David J. Green

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Cohomology of group number 59 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 2 minimal generators and exponent 16.
• It is non-abelian.
• It has p-Rank 3.
• Its center has rank 1.
• 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 1.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  ( − 1) · (t4  −  t3  +  t2  −  t  +  1)

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

• The a-invariants are -∞,-5,-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 19 minimal generators of maximal degree 10:

1. a_1_0, a nilpotent element of degree 1
2. a_1_1, a nilpotent element of degree 1
3. a_2_1, a nilpotent element of degree 2
4. b_2_2, an element of degree 2
5. a_3_3, a nilpotent element of degree 3
6. b_3_2, an element of degree 3
7. a_4_2, a nilpotent element of degree 4
8. b_4_4, an element of degree 4
9. a_5_3, a nilpotent element of degree 5
10. a_5_4, a nilpotent element of degree 5
11. b_5_5, an element of degree 5
12. a_6_5, a nilpotent element of degree 6
13. b_6_8, an element of degree 6
14. a_7_8, a nilpotent element of degree 7
15. b_7_10, an element of degree 7
16. a_8_6, a nilpotent element of degree 8
17. c_8_13, a Duflot regular element of degree 8
18. a_9_11, a nilpotent element of degree 9
19. a_10_12, a nilpotent element of degree 10

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

Ring relations

There are 136 minimal relations of maximal degree 20:

