1_1²·b_1_2⁷
        + c_4_9·a_6_19·a_1_1·b_1_2² + a_4_7·c_8_29·a_1_1 + c_4_9·a_1_1²·a_7_17
        + c_4_9·a_6_19·a_1_1²·b_1_2 + a_2_4·c_4_9·a_6_19·a_1_1 + c_4_9²·a_1_1·b_1_2·a_3_6
        + c_4_9²·a_1_1²·b_1_2³ + a_4_7·c_4_9²·a_1_1
105. a_8_26·a_5_11 + a_6_19·a_1_1·b_1_2³·a_3_6 + a_2_4·a_6_19·a_5_13
        + a_2_4·c_4_9·a_6_19·a_1_1
106. a_8_26·a_5_13 + a_6_19·b_1_2²·a_5_13 + a_1_1²·b_1_2⁴·a_7_17
        + a_6_19·a_1_1·b_1_2·a_5_13 + a_6_19·a_1_1·b_1_2³·a_3_6 + a_6_19²·a_1_1
        + c_4_9·a_1_1·b_1_2·a_7_17 + c_4_9·a_8_25·a_1_1 + c_4_9·a_6_19·a_3_6
        + a_4_8·c_8_29·a_1_1 + a_4_8·c_4_9·a_5_13 + a_4_7·c_8_29·a_1_1 + a_4_7·c_4_9·a_5_13
        + c_4_9·a_1_1·a_3_6·a_5_13 + c_4_9·a_6_19·a_1_1²·b_1_2 + a_2_4·c_4_9·a_6_19·a_1_1
        + c_4_9²·a_1_1·b_1_2·a_3_6 + c_4_9²·a_1_1²·b_1_2³ + a_4_7·c_4_9²·a_1_1
107. a_8_26·a_5_14 + a_6_19·b_1_2²·a_5_13 + a_6_19²·b_1_2 + a_6_19·a_1_1·b_1_2·a_5_13
        + c_4_9·a_1_1·b_1_2·a_7_17 + c_4_9·a_8_25·a_1_1 + c_4_9·a_6_19·a_1_1·b_1_2²
        + a_4_8·c_8_29·a_1_1 + a_4_7·c_8_29·a_1_1 + c_4_9·a_1_1·a_3_6·a_5_13
        + c_4_9·a_1_1²·a_7_17 + c_4_9·a_6_19·a_1_1²·b_1_2 + c_4_9²·a_1_1²·b_1_2³
        + a_4_8·c_4_9²·a_1_1 + a_4_7·c_4_9²·a_1_1
108. a_7_17² + a_7_22² + a_6_19²·b_1_2² + a_6_19·a_3_6·a_5_13 + a_6_19·a_1_1·a_7_17
        + a_6_19·a_1_1·b_1_2⁴·a_3_6 + a_6_19²·a_1_1·b_1_2 + c_8_29·a_1_1·b_1_2²·a_3_6
        + c_4_9·a_1_1²·b_1_2⁸ + c_4_9·a_6_19·a_1_1·a_3_6 + c_4_9·a_6_19·a_1_1²·b_1_2²
        + c_4_9²·a_1_1²·b_1_2⁴
109. a_7_17² + a_7_22·a_7_17 + a_8_25·a_1_1·b_1_2⁵ + a_6_19·b_1_2³·a_5_13
        + a_6_19·b_1_2⁵·a_3_6 + a_6_19²·b_1_2² + a_6_19·a_3_6·a_5_13 + a_6_19·a_1_1·a_7_17
        + a_6_19·a_1_1·b_1_2²·a_5_13 + a_6_19·a_1_1·b_1_2⁴·a_3_6 + a_1_1²·a_5_13·a_7_17
        + c_8_29·a_1_1²·b_1_2⁴ + c_4_9·a_1_1²·b_1_2⁸ + c_4_9·a_8_25·a_1_1·b_1_2
        + c_4_9·a_6_19·b_1_2·a_3_6 + c_4_9·a_6_19·a_1_1·b_1_2³ + a_4_7·c_4_9·b_1_2·a_5_13
        + c_4_9·a_1_1²·b_1_2·a_7_17 + a_4_8·c_8_29·a_1_1² + a_4_8·c_4_9·a_1_1·a_5_13
        + c_4_9²·a_1_1·b_1_2²·a_3_6 + c_4_9²·a_1_1²·b_1_2⁴ + a_4_7·c_4_9²·a_1_1·b_1_2
        + a_4_8·c_4_9²·a_1_1²
110. a_7_17² + a_7_22·a_7_17 + a_7_22² + a_8_25·b_1_2³·a_3_6 + a_8_25·a_1_1·b_1_2⁵
        + a_6_19·b_1_2³·a_5_13 + a_6_19²·b_1_2² + a_6_19·a_3_6·a_5_13
        + a_6_19·a_1_1·b_1_2²·a_5_13 + a_6_19·a_1_1·b_1_2⁴·a_3_6 + a_1_1²·a_5_13·a_7_17
        + c_4_9·a_6_19·b_1_2·a_3_6 + a_4_8·c_8_29·a_1_1² + a_4_8·c_4_9·a_1_1·a_5_13
        + a_4_8·c_4_9²·a_1_1²
111. a_6_19·b_1_2⁵·a_3_6 + a_6_19·a_8_25 + a_6_19²·b_1_2² + a_6_19·a_3_6·a_5_13
        + a_6_19·a_1_1·b_1_2⁴·a_3_6 + a_6_19²·a_1_1·b_1_2 + c_4_9·a_1_1²·b_1_2·a_7_17
