David J. Green

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Singular

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Cohomology of group number 29 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 8.
• It is non-abelian.
• It has p-Rank 4.
• Its center has rank 2.
• It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 4.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 4 and depth 2.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  (2) · (t2  −  1/2·t  +  1/2)

  (t  +  1) · (t  −  1)4 · (t2  +  1)2

• The a-invariants are -∞,-∞,-4,-4,-4. 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 22 minimal generators of maximal degree 6:

1. a_1_0, a nilpotent element of degree 1
2. a_1_1, a nilpotent element of degree 1
3. a_2_0, a nilpotent element of degree 2
4. b_2_1, an element of degree 2
5. b_2_2, an element of degree 2
6. b_2_3, an element of degree 2
7. a_3_2, a nilpotent element of degree 3
8. a_3_3, a nilpotent element of degree 3
9. a_3_5, a nilpotent element of degree 3
10. a_3_4, a nilpotent element of degree 3
11. b_3_6, an element of degree 3
12. b_3_7, an element of degree 3
13. a_4_3, a nilpotent element of degree 4
14. a_4_4, a nilpotent element of degree 4
15. b_4_10, an element of degree 4
16. b_4_11, an element of degree 4
17. c_4_12, a Duflot regular element of degree 4
18. c_4_13, a Duflot regular element of degree 4
19. a_5_15, a nilpotent element of degree 5
20. a_5_14, a nilpotent element of degree 5
21. b_5_19, an element of degree 5
22. a_6_9, a nilpotent element of degree 6

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

Ring relations

There are 169 minimal relations of maximal degree 12:

1. a_1_0²
2. a_1_1²
3. a_1_0·a_1_1
4. a_2_0·a_1_1
5. a_2_0·a_1_0
6. b_2_1·a_1_1 + b_2_1·a_1_0
7. b_2_2·a_1_1
8. b_2_2·a_1_0
9. b_2_3·a_1_0 + b_2_1·a_1_1
10. a_2_0²
11. b_2_2² + b_2_1·b_2_3 + b_2_1²
12. a_2_0·b_2_1 + a_1_1·a_3_2
13. a_2_0·b_2_1 + a_1_0·a_3_2
14. a_2_0·b_2_2 + a_2_0·b_2_1 + a_1_1·a_3_3
15. a_2_0·b_2_1 + a_1_0·a_3_3
16. a_1_1·a_3_5
17. a_2_0·b_2_2 + a_1_0·a_3_5
18. a_2_0·b_2_3 + a_2_0·b_2_2 + a_2_0·b_2_1 + a_1_1·a_3_4
19. a_2_0·b_2_2 + a_1_0·a_3_4
20. a_1_1·b_3_6
21. a_1_0·b_3_6
22. a_1_1·b_3_7 + a_2_0·b_2_3
23. a_1_0·b_3_7 + a_2_0·b_2_2 + a_2_0·b_2_1
24. a_2_0·a_3_2
25. b_2_2·a_3_2 + b_2_1·a_3_3 + b_2_1·a_3_2 + b_2_1²·a_1_0
26. b_2_3·a_3_2 + b_2_2·a_3_3 + b_2_2·a_3_2 + b_2_1·a_3_2