1. a_1_0²
2. a_1_0·a_1_1
3. a_2_1·a_1_1 + a_1_1³
4. a_2_1·a_1_0 + a_1_1³
5. b_2_2·a_1_0 + a_1_1³
6. a_2_1²
7. b_2_2·a_1_1²
8. a_1_0·a_3_3
9. a_1_1·b_3_2 + a_2_1·b_2_2 + a_1_1·a_3_3
10. a_1_0·b_3_2
11. b_2_2·a_3_3 + a_2_1·a_3_3
12. a_1_1²·a_3_3
13. b_2_2²·a_1_1 + a_2_1·b_3_2
14. b_2_2·a_3_3 + a_4_2·a_1_1
15. a_4_2·a_1_0
16. b_4_4·a_1_0
17. a_3_3²
18. b_3_2² + b_2_2³ + a_3_3·b_3_2 + a_2_1·b_2_2²
19. a_2_1·a_4_2
20. a_3_3·b_3_2 + b_4_4·a_1_1²
21. a_3_3·b_3_2 + a_1_1·a_5_3
22. a_1_0·a_5_3
23. a_3_3·b_3_2 + b_2_2·a_4_2 + a_2_1·b_2_2² + a_1_1·a_5_4
24. a_1_0·a_5_4
25. a_1_0·b_5_5
26. a_4_2·a_3_3
27. b_2_2·a_5_3 + a_2_1·b_2_2·b_3_2 + a_2_1·a_5_3
28. a_4_2·b_3_2 + a_2_1·b_2_2·b_3_2 + a_2_1·a_5_4
29. b_2_2·a_5_3 + b_2_2·b_4_4·a_1_1 + a_2_1·b_5_5 + a_2_1·b_2_2·b_3_2
30. b_2_2·a_5_3 + a_2_1·b_2_2·b_3_2 + a_1_1²·b_5_5
31. a_4_2·b_3_2 + a_2_1·b_2_2·b_3_2 + a_6_5·a_1_1
32. a_6_5·a_1_0
33. b_4_4·b_3_2 + b_2_2·b_5_5 + b_6_8·a_1_1 + a_4_2·b_3_2 + b_2_2·a_5_3 + a_2_1·b_2_2·b_3_2
34. b_6_8·a_1_0
35. a_4_2²
36. a_3_3·a_5_3
37. b_3_2·a_5_3 + a_2_1·b_2_2³ + b_4_4·a_1_1·a_3_3
38. a_3_3·a_5_4
39. b_3_2·a_5_4 + b_2_2·a_6_5 + b_4_4·a_1_1·a_3_3 + b_2_2·a_1_1·a_5_4
40. b_2_2·a_1_1·a_5_4 + a_2_1·a_6_5
41. b_3_2·b_5_5 + b_2_2²·b_4_4 + a_3_3·b_5_5 + a_2_1·b_6_8 + a_2_1·b_2_2·b_4_4
       + a_2_1·b_2_2³ + b_2_2·a_1_1·a_5_4
42. b_6_8·a_1_1² + b_4_4·a_1_1·a_3_3
43. a_3_3·b_5_5 + a_4_2·b_4_4 + a_2_1·b_2_2·b_4_4 + a_1_1·a_7_8 + b_4_4·a_1_1·a_3_3
44. a_1_0·a_7_8
45. b_3_2·b_5_5 + b_2_2²·b_4_4 + a_3_3·b_5_5 + a_1_1·b_7_10 + a_4_2·b_4_4 + a_2_1·b_2_2·b_4_4
       + b_4_4·a_1_1·a_3_3 + b_2_2·a_1_1·a_5_4
46. a_1_0·b_7_10
47. a_4_2·a_5_3
48. a_4_2·a_5_4 + a_2_1·b_2_2·a_5_4
49. a_6_5·a_3_3
50. a_6_5·b_3_2 + b_2_2²·a_5_4 + a_2_1·b_4_4·a_3_3
51. b_6_8·a_3_3 + b_4_4·a_5_3 + a_2_1·b_2_2·b_5_5
52. b_4_4·a_5_4 + b_4_4·a_5_3 + a_4_2·b_5_5 + b_2_2·a_7_8 + b_2_2·b_6_8·a_1_1
       + a_2_1·b_4_4·a_3_3
53. a_4_2·b_5_5 + a_2_1·b_2_2·b_5_5 + a_2_1·a_7_8 + a_2_1·b_4_4·a_3_3
54. a_1_1²·a_7_8
55. b_6_8·b_3_2 + b_2_2·b_7_10 + b_2_2²·b_5_5 + b_2_2³·b_3_2 + b_4_4·a_5_4 + b_4_4·a_5_3