        + a_4_8·c_8_29·a_1_1² + a_4_8·c_4_9·a_1_1·a_5_13
112. a_6_19·a_8_26 + a_6_19²·b_1_2² + a_6_19·a_3_6·a_5_13 + a_6_19·a_1_1·b_1_2²·a_5_13
        + c_4_9·a_6_19·b_1_2·a_3_6 + c_4_9·a_1_1²·b_1_2·a_7_17
        + c_4_9·a_6_19·a_1_1²·b_1_2² + a_4_8·c_8_29·a_1_1²
113. a_8_25·a_7_17 + a_8_25·a_7_22 + a_8_25·a_1_1·b_1_2⁶ + a_6_19·b_1_2²·a_7_17
        + a_6_19·b_1_2⁴·a_5_13 + a_6_19·a_1_1·b_1_2³·a_5_13 + a_6_19·a_8_25·a_1_1
        + c_4_9·a_6_19·b_1_2²·a_3_6 + a_4_7·c_4_9·a_7_17 + a_2_4·c_8_29·a_5_14
        + a_2_4·c_8_29·a_5_13 + c_8_29·a_1_1²·a_5_13 + c_4_9·a_6_19·a_1_1²·b_1_2³
        + a_4_8·c_4_9·a_1_1·b_1_2·a_5_13 + a_2_4·c_4_9·c_8_29·a_1_1 + a_2_4·c_4_9²·a_5_14
        + a_2_4·c_4_9²·a_5_13 + a_2_4·c_4_9³·a_1_1
114. a_8_25·a_7_17 + a_8_25·b_1_2²·a_5_13 + a_8_25·b_1_2⁴·a_3_6 + a_8_25·a_1_1·b_1_2⁶
        + a_6_19·b_1_2²·a_7_17 + a_6_19·b_1_2⁴·a_5_13 + a_6_19·a_8_25·b_1_2 + a_6_19²·a_3_6
        + a_6_19·a_1_1²·a_7_17 + c_4_9·a_8_25·a_3_6 + c_4_9·a_6_19·b_1_2²·a_3_6
        + a_4_7·c_4_9·a_7_17 + a_2_4·c_8_29·a_5_14 + a_2_4·c_8_29·a_5_13
        + c_8_29·a_1_1²·a_5_13 + a_4_8·c_4_9·a_1_1·b_1_2·a_5_13 + a_2_4·c_4_9·c_8_29·a_1_1
        + a_2_4·c_4_9²·a_5_14 + a_2_4·c_4_9²·a_5_13 + a_2_4·c_4_9³·a_1_1
115. a_8_26·a_7_17 + a_6_19·b_1_2²·a_7_17 + a_6_19·a_1_1·b_1_2³·a_5_13
        + a_6_19²·a_1_1·b_1_2² + c_4_9·a_1_1·b_1_2³·a_7_17 + c_4_9·a_8_25·a_3_6
        + c_4_9·a_6_19·a_5_14 + c_4_9·a_6_19·a_5_13 + c_8_29·a_1_1²·a_5_13
        + c_4_9·a_1_1²·b_1_2²·a_7_17 + a_4_8·c_8_29·a_1_1²·b_1_2
        + a_4_8·c_4_9·a_1_1·b_1_2·a_5_13 + c_4_9²·a_6_19·a_1_1 + a_2_4·c_4_9²·a_5_14
        + a_2_4·c_4_9²·a_5_13 + c_4_9²·a_1_1²·a_5_13 + a_4_8·c_4_9²·a_1_1²·b_1_2
        + a_2_4·c_4_9³·a_1_1
116. a_8_26·a_7_22 + a_6_19·b_1_2⁴·a_5_13 + a_6_19·a_8_25·b_1_2 + a_6_19²·a_3_6
        + c_4_9·a_1_1·b_1_2³·a_7_17 + c_4_9·a_8_25·a_3_6 + c_4_9·a_8_25·a_1_1·b_1_2²
        + a_2_4·c_8_29·a_5_14 + a_2_4·c_8_29·a_5_13 + c_8_29·a_1_1²·a_5_13
        + a_2_4·c_4_9·c_8_29·a_1_1 + a_4_8·c_4_9²·a_1_1²·b_1_2
117. a_8_25·b_1_2⁵·a_3_6 + a_8_25² + a_6_19·a_8_25·b_1_2² + a_6_19²·b_1_2·a_3_6
        + a_6_19²·a_1_1·b_1_2³ + c_4_9·a_1_1²·b_1_2³·a_7_17
        + c_4_9·a_6_19·a_1_1·b_1_2²·a_3_6 + a_2_4·c_8_29·a_1_1·a_5_14
        + a_2_4·c_4_9·a_5_13·a_5_14
118. a_8_25·b_1_2⁵·a_3_6 + a_8_25·a_8_26 + a_8_25² + a_6_19·a_1_1·b_1_2²·a_7_17
        + c_4_9·a_8_25·b_1_2·a_3_6 + c_4_9·a_6_19·a_1_1·b_1_2²·a_3_6
        + c_4_9·a_6_19·a_1_1²·b_1_2⁴ + a_2_4·c_8_29·a_1_1·a_5_14
119. a_8_26² + a_6_19²·b_1_2⁴ + c_4_9²·a_1_1·b_1_2⁴·a_3_6 + c_4_9²·a_1_1²·b_1_2⁶