27. a_2_0·a_3_3
28. b_2_2·a_3_5
29. a_2_0·a_3_5
30. b_2_3·a_3_2 + b_2_2·a_3_2 + b_2_1·a_3_4 + b_2_1·a_3_5 + b_2_1·a_3_2 + b_2_1²·a_1_0
31. b_2_3·a_3_5 + b_2_3·a_3_3 + b_2_2·a_3_4 + b_2_2·a_3_2 + b_2_1·a_3_5 + b_2_1·a_3_2
       + b_2_1²·a_1_0
32. a_2_0·a_3_4
33. a_2_0·b_3_6
34. b_2_3·b_3_7 + b_2_3·b_3_6 + b_2_2·b_3_6 + b_2_1·b_3_6 + b_2_3·a_3_4 + b_2_3·a_3_5
       + b_2_3·a_3_3 + b_2_1²·a_1_0
35. b_2_2·b_3_7 + b_2_2·b_3_6 + b_2_1·b_3_7 + b_2_3·a_3_5 + b_2_3·a_3_3 + b_2_1·a_3_5
36. a_2_0·b_3_7
37. a_4_3·a_1_1
38. a_4_3·a_1_0
39. a_4_4·a_1_1
40. a_4_4·a_1_0
41. b_4_10·a_1_1 + b_2_1·a_3_5 + b_2_1²·a_1_0
42. b_4_10·a_1_0 + b_2_1·a_3_5 + b_2_1²·a_1_0
43. b_4_11·a_1_1 + b_2_3·a_3_5 + b_2_1·a_3_5
44. b_4_11·a_1_0
45. a_3_2²
46. a_3_2·a_3_3 + b_2_1·a_1_0·a_3_2
47. a_3_3²
48. a_3_5²
49. a_3_3·a_3_5 + a_3_2·a_3_5
50. a_3_2·a_3_4 + a_3_2·a_3_5 + b_2_1·a_1_0·a_3_2
51. a_3_4²
52. a_3_5·a_3_4 + a_3_3·a_3_4 + a_3_2·a_3_5 + b_2_1·a_1_0·a_3_2
53. a_3_5·b_3_6
54. b_3_6² + b_2_1·b_2_3² + b_2_1²·b_2_3
55. a_3_5·b_3_7 + a_3_5·a_3_4 + a_3_2·a_3_5
56. a_3_4·b_3_7 + a_3_4·b_3_6 + a_3_3·b_3_6 + a_3_2·b_3_6 + a_3_2·a_3_5 + b_2_1·a_1_0·a_3_2
57. a_3_3·b_3_7 + a_3_3·b_3_6 + a_3_2·b_3_6 + a_3_5·a_3_4
58. b_3_6·b_3_7 + b_2_1·b_2_3² + b_2_1·b_2_2·b_2_3 + b_2_1²·b_2_2 + b_2_1³ + a_3_4·b_3_6
       + a_3_3·b_3_6
59. b_3_7² + b_2_1·b_2_3² + b_2_1³
60. a_3_2·b_3_7 + a_3_2·b_3_6 + b_2_1·a_4_3 + a_3_2·a_3_5
61. a_3_3·b_3_6 + b_2_3·a_4_3 + a_3_5·a_3_4 + a_3_2·a_3_5 + b_2_3·a_1_1·a_3_4
       + b_2_1·a_1_0·a_3_2
62. a_3_2·b_3_7 + b_2_2·a_4_3 + b_2_1·a_1_0·a_3_2
63. a_2_0·a_4_3
64. a_3_2·b_3_6 + b_2_1·a_4_4
65. a_3_4·b_3_6 + a_3_3·b_3_6 + b_2_3·a_4_4 + b_2_3·a_1_1·a_3_4
66. a_3_3·b_3_6 + a_3_2·b_3_6 + b_2_2·a_4_4
67. a_2_0·a_4_4
68. a_2_0·b_4_10 + a_3_2·a_3_5 + b_2_1·a_1_0·a_3_2
69. b_2_2·b_4_10 + b_2_1·b_4_11 + b_2_1²·b_2_2 + a_3_3·b_3_6 + a_3_2·b_3_7 + a_3_2·b_3_6
       + a_3_2·a_3_5
70. b_2_3·b_4_10 + b_2_2·b_4_11 + b_2_1·b_4_10 + b_2_1²·b_2_3 + b_2_1³ + a_3_4·b_3_6
       + a_3_2·b_3_7 + b_2_1·a_1_0·a_3_2
71. a_2_0·b_4_11 + a_3_5·a_3_4
72. a_3_2·a_3_5 + a_1_1·a_5_15 + b_2_1·a_1_0·a_3_2
73. a_3_2·a_3_5 + a_1_0·a_5_15 + b_2_1·a_1_0·a_3_2
74. a_3_5·a_3_4 + a_1_1·a_5_14 + b_2_3·a_1_1·a_3_4
75. a_1_0·a_5_14
76. a_1_1·b_5_19 + a_3_5·a_3_4 + b_2_1·a_1_0·a_3_2
77. a_1_0·b_5_19 + b_2_1·a_1_0·a_3_2
78. a_4_3·a_3_2
79. a_4_3·a_3_5
80. a_4_3·a_3_4
81. a_4_3·a_3_3