       + b_4_4²·a_1_1 + a_4_2·b_5_5 + b_2_2·b_6_8·a_1_1 + a_2_1·b_2_2²·b_3_2 + a_4_2·a_5_4
       + a_2_1·b_4_4·a_3_3
56. a_4_2·b_5_5 + b_2_2·b_6_8·a_1_1 + a_2_1·b_7_10 + a_2_1·b_2_2²·b_3_2 + a_2_1·b_4_4·a_3_3
57. a_4_2·b_5_5 + a_2_1·b_2_2·b_5_5 + a_8_6·a_1_1
58. a_8_6·a_1_0
59. a_5_3²
60. a_5_4²
61. a_5_3·b_5_5 + b_4_4·a_1_1·b_5_5 + a_2_1·b_4_4² + a_2_1·b_2_2²·b_4_4
62. a_5_3·a_5_4 + a_4_2·a_6_5
63. a_5_3·a_5_4 + a_2_1·b_2_2·a_6_5
64. a_3_3·a_7_8
65. a_5_4·b_5_5 + a_5_3·b_5_5 + b_4_4·a_6_5 + a_4_2·b_6_8 + a_2_1·b_2_2·b_6_8
       + a_2_1·b_2_2²·b_4_4 + a_5_3·a_5_4 + b_4_4²·a_1_1² + b_2_2·a_1_1·a_7_8
66. a_5_3·b_5_5 + b_3_2·a_7_8 + b_4_4·a_6_5 + a_2_1·b_2_2·b_6_8 + a_2_1·b_2_2²·b_4_4
67. a_5_3·b_5_5 + a_3_3·b_7_10 + a_2_1·b_2_2²·b_4_4 + b_4_4²·a_1_1²
68. b_3_2·b_7_10 + b_2_2²·b_6_8 + b_2_2³·b_4_4 + b_2_2⁵ + b_4_4·a_6_5 + a_2_1·b_4_4²
       + a_2_1·b_2_2·b_6_8 + a_2_1·b_2_2²·b_4_4 + a_2_1·b_2_2⁴
69. b_5_5² + b_2_2·b_4_4² + a_5_3·b_5_5 + a_2_1·b_4_4² + a_2_1·b_2_2²·b_4_4
       + b_4_4²·a_1_1² + c_8_13·a_1_1²
70. b_4_4·a_6_5 + a_4_2·b_6_8 + b_2_2·a_8_6 + a_2_1·b_2_2²·b_4_4 + a_2_1·b_2_2⁴
       + b_4_4²·a_1_1²
71. a_5_4·b_5_5 + a_5_3·b_5_5 + b_4_4·a_6_5 + a_4_2·b_6_8 + a_2_1·b_2_2·b_6_8
       + a_2_1·b_2_2²·b_4_4 + a_5_3·a_5_4 + b_4_4²·a_1_1² + a_2_1·a_8_6
72. a_5_4·b_5_5 + b_4_4·a_6_5 + a_5_3·a_5_4 + a_1_1·a_9_11 + b_4_4²·a_1_1²
73. a_1_0·a_9_11
74. a_6_5·a_5_3 + a_2_1·b_2_2²·a_5_4
75. a_6_5·a_5_4
76. a_4_2·a_7_8 + a_2_1·b_2_2·a_7_8
77. a_6_5·b_5_5 + a_4_2·b_7_10 + b_2_2²·a_7_8 + a_2_1·b_2_2²·b_5_5 + a_2_1·b_2_2³·b_3_2
       + a_4_2·a_7_8
78. b_6_8·a_5_3 + b_4_4²·a_3_3 + a_2_1·b_2_2·b_7_10 + a_2_1·b_2_2²·b_5_5
       + a_2_1·b_2_2³·b_3_2 + a_4_2·a_7_8
79. b_6_8·b_5_5 + b_4_4·b_7_10 + b_2_2·b_4_4·b_5_5 + b_2_2³·b_5_5 + b_6_8·a_5_3 + a_6_5·b_5_5
       + b_4_4·a_7_8 + b_4_4·b_6_8·a_1_1 + b_2_2²·a_7_8 + a_2_1·b_4_4·b_5_5
       + a_2_1·b_2_2³·b_3_2 + b_2_2·c_8_13·a_1_1 + c_8_13·a_1_1³
80. a_8_6·a_3_3
81. a_8_6·b_3_2 + b_6_8·a_5_3 + a_6_5·b_5_5 + b_4_4²·a_3_3 + a_2_1·b_2_2²·b_5_5
       + a_2_1·b_2_2³·b_3_2 + a_2_1·b_4_4·a_5_3 + a_2_1·b_2_2²·a_5_4