About the group

Ring generators

Ring relations

Completion information

Restriction maps

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 16.
• The completion test was perfect: It applied in the last degree in which a generator or relation was found.
• The following is a filter regular homogeneous system of parameters:
  1. c_4_9, a Duflot regular element of degree 4
  2. c_8_29, a Duflot regular element of degree 8
  3. b_1_2², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, 9, 11].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 2

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. b_1_2 → 0, an element of degree 1
4. a_2_4 → 0, an element of degree 2
5. a_3_5 → 0, an element of degree 3
6. a_3_6 → 0, an element of degree 3
7. a_4_7 → 0, an element of degree 4
8. a_4_8 → 0, an element of degree 4
9. c_4_9 → c_1_0⁴, an element of degree 4
10. a_5_11 → 0, an element of degree 5
11. a_5_13 → 0, an element of degree 5
12. a_5_14 → 0, an element of degree 5
13. a_6_19 → 0, an element of degree 6
14. a_7_22 → 0, an element of degree 7
15. a_7_17 → 0, an element of degree 7
16. a_8_25 → 0, an element of degree 8
17. a_8_26 → 0, an element of degree 8
18. c_8_29 → c_1_1⁸, an element of degree 8

Restriction map to a maximal el. ab. subgp. of rank 3

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. b_1_2 → c_1_2, an element of degree 1
4. a_2_4 → 0, an element of degree 2
5. a_3_5 → 0, an element of degree 3
6. a_3_6 → 0, an element of degree 3
7. a_4_7 → 0, an element of degree 4
8. a_4_8 → 0, an element of degree 4
9. c_4_9 → c_1_0⁴, an element of degree 4
10. a_5_11 → 0, an element of degree 5
11. a_5_13 → 0, an element of degree 5
12. a_5_14 → 0, an element of degree 5
13. a_6_19 → 0, an element of degree 6
14. a_7_22 → 0, an element of degree 7
15. a_7_17 → 0, an element of degree 7
16. a_8_25 → 0, an element of degree 8
17. a_8_26 → 0, an element of degree 8
18. c_8_29 → c_1_1⁴·c_1_2⁴ + c_1_1⁸ + c_1_0⁴·c_1_2⁴, an element of degree 8

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128