82. a_4_3·b_3_6 + b_2_1·b_2_2·a_3_4
83. a_4_3·b_3_7 + b_2_1·b_2_2·a_3_4 + b_2_1²·a_3_4 + b_2_1²·a_3_5 + b_2_1²·a_3_3
       + b_2_1²·a_3_2
84. a_4_4·a_3_2
85. a_4_4·a_3_5
86. a_4_4·a_3_4
87. a_4_4·a_3_3
88. a_4_4·b_3_6 + b_2_1·b_2_3·a_3_4 + b_2_1·b_2_2·a_3_4 + b_2_1²·a_3_4
89. a_4_4·b_3_7 + b_2_1·b_2_3·a_3_4 + b_2_1²·a_3_4
90. b_4_10·a_3_5 + b_2_1·c_4_13·a_1_0 + b_2_1·c_4_12·a_1_0
91. b_4_11·a_3_2 + b_4_10·a_3_3 + b_4_10·a_3_2 + b_2_1²·a_3_5 + b_2_1²·a_3_3 + b_2_1²·a_3_2
92. b_4_11·a_3_5 + b_2_3·c_4_13·a_1_1 + b_2_1·c_4_13·a_1_0
93. b_4_11·a_3_3 + b_4_10·a_3_4 + b_2_1²·a_3_4 + b_2_3·c_4_13·a_1_1 + b_2_1·c_4_12·a_1_0
94. b_4_11·b_3_7 + b_4_11·b_3_6 + b_4_10·b_3_7 + b_2_1²·b_3_7 + b_4_11·a_3_4 + b_4_10·a_3_3
       + b_2_1·b_2_3·a_3_4 + b_2_1·b_2_2·a_3_4 + b_2_1²·a_3_5 + b_2_1²·a_3_2
95. b_4_10·a_3_2 + b_2_1·a_5_15 + b_2_1²·a_3_4 + b_2_1²·a_3_5 + b_2_1³·a_1_0
       + b_2_1·c_4_13·a_1_0 + b_2_1·c_4_12·a_1_0
96. b_4_10·a_3_4 + b_4_10·a_3_3 + b_2_3·a_5_15 + b_2_1·b_2_3·a_3_4 + b_2_1²·a_3_5
       + b_2_1³·a_1_0
97. b_4_10·a_3_3 + b_4_10·a_3_2 + b_2_2·a_5_15 + b_2_1·b_2_2·a_3_4 + b_2_1²·a_3_5
       + b_2_1³·a_1_0
98. a_2_0·a_5_15
99. b_4_10·a_3_3 + b_4_10·a_3_2 + b_2_1·a_5_14 + b_2_1·b_2_3·a_3_4 + b_2_1²·a_3_3
       + b_2_1²·a_3_2 + b_2_1³·a_1_0 + b_2_1·c_4_12·a_1_0
100. b_4_11·a_3_4 + b_4_10·a_3_4 + b_2_3·a_5_14 + b_2_3²·a_3_4 + b_2_1²·a_3_4 + b_2_1²·a_3_5
        + b_2_1³·a_1_0 + b_2_1·c_4_13·a_1_0
101. b_4_10·a_3_4 + b_4_10·a_3_3 + b_4_10·a_3_2 + b_2_2·a_5_14 + b_2_2·b_2_3·a_3_4
        + b_2_1²·a_3_4 + b_2_1²·a_3_5 + b_2_1²·a_3_3 + b_2_1²·a_3_2 + b_2_1·c_4_13·a_1_0
        + b_2_1·c_4_12·a_1_0
102. a_2_0·a_5_14
103. b_4_10·b_3_7 + b_4_10·b_3_6 + b_2_1·b_5_19 + b_2_1·b_2_2·b_3_6 + b_2_1²·b_3_7
        + b_4_10·a_3_4 + b_4_10·a_3_2 + b_2_1²·a_3_4 + b_2_1²·a_3_5 + b_2_1³·a_1_0
        + b_2_1·c_4_13·a_1_0
104. b_4_11·b_3_6 + b_4_10·b_3_6 + b_2_3·b_5_19 + b_2_2·b_2_3·b_3_6 + b_2_1·b_2_3·b_3_6
        + b_2_1²·b_3_6 + b_4_11·a_3_4 + b_4_10·a_3_4 + b_2_2·b_2_3·a_3_4 + b_2_1·b_2_2·a_3_4
        + b_2_1²·a_3_5 + b_2_1²·a_3_3 + b_2_1³·a_1_0 + b_2_3·c_4_13·a_1_1
105. b_4_10·b_3_7 + b_2_2·b_5_19 + b_2_1·b_2_3·b_3_6 + b_2_1·b_2_2·b_3_6 + b_2_1²·b_3_7
        + b_2_1²·b_3_6 + b_4_10·a_3_2 + b_2_1²·a_3_5 + b_2_1²·a_3_3 + b_2_1³·a_1_0
106. a_2_0·b_5_19
107. a_6_9·a_1_1
108. a_6_9·a_1_0
109. a_4_3²
110. a_4_3·a_4_4
111. a_4_4²
112. b_4_11² + b_2_1·b_2_3³ + b_2_1·b_2_2·b_4_11 + b_2_1·b_2_2·b_2_3²