82. b_6_8·a_5_4 + b_6_8·a_5_3 + a_6_5·b_5_5 + b_2_2·a_9_11 + a_2_1·b_4_4·b_5_5
       + a_2_1·b_2_2²·b_5_5 + a_2_1·b_2_2³·b_3_2 + a_4_2·a_7_8
83. b_6_8·a_5_3 + a_6_5·b_5_5 + b_4_4²·a_3_3 + b_2_2²·a_7_8 + a_2_1·a_9_11
       + a_2_1·b_4_4·a_5_3 + a_2_1·b_2_2²·a_5_4
84. b_6_8·b_5_5 + b_4_4·b_7_10 + b_2_2·b_4_4·b_5_5 + b_2_2³·b_5_5 + b_4_4·a_7_8
       + b_4_4·b_6_8·a_1_1 + b_4_4²·a_3_3 + a_2_1·b_4_4·b_5_5 + a_2_1·b_2_2³·b_3_2
       + a_10_12·a_1_1 + a_4_2·a_7_8 + a_2_1·b_2_2²·a_5_4 + b_2_2·c_8_13·a_1_1
85. b_6_8·b_5_5 + b_4_4·b_7_10 + b_2_2·b_4_4·b_5_5 + b_2_2³·b_5_5 + b_6_8·a_5_3 + a_6_5·b_5_5
       + b_4_4·a_7_8 + b_4_4·b_6_8·a_1_1 + b_2_2²·a_7_8 + a_2_1·b_4_4·b_5_5
       + a_2_1·b_2_2³·b_3_2 + a_10_12·a_1_0 + b_2_2·c_8_13·a_1_1
86. a_6_5²
87. a_5_3·b_7_10 + a_4_2·b_4_4² + a_2_1·b_2_2·b_4_4² + a_2_1·b_2_2²·b_6_8
       + a_2_1·b_2_2³·b_4_4 + a_2_1·b_2_2⁵ + a_5_3·a_7_8 + b_4_4·a_1_1·a_7_8
88. b_6_8² + b_4_4³ + b_2_2·b_4_4·b_6_8 + b_2_2²·b_4_4² + b_2_2⁴·b_4_4
       + a_4_2·b_4_4² + a_2_1·b_4_4·b_6_8 + a_2_1·b_2_2²·b_6_8 + a_2_1·b_2_2⁵ + a_5_4·a_7_8
       + a_5_3·a_7_8 + b_2_2²·c_8_13
89. b_5_5·b_7_10 + b_2_2·b_4_4·b_6_8 + b_2_2²·b_4_4² + b_2_2⁴·b_4_4 + b_5_5·a_7_8
       + a_4_2·b_4_4² + a_2_1·b_2_2·b_4_4² + a_2_1·b_2_2³·b_4_4 + a_2_1·b_2_2⁵
       + a_5_4·a_7_8 + a_5_3·a_7_8 + b_4_4·a_1_1·a_7_8 + a_2_1·b_2_2²·a_6_5
       + a_2_1·b_2_2·c_8_13 + c_8_13·a_1_1·a_3_3
90. a_5_3·a_7_8 + a_4_2·a_8_6
91. a_5_4·b_7_10 + a_6_5·b_6_8 + b_2_2²·a_8_6 + b_2_2³·a_6_5 + a_2_1·b_2_2²·b_6_8
       + a_2_1·b_2_2³·b_4_4 + a_2_1·b_2_2⁵ + a_5_3·a_7_8 + b_4_4·a_1_1·a_7_8
       + b_4_4²·a_1_1·a_3_3 + a_2_1·b_2_2²·a_6_5
92. a_5_3·a_7_8 + a_2_1·b_2_2·a_8_6
93. b_5_5·a_7_8 + b_4_4·a_8_6 + a_4_2·b_4_4² + a_2_1·b_2_2³·b_4_4 + a_5_3·a_7_8
       + b_4_4·a_1_1·a_7_8 + b_4_4²·a_1_1·a_3_3 + c_8_13·a_1_1·a_3_3
94. a_3_3·a_9_11
95. a_5_4·a_7_8 + a_5_3·a_7_8 + b_2_2·a_1_1·a_9_11
96. a_5_4·b_7_10 + b_3_2·a_9_11 + a_4_2·b_4_4² + b_2_2³·a_6_5 + a_2_1·b_2_2²·b_6_8
       + a_2_1·b_2_2³·b_4_4 + a_2_1·b_2_2⁵ + a_5_4·a_7_8 + a_5_3·a_7_8 + b_4_4²·a_1_1·a_3_3