        + b_2_1²·b_2_3² + b_2_1²·b_2_2·b_2_3 + b_2_1³·b_2_3 + b_2_1⁴ + a_4_4·b_4_10
        + a_4_3·b_4_11 + a_4_3·b_4_10 + b_2_2·b_2_3·a_4_4 + b_2_1·b_2_3·a_4_4 + b_2_1·b_2_2·a_4_4
        + b_2_1²·a_4_3 + b_2_3²·a_1_1·a_3_4 + b_2_3²·c_4_13 + b_2_1·b_2_3·c_4_13
        + b_2_1·b_2_3·c_4_12 + b_2_1²·c_4_12 + c_4_13·a_1_1·a_3_4 + c_4_13·a_1_0·a_3_5
        + c_4_13·a_1_0·a_3_2 + c_4_12·a_1_0·a_3_2
113. b_4_10·b_4_11 + b_4_10² + b_2_1·b_2_2·b_2_3² + b_2_1²·b_4_10 + b_2_1²·b_2_2·b_2_3
        + b_2_1³·b_2_3 + b_2_1³·b_2_2 + b_2_1⁴ + a_4_4·b_4_11 + a_4_3·b_4_11
        + b_2_1·b_2_3·a_4_4 + b_2_1²·a_4_4 + b_2_1²·a_4_3 + b_2_2·b_2_3·c_4_13
        + b_2_1·b_2_3·c_4_13 + b_2_1·b_2_2·c_4_12 + b_2_1²·c_4_12 + c_4_13·a_1_1·a_3_4
        + c_4_13·a_1_0·a_3_5 + c_4_13·a_1_0·a_3_2 + c_4_12·a_1_0·a_3_2
114. b_4_10² + b_2_1²·b_4_10 + b_2_1²·b_2_3² + b_2_1²·b_2_2·b_2_3 + b_2_1⁴
        + b_2_1·b_2_2·a_4_4 + b_2_1²·a_4_3 + b_2_1·a_1_0·a_5_15 + b_2_1·b_2_3·c_4_13
        + b_2_1²·c_4_12
115. a_3_2·a_5_15 + c_4_13·a_1_0·a_3_5 + c_4_13·a_1_0·a_3_2 + c_4_12·a_1_0·a_3_2
116. a_3_5·a_5_15 + c_4_13·a_1_0·a_3_5 + c_4_13·a_1_0·a_3_2 + c_4_12·a_1_0·a_3_5
        + c_4_12·a_1_0·a_3_2
117. b_4_10² + b_2_1²·b_4_10 + b_2_1²·b_2_3² + b_2_1²·b_2_2·b_2_3 + b_2_1⁴
        + b_2_1·b_2_2·a_4_4 + b_2_1²·a_4_3 + a_3_4·a_5_15 + b_2_1·b_2_3·c_4_13 + b_2_1²·c_4_12
        + c_4_13·a_1_0·a_3_5 + c_4_13·a_1_0·a_3_2 + c_4_12·a_1_0·a_3_5 + c_4_12·a_1_0·a_3_2
118. b_4_10² + b_2_1²·b_4_10 + b_2_1²·b_2_3² + b_2_1²·b_2_2·b_2_3 + b_2_1⁴
        + b_2_1·b_2_2·a_4_4 + b_2_1²·a_4_3 + a_3_3·a_5_15 + b_2_1·b_2_3·c_4_13 + b_2_1²·c_4_12
        + c_4_13·a_1_0·a_3_5 + c_4_13·a_1_0·a_3_2 + c_4_12·a_1_0·a_3_2
119. b_3_6·a_5_15 + a_4_4·b_4_10 + b_2_1·b_2_3·a_4_4 + b_2_1·b_2_2·a_4_4 + b_2_1²·a_4_4
120. b_3_7·a_5_15 + a_4_4·b_4_10 + a_4_3·b_4_10 + b_2_1·b_2_3·a_4_4 + b_2_1²·a_4_4
        + c_4_12·a_1_0·a_3_5
121. a_3_2·a_5_14 + c_4_12·a_1_0·a_3_2
122. a_4_4·b_4_10 + a_4_3·b_4_11 + a_4_3·b_4_10 + b_2_1²·a_4_4 + b_2_1²·a_4_3
        + b_2_3·a_1_1·a_5_14 + b_2_3²·a_1_1·a_3_4 + c_4_13·a_1_1·a_3_4 + c_4_13·a_1_0·a_3_5
        + c_4_13·a_1_0·a_3_2 + c_4_12·a_1_0·a_3_2
123. a_4_4·b_4_10 + a_4_3·b_4_11 + a_4_3·b_4_10 + b_2_1²·a_4_4 + b_2_1²·a_4_3 + a_3_5·a_5_14
        + c_4_13·a_1_0·a_3_2 + c_4_12·a_1_0·a_3_5 + c_4_12·a_1_0·a_3_2
124. a_3_4·a_5_14 + c_4_13·a_1_0·a_3_5
125. a_4_4·b_4_10 + a_4_3·b_4_11 + a_4_3·b_4_10 + b_2_1²·a_4_4 + b_2_1²·a_4_3 + a_3_3·a_5_14
        + c_4_13·a_1_0·a_3_2
126. b_3_6·a_5_14 + a_4_4·b_4_11 + a_4_4·b_4_10 + a_4_3·b_4_11 + a_4_3·b_4_10 + b_2_3²·a_4_4
        + b_2_2·b_2_3·a_4_4 + b_2_1·b_2_3·a_4_4 + b_2_1²·a_4_4 + b_2_1²·a_4_3
        + b_2_3²·a_1_1·a_3_4 + c_4_13·a_1_1·a_3_4 + c_4_13·a_1_0·a_3_5 + c_4_13·a_1_0·a_3_2
        + c_4_12·a_1_0·a_3_2
127. b_3_7·a_5_14 + a_4_4·b_4_11 + a_4_3·b_4_11 + b_2_3²·a_4_4 + b_2_1·b_2_3·a_4_4
        + b_2_3²·a_1_1·a_3_4 + c_4_13·a_1_1·a_3_4 + c_4_12·a_1_0·a_3_2
128. a_3_2·b_5_19 + a_4_3·b_4_10 + b_2_1·b_2_2·a_4_4 + b_2_1²·a_4_4 + b_2_1²·a_4_3
        + b_2_1²·a_1_0·a_3_2 + c_4_13·a_1_0·a_3_5 + c_4_13·a_1_0·a_3_2
129. b_4_10² + b_2_1²·b_4_10 + b_2_1²·b_2_3² + b_2_1²·b_2_2·b_2_3 + b_2_1⁴
        + a_3_5·b_5_19 + b_2_1·b_2_2·a_4_4 + b_2_1²·a_4_3 + b_2_1²·a_1_0·a_3_2
        + b_2_1·b_2_3·c_4_13 + b_2_1²·c_4_12 + c_4_13·a_1_1·a_3_4
130. b_4_10² + b_2_1²·b_4_10 + b_2_1²·b_2_3² + b_2_1²·b_2_2·b_2_3 + b_2_1⁴
        + a_3_4·b_5_19 + a_4_4·b_4_11 + a_4_4·b_4_10 + a_4_3·b_4_11 + a_4_3·b_4_10
        + b_2_2·b_2_3·a_4_4 + b_2_1·b_2_2·a_4_4 + b_2_1²·a_4_4 + b_2_1·b_2_3·c_4_13
        + b_2_1²·c_4_12 + c_4_13·a_1_0·a_3_5 + c_4_13·a_1_0·a_3_2 + c_4_12·a_1_0·a_3_2
131. a_3_3·b_5_19 + a_4_4·b_4_10 + b_2_1·b_2_3·a_4_4 + b_2_1²·a_4_4 + c_4_13·a_1_1·a_3_4
        + c_4_12·a_1_0·a_3_2
132. b_3_6·b_5_19 + b_2_1·b_2_3·b_4_11 + b_2_1·b_2_2·b_4_11 + b_2_1·b_2_2·b_2_3²
        + b_2_1²·b_4_11 + b_2_1²·b_2_3² + b_2_1²·b_2_2·b_2_3 + b_2_1³·b_2_3 + a_4_4·b_4_11
        + a_4_4·b_4_10 + a_4_3·b_4_11 + a_4_3·b_4_10 + b_2_2·b_2_3·a_4_4 + b_2_1·b_2_2·a_4_4
        + b_2_1²·a_4_3 + c_4_13·a_1_1·a_3_4 + c_4_13·a_1_0·a_3_5 + c_4_13·a_1_0·a_3_2
        + c_4_12·a_1_0·a_3_2
133. b_3_7·b_5_19 + b_2_1·b_2_3·b_4_11 + b_2_1·b_2_2·b_2_3² + b_2_1²·b_4_11
        + b_2_1²·b_2_2·b_2_3 + a_4_3·b_4_10 + b_2_1²·a_4_4 + b_2_1²·a_4_3 + c_4_13·a_1_1·a_3_4
        + c_4_13·a_1_0·a_3_5 + c_4_13·a_1_0·a_3_2
134. b_4_10² + b_2_1²·b_4_10 + b_2_1²·b_2_3² + b_2_1²·b_2_2·b_2_3 + b_2_1⁴
        + a_4_3·b_4_10 + b_2_1·a_6_9 + b_2_1·b_2_2·a_4_4 + b_2_1²·a_1_0·a_3_2
        + b_2_1·b_2_3·c_4_13 + b_2_1²·c_4_12 + c_4_13·a_1_0·a_3_5 + c_4_12·a_1_0·a_3_2
135. b_4_10² + b_2_1²·b_4_10 + b_2_1²·b_2_3² + b_2_1²·b_2_2·b_2_3 + b_2_1⁴
        + a_4_4·b_4_11 + a_4_4·b_4_10 + b_2_3·a_6_9 + b_2_1·b_2_2·a_4_4 + b_2_1²·a_4_4