97. b_5_5·b_7_10 + b_2_2·b_4_4·b_6_8 + b_2_2²·b_4_4² + b_2_2⁴·b_4_4 + b_5_5·a_7_8
       + a_6_5·b_6_8 + b_2_2·a_10_12 + a_2_1·b_2_2³·b_4_4 + a_2_1·b_2_2⁵ + a_5_3·a_7_8
       + b_4_4·a_1_1·a_7_8 + b_4_4²·a_1_1·a_3_3 + c_8_13·a_1_1·a_3_3
98. a_5_4·a_7_8 + a_2_1·a_10_12
99. b_6_8·b_7_10 + b_4_4²·b_5_5 + b_2_2²·b_4_4·b_5_5 + b_2_2³·b_7_10 + b_2_2⁵·b_3_2
       + b_6_8·a_7_8 + b_4_4²·a_5_3 + b_4_4³·a_1_1 + b_2_2³·a_7_8 + a_2_1·b_4_4·b_7_10
       + a_2_1·b_2_2³·b_5_5 + a_2_1·b_2_2⁴·b_3_2 + a_6_5·a_7_8 + a_2_1·b_4_4·a_7_8
       + b_2_2·c_8_13·b_3_2 + a_2_1·c_8_13·b_3_2
100. a_8_6·a_5_3 + a_2_1·b_2_2²·a_7_8
101. a_8_6·a_5_4 + a_6_5·a_7_8 + a_2_1·b_2_2²·a_7_8 + a_2_1·b_2_2³·a_5_4
102. a_8_6·b_5_5 + b_2_2·b_4_4·a_7_8 + a_2_1·b_2_2·b_4_4·b_5_5 + a_2_1·b_2_2³·b_5_5
        + a_2_1·b_4_4·a_7_8 + a_2_1·b_2_2²·a_7_8 + a_2_1·c_8_13·a_3_3
103. a_6_5·a_7_8 + a_4_2·a_9_11 + a_2_1·b_2_2²·a_7_8
104. a_6_5·b_7_10 + b_2_2²·a_9_11 + b_2_2⁴·a_5_4 + a_2_1·b_2_2·b_4_4·b_5_5
        + a_2_1·b_2_2²·b_7_10 + a_6_5·a_7_8 + a_2_1·b_4_4·a_7_8 + a_2_1·b_4_4²·a_3_3
        + a_2_1·b_2_2³·a_5_4
105. a_6_5·a_7_8 + a_2_1·b_2_2·a_9_11 + a_2_1·b_2_2²·a_7_8
106. b_6_8·b_7_10 + b_4_4²·b_5_5 + b_2_2²·b_4_4·b_5_5 + b_2_2³·b_7_10 + b_2_2⁵·b_3_2
        + b_4_4·a_9_11 + b_4_4³·a_1_1 + b_2_2·b_4_4·a_7_8 + b_2_2³·a_7_8 + a_2_1·b_2_2⁴·b_3_2
        + a_6_5·a_7_8 + a_2_1·b_4_4²·a_3_3 + a_2_1·b_2_2²·a_7_8 + b_2_2·c_8_13·b_3_2
107. a_10_12·a_3_3 + a_2_1·b_4_4²·a_3_3 + a_2_1·c_8_13·a_3_3
108. b_6_8·b_7_10 + b_4_4²·b_5_5 + b_2_2²·b_4_4·b_5_5 + b_2_2³·b_7_10 + b_2_2⁵·b_3_2
        + a_10_12·b_3_2 + b_6_8·a_7_8 + a_6_5·b_7_10 + b_4_4²·a_5_3 + b_4_4³·a_1_1
        + b_2_2⁴·a_5_4 + a_2_1·b_4_4·b_7_10 + a_2_1·b_2_2²·b_7_10 + a_2_1·b_4_4·a_7_8
        + a_2_1·b_4_4²·a_3_3 + a_2_1·b_2_2³·a_5_4 + b_2_2·c_8_13·b_3_2
109. a_7_8²
110. a_7_8·b_7_10 + b_6_8·a_8_6 + a_4_2·b_4_4·b_6_8 + b_2_2·b_4_4·a_8_6 + b_2_2³·a_8_6
        + a_2_1·b_2_2²·b_4_4² + a_2_1·b_2_2³·b_6_8 + a_2_1·b_2_2⁶ + a_6_5·a_8_6
        + b_4_4³·a_1_1² + a_2_1·b_4_4·a_8_6 + a_2_1·b_2_2³·a_6_5 + c_8_13·a_1_1·a_5_4
111. a_5_3·a_9_11 + a_6_5·a_8_6 + a_2_1·b_2_2³·a_6_5