        + b_2_1²·a_4_3 + b_2_3²·a_1_1·a_3_4 + b_2_1²·a_1_0·a_3_2 + b_2_1·b_2_3·c_4_13
        + b_2_1²·c_4_12 + c_4_13·a_1_1·a_3_4 + c_4_13·a_1_0·a_3_5 + c_4_13·a_1_0·a_3_2
        + c_4_12·a_1_1·a_3_4
136. a_4_4·b_4_10 + a_4_3·b_4_10 + b_2_2·a_6_9 + b_2_1²·a_4_4 + b_2_1²·a_4_3
        + c_4_13·a_1_0·a_3_2 + c_4_12·a_1_0·a_3_5 + c_4_12·a_1_0·a_3_2
137. a_2_0·a_6_9
138. a_4_3·a_5_15
139. a_4_4·a_5_15
140. a_4_3·a_5_14
141. b_4_11·a_5_15 + b_4_10·a_5_15 + b_2_1·b_2_3·a_5_14 + b_2_1·b_2_3²·a_3_4
        + b_2_1·b_2_2·b_2_3·a_3_4 + b_2_1²·a_5_15 + b_2_1²·b_2_3·a_3_4 + b_2_1²·b_2_2·a_3_4
        + b_2_2·c_4_13·a_3_4 + b_2_1·c_4_13·a_3_5 + b_2_1·c_4_13·a_3_3 + b_2_1·c_4_12·a_3_5
        + b_2_1·c_4_12·a_3_3 + b_2_1²·c_4_13·a_1_0
142. b_4_10·a_5_15 + b_2_1·b_2_2·a_5_14 + b_2_1·b_2_2·b_2_3·a_3_4 + b_2_1²·a_5_14
        + b_2_1²·a_5_15 + b_2_1³·a_3_4 + b_2_1³·a_3_5 + b_2_1³·a_3_3 + b_2_1³·a_3_2
        + b_2_1·c_4_13·a_3_4 + b_2_1·c_4_13·a_3_3 + b_2_1·c_4_12·a_3_5 + b_2_1·c_4_12·a_3_2
        + b_2_1²·c_4_12·a_1_0
143. a_4_4·a_5_14
144. b_4_11·a_5_14 + b_4_11·a_5_15 + b_2_3²·a_5_14 + b_2_3³·a_3_4 + b_2_2·b_2_3·a_5_14
        + b_2_2·b_2_3²·a_3_4 + b_2_1·b_2_3²·a_3_4 + b_2_1·b_2_2·b_2_3·a_3_4 + b_2_1²·a_5_14
        + b_2_1²·b_2_3·a_3_4 + b_2_1³·a_3_4 + b_2_3·c_4_13·a_3_4 + b_2_1·c_4_13·a_3_5
        + b_2_1·c_4_12·a_3_4 + b_2_1·c_4_12·a_3_5 + b_2_1²·c_4_13·a_1_0 + b_2_1²·c_4_12·a_1_0
145. b_4_11·a_5_15 + b_4_10·a_5_14 + b_4_10·a_5_15 + b_2_2·b_2_3·a_5_14 + b_2_2·b_2_3²·a_3_4
        + b_2_1²·a_5_15 + b_2_1³·a_3_4 + b_2_1³·a_3_5 + b_2_1³·a_3_3 + b_2_1³·a_3_2
        + b_2_1·c_4_13·a_3_4 + b_2_1·c_4_13·a_3_3 + b_2_1·c_4_12·a_3_2 + b_2_1²·c_4_12·a_1_0
146. b_4_10·a_5_15 + a_4_3·b_5_19 + b_2_1²·a_5_14 + b_2_1²·a_5_15 + b_2_1²·b_2_3·a_3_4
        + b_2_1²·b_2_2·a_3_4 + b_2_1³·a_3_5 + b_2_1³·a_3_3 + b_2_1³·a_3_2
        + b_2_1·c_4_13·a_3_4 + b_2_1·c_4_13·a_3_3 + b_2_1·c_4_12·a_3_5 + b_2_1·c_4_12·a_3_2
        + b_2_1²·c_4_12·a_1_0
147. b_4_11·a_5_15 + a_4_4·b_5_19 + b_2_1²·b_2_2·a_3_4 + b_2_1³·a_3_4 + b_2_1³·a_3_5
        + b_2_1³·a_3_3 + b_2_1³·a_3_2 + b_2_2·c_4_13·a_3_4 + b_2_1·c_4_13·a_3_4
        + b_2_1·c_4_13·a_3_5 + b_2_1·c_4_12·a_3_3 + b_2_1·c_4_12·a_3_2 + b_2_1²·c_4_13·a_1_0
        + b_2_1²·c_4_12·a_1_0
148. b_4_11·b_5_19 + b_2_2·b_2_3·b_5_19 + b_2_1·b_2_2·b_5_19 + b_2_1·b_2_2·b_2_3·b_3_6
        + b_2_1³·b_3_7 + b_4_10·a_5_15 + b_2_1²·a_5_15 + b_2_1²·b_2_3·a_3_4
        + b_2_1²·b_2_2·a_3_4 + b_2_1³·a_3_4 + b_2_1³·a_3_3 + b_2_3·c_4_13·b_3_6
        + b_2_2·c_4_13·b_3_6 + b_2_1·c_4_13·b_3_6 + b_2_1·c_4_12·b_3_7 + b_2_3·c_4_13·a_3_4
        + b_2_3·c_4_13·a_3_3 + b_2_1·c_4_12·a_3_5 + b_2_1²·c_4_12·a_1_0
149. b_4_10·b_5_19 + b_2_1·b_2_3·b_5_19 + b_2_1²·b_2_3·b_3_6 + b_2_1³·b_3_7 + b_2_1³·b_3_6
        + b_4_11·a_5_15 + b_4_10·a_5_15 + b_2_1³·a_3_5 + b_2_1⁴·a_1_0 + b_2_2·c_4_13·b_3_6
        + b_2_1·c_4_13·b_3_6 + b_2_1·c_4_12·b_3_7 + b_2_1·c_4_12·b_3_6 + b_2_1·c_4_13·a_3_4
        + b_2_1·c_4_13·a_3_3 + b_2_1·c_4_12·a_3_4 + b_2_1·c_4_12·a_3_5 + b_2_1·c_4_12·a_3_3
        + b_2_1·c_4_12·a_3_2 + b_2_1²·c_4_13·a_1_0
150. a_6_9·a_3_2
151. a_6_9·a_3_5
152. a_6_9·a_3_4
153. a_6_9·a_3_3
154. a_6_9·b_3_6 + b_4_11·a_5_15 + b_2_1·b_2_2·b_2_3·a_3_4 + b_2_1²·b_2_2·a_3_4
        + b_2_1³·a_3_4 + b_2_1³·a_3_5 + b_2_1³·a_3_3 + b_2_1³·a_3_2 + b_2_2·c_4_13·a_3_4
        + b_2_1·c_4_13·a_3_4 + b_2_1·c_4_13·a_3_5 + b_2_1·c_4_12·a_3_3 + b_2_1·c_4_12·a_3_2
        + b_2_1²·c_4_13·a_1_0 + b_2_1²·c_4_12·a_1_0
155. a_6_9·b_3_7 + b_4_11·a_5_15 + b_4_10·a_5_15 + b_2_1·b_2_2·b_2_3·a_3_4 + b_2_1²·a_5_14
        + b_2_1²·a_5_15 + b_2_1²·b_2_2·a_3_4 + b_2_2·c_4_13·a_3_4 + b_2_1·c_4_13·a_3_5
        + b_2_1·c_4_13·a_3_3 + b_2_1·c_4_12·a_3_5 + b_2_1·c_4_12·a_3_3 + b_2_1²·c_4_13·a_1_0
156. a_5_15²
157. a_5_15·a_5_14 + c_4_12·a_1_0·a_5_15
158. a_5_14²
159. b_5_19² + b_2_1²·b_2_2·b_4_11 + b_2_1²·b_2_2·b_2_3² + b_2_1³·b_2_2·b_2_3
        + b_2_1⁴·b_2_3 + b_2_1⁵ + b_2_1·b_2_2·b_2_3·a_4_4 + b_2_1²·b_2_3·a_4_4
        + b_2_1²·b_2_2·a_4_4 + b_2_1³·a_4_4 + b_2_1·b_2_3²·c_4_13 + b_2_1²·b_2_3·c_4_13
        + b_2_1²·b_2_3·c_4_12 + b_2_1³·c_4_12
160. a_4_3·a_6_9
161. a_5_14·b_5_19 + a_5_15·b_5_19 + b_2_3²·a_6_9 + b_2_2·b_2_3²·a_4_4 + b_2_1·b_2_3·a_6_9
        + b_2_1·b_2_3²·a_4_4 + b_2_1²·a_6_9 + b_2_1²·b_2_2·a_4_4 + b_2_3²·a_1_1·a_5_14
        + b_2_1³·a_1_0·a_3_2 + b_2_3·a_4_4·c_4_13 + b_2_1·a_4_4·c_4_12 + c_4_13·a_1_1·a_5_14
        + c_4_12·a_1_0·a_5_15 + b_2_3·c_4_12·a_1_1·a_3_4
162. a_5_15·b_5_19 + b_2_1·b_2_2·a_6_9 + b_2_1²·a_6_9 + b_2_1³·a_4_3 + b_2_1²·a_1_0·a_5_15
        + b_2_1³·a_1_0·a_3_2 + b_2_2·a_4_4·c_4_13 + b_2_1·a_4_4·c_4_13 + b_2_1·a_4_3·c_4_12
        + b_2_1·c_4_12·a_1_0·a_3_2
163. a_4_4·a_6_9
164. a_5_15·b_5_19 + b_4_11·a_6_9 + b_2_1·b_2_3²·a_4_4 + b_2_1²·a_6_9 + b_2_1²·b_2_2·a_4_4