112. a_7_8·b_7_10 + b_6_8·a_8_6 + a_4_2·b_4_4·b_6_8 + b_2_2·b_4_4·a_8_6 + b_2_2³·a_8_6
        + a_2_1·b_2_2²·b_4_4² + a_2_1·b_2_2³·b_6_8 + a_2_1·b_2_2⁶ + a_5_4·a_9_11
        + b_4_4³·a_1_1² + a_2_1·b_2_2³·a_6_5 + c_8_13·a_1_1·a_5_4
113. b_7_10² + b_2_2·b_4_4³ + b_2_2²·b_4_4·b_6_8 + b_2_2⁵·b_4_4 + b_2_2⁷
        + a_4_2·b_4_4·b_6_8 + a_2_1·b_4_4³ + a_2_1·b_2_2·b_4_4·b_6_8 + a_2_1·b_2_2²·b_4_4²
        + a_2_1·b_2_2³·b_6_8 + a_2_1·b_2_2⁴·b_4_4 + a_6_5·a_8_6 + b_4_4·a_1_1·a_9_11
        + a_2_1·b_2_2³·a_6_5 + b_2_2³·c_8_13 + a_2_1·b_2_2²·c_8_13 + b_4_4·c_8_13·a_1_1²
114. b_5_5·a_9_11 + b_6_8·a_8_6 + b_2_2·b_4_4·a_8_6 + a_2_1·b_2_2²·b_4_4²
        + a_2_1·b_2_2³·b_6_8 + a_2_1·b_2_2⁴·b_4_4 + a_6_5·a_8_6 + a_2_1·b_2_2²·a_8_6
        + a_2_1·b_2_2³·a_6_5 + a_2_1·b_2_2²·c_8_13 + b_4_4·c_8_13·a_1_1²
115. a_6_5·a_8_6 + a_4_2·a_10_12 + a_2_1·b_2_2²·a_8_6 + a_2_1·b_2_2³·a_6_5
116. a_6_5·a_8_6 + a_2_1·b_2_2·a_10_12 + a_2_1·b_2_2²·a_8_6 + a_2_1·b_2_2³·a_6_5
117. b_7_10² + b_2_2·b_4_4³ + b_2_2²·b_4_4·b_6_8 + b_2_2⁵·b_4_4 + b_2_2⁷ + b_6_8·a_8_6
        + b_4_4·a_10_12 + a_6_5·a_8_6 + a_2_1·b_2_2³·a_6_5 + b_2_2³·c_8_13 + a_2_1·b_4_4·c_8_13
        + c_8_13·a_1_1·a_5_4
118. a_8_6·a_7_8 + a_2_1·b_2_2·b_4_4·a_7_8 + a_2_1·b_2_2³·a_7_8
119. a_6_5·a_9_11 + a_2_1·b_2_2·b_4_4·a_7_8 + a_2_1·b_2_2²·a_9_11 + a_2_1·b_2_2⁴·a_5_4
120. a_8_6·b_7_10 + b_2_2·b_4_4·a_9_11 + b_2_2⁴·a_7_8 + a_2_1·b_2_2²·b_4_4·b_5_5
        + a_2_1·b_2_2³·b_7_10 + a_2_1·b_2_2⁴·b_5_5 + a_6_5·a_9_11 + a_2_1·b_2_2⁴·a_5_4
        + a_2_1·b_2_2·c_8_13·b_3_2 + a_2_1·c_8_13·a_5_4
121. b_6_8·a_9_11 + b_4_4²·a_7_8 + b_4_4³·a_3_3 + b_2_2²·b_4_4·a_7_8 + b_2_2⁴·a_7_8
        + a_2_1·b_4_4²·b_5_5 + a_2_1·b_2_2·b_4_4·b_7_10 + a_2_1·b_4_4·a_9_11
        + a_2_1·b_4_4²·a_5_3 + a_2_1·b_2_2·b_4_4·a_7_8 + a_2_1·b_2_2⁴·a_5_4
        + b_2_2·c_8_13·a_5_4 + a_2_1·b_2_2·c_8_13·b_3_2 + a_2_1·c_8_13·a_5_4
122. a_10_12·a_5_3 + a_6_5·a_9_11 + a_2_1·b_4_4²·a_5_3 + a_2_1·b_2_2·b_4_4·a_7_8
        + a_2_1·b_2_2³·a_7_8 + a_2_1·b_2_2⁴·a_5_4 + a_2_1·c_8_13·a_5_3
123. a_10_12·a_5_4 + a_2_1·b_4_4²·a_5_3 + a_2_1·c_8_13·a_5_4
124. a_10_12·b_5_5 + a_8_6·b_7_10 + b_6_8·a_9_11 + b_4_4²·a_7_8 + b_4_4³·a_3_3
        + a_2_1·b_2_2·b_4_4·b_7_10 + a_2_1·b_2_2³·b_7_10 + a_2_1·b_4_4²·a_5_3
        + a_2_1·b_2_2⁴·a_5_4 + b_2_2·c_8_13·a_5_4 + a_2_1·c_8_13·b_5_5 + a_2_1·c_8_13·a_5_4
125. a_8_6²
126. a_6_5·a_10_12 + a_2_1·a_6_5·c_8_13
127. b_7_10·a_9_11 + b_6_8·a_10_12 + a_4_2·b_4_4³ + b_2_2²·b_4_4·a_8_6 + b_2_2³·a_10_12
        + b_2_2⁴·a_8_6 + a_2_1·b_4_4²·b_6_8 + a_2_1·b_2_2·b_4_4³ + a_2_1·b_2_2⁴·b_6_8
        + a_2_1·b_2_2⁵·b_4_4 + a_7_8·a_9_11 + a_2_1·b_2_2·b_4_4·a_8_6 + a_2_1·b_2_2³·a_8_6
        + a_2_1·b_6_8·c_8_13 + b_4_4·c_8_13·a_1_1·a_3_3
128. b_6_8·a_10_12 + b_4_4²·a_8_6 + a_4_2·b_4_4³ + b_2_2·b_4_4·a_10_12
        + b_2_2²·b_4_4·a_8_6 + b_2_2⁴·a_8_6 + a_2_1·b_4_4²·b_6_8 + a_2_1·b_2_2²·b_4_4·b_6_8
        + a_2_1·b_2_2⁴·b_6_8 + a_2_1·b_2_2⁷ + b_4_4²·a_1_1·a_7_8 + b_4_4³·a_1_1·a_3_3
        + a_2_1·b_2_2²·a_10_12 + a_2_1·b_2_2⁴·a_6_5 + b_2_2·a_6_5·c_8_13 + a_2_1·b_6_8·c_8_13
        + a_2_1·b_2_2·b_4_4·c_8_13 + a_2_1·a_6_5·c_8_13
129. a_7_8·a_9_11 + a_2_1·b_4_4·a_10_12 + a_2_1·a_6_5·c_8_13
130. a_8_6·a_9_11 + a_2_1·b_2_2²·b_4_4·a_7_8 + a_2_1·b_2_2³·a_9_11
        + a_2_1·b_2_2·c_8_13·a_5_4
131. a_10_12·a_7_8 + a_2_1·b_4_4²·a_7_8 + a_2_1·b_2_2·b_4_4·a_9_11 + a_2_1·b_2_2⁴·a_7_8
        + a_2_1·c_8_13·a_7_8 + a_2_1·b_2_2·c_8_13·a_5_4
132. a_10_12·b_7_10 + b_2_2·b_4_4²·a_7_8 + b_2_2³·b_4_4·a_7_8 + b_2_2⁴·a_9_11
        + a_2_1·b_4_4²·b_7_10 + a_2_1·b_2_2·b_4_4²·b_5_5 + a_2_1·b_2_2²·b_4_4·b_7_10
        + a_2_1·b_2_2³·b_4_4·b_5_5 + a_2_1·b_2_2⁴·b_7_10 + a_2_1·b_2_2⁵·b_5_5
        + a_2_1·b_2_2·b_4_4·a_9_11 + a_2_1·b_2_2⁵·a_5_4 + b_2_2²·c_8_13·a_5_4
        + a_2_1·c_8_13·b_7_10 + a_2_1·b_4_4·c_8_13·a_3_3 + a_2_1·b_2_2·c_8_13·a_5_4
133. a_9_11²
134. a_8_6·a_10_12 + a_2_1·b_4_4²·a_8_6 + a_2_1·b_2_2²·b_4_4·a_8_6
        + a_2_1·b_2_2³·a_10_12 + a_2_1·b_2_2⁴·a_8_6 + a_2_1·a_8_6·c_8_13
        + a_2_1·b_2_2·a_6_5·c_8_13
135. a_10_12·a_9_11 + a_2_1·b_4_4²·a_9_11 + a_2_1·b_2_2³·b_4_4·a_7_8
        + a_2_1·b_2_2⁴·a_9_11 + a_2_1·c_8_13·a_9_11 + a_2_1·b_2_2²·c_8_13·a_5_4
136. a_10_12²

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 20.
• 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_8_13, a Duflot regular element of degree 8
  2. b_4_4 + b_2_2², an element of degree 4
  3. b_2_2, an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, 3, 9, 11].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].

About the group

Ring generators

Ring relations

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 1

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. a_2_1 → 0, an element of degree 2
4. b_2_2 → 0, an element of degree 2
5. a_3_3 → 0, an element of degree 3
6. b_3_2 → 0, an element of degree 3
7. a_4_2 → 0, an element of degree 4
8. b_4_4 → 0, an element of degree 4
9. a_5_3 → 0, an element of degree 5
10. a_5_4 → 0, an element of degree 5
11. b_5_5 → 0, an element of degree 5
12. a_6_5 → 0, an element of degree 6
13. b_6_8 → 0, an element of degree 6
14. a_7_8 → 0, an element of degree 7
15. b_7_10 → 0, an element of degree 7
16. a_8_6 → 0, an element of degree 8
17. c_8_13 → c_1_0⁸, an element of degree 8
18. a_9_11 → 0, an element of degree 9
19. a_10_12 → 0, an element of degree 10

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. a_2_1 → 0, an element of degree 2
4. b_2_2 → c_1_2², an element of degree 2
5. a_3_3 → 0, an element of degree 3
6. b_3_2 → c_1_2³, an element of degree 3
7. a_4_2 → 0, an element of degree 4
8. b_4_4 → c_1_1²·c_1_2² + c_1_1⁴, an element of degree 4
9. a_5_3 → 0, an element of degree 5
10. a_5_4 → 0, an element of degree 5
11. b_5_5 → c_1_1²·c_1_2³ + c_1_1⁴·c_1_2, an element of degree 5
12. a_6_5 → 0, an element of degree 6
13. b_6_8 → c_1_1²·c_1_2⁴ + c_1_1⁶ + c_1_0²·c_1_2⁴ + c_1_0⁴·c_1_2², an element of degree 6
14. a_7_8 → 0, an element of degree 7
15. b_7_10 → c_1_2⁷ + c_1_1⁴·c_1_2³ + c_1_1⁶·c_1_2 + c_1_0²·c_1_2⁵ + c_1_0⁴·c_1_2³, an element of degree 7
16. a_8_6 → 0, an element of degree 8
17. c_8_13 → c_1_1²·c_1_2⁶ + c_1_1⁸ + c_1_0²·c_1_1²·c_1_2⁴ + c_1_0²·c_1_1⁴·c_1_2²
       + c_1_0⁴·c_1_2⁴ + c_1_0⁴·c_1_1²·c_1_2² + c_1_0⁴·c_1_1⁴ + c_1_0⁸, an element of degree 8
18. a_9_11 → 0, an element of degree 9
19. a_10_12 → 0, an element of degree 10

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128