        + b_2_3²·a_1_1·a_5_14 + b_2_3³·a_1_1·a_3_4 + b_2_1³·a_1_0·a_3_2 + b_2_3·a_4_4·c_4_13
        + b_2_1·a_4_4·c_4_12 + c_4_13·a_1_1·a_5_14 + c_4_12·a_1_1·a_5_14 + c_4_12·a_1_0·a_5_15
        + b_2_3·c_4_13·a_1_1·a_3_4 + b_2_3·c_4_12·a_1_1·a_3_4 + b_2_1·c_4_12·a_1_0·a_3_2
165. b_4_10·a_6_9 + b_2_1·b_2_2·b_2_3·a_4_4 + b_2_1²·b_2_2·a_4_4 + b_2_1³·a_4_4
        + b_2_1³·a_4_3 + b_2_1²·a_1_0·a_5_15 + b_2_2·a_4_4·c_4_13 + b_2_1·a_4_4·c_4_13
        + b_2_1·a_4_3·c_4_12 + c_4_13·a_1_0·a_5_15 + c_4_12·a_1_0·a_5_15
        + b_2_1·c_4_13·a_1_0·a_3_2 + b_2_1·c_4_12·a_1_0·a_3_2
166. a_6_9·b_5_19 + b_2_1·b_2_2·b_2_3·a_5_14 + b_2_1·b_2_2·b_2_3²·a_3_4
        + b_2_1²·b_2_3²·a_3_4 + b_2_1²·b_2_2·a_5_14 + b_2_1²·b_2_2·b_2_3·a_3_4
        + b_2_1³·b_2_2·a_3_4 + b_2_1⁴·a_3_5 + b_2_1⁴·a_3_3 + b_2_1⁴·a_3_2
        + b_2_1·b_2_3·c_4_13·a_3_4 + b_2_1·b_2_2·c_4_13·a_3_4 + b_2_1²·c_4_13·a_3_4
        + b_2_1²·c_4_12·a_3_4 + b_2_1²·c_4_12·a_3_5 + b_2_1²·c_4_12·a_3_3
        + b_2_1²·c_4_12·a_3_2
167. a_6_9·a_5_15
168. a_6_9·a_5_14
169. a_6_9²

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

Data used for Benson′s test

• Benson′s completion test succeeded in degree 12.
• 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_12, a Duflot regular element of degree 4
  2. c_4_13, a Duflot regular element of degree 4
  3. b_2_3 + b_2_2, an element of degree 2
  4. b_3_6, an element of degree 3
• The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 6, 9].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

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. a_2_0 → 0, an element of degree 2
4. b_2_1 → 0, an element of degree 2
5. b_2_2 → 0, an element of degree 2
6. b_2_3 → 0, an element of degree 2
7. a_3_2 → 0, an element of degree 3
8. a_3_3 → 0, an element of degree 3
9. a_3_5 → 0, an element of degree 3
10. a_3_4 → 0, an element of degree 3
11. b_3_6 → 0, an element of degree 3
12. b_3_7 → 0, an element of degree 3
13. a_4_3 → 0, an element of degree 4
14. a_4_4 → 0, an element of degree 4
15. b_4_10 → 0, an element of degree 4
16. b_4_11 → 0, an element of degree 4
17. c_4_12 → c_1_1⁴, an element of degree 4
18. c_4_13 → c_1_1⁴ + c_1_0⁴, an element of degree 4
19. a_5_15 → 0, an element of degree 5
20. a_5_14 → 0, an element of degree 5
21. b_5_19 → 0, an element of degree 5
22. a_6_9 → 0, an element of degree 6

Restriction map to a maximal el. ab. subgp. of rank 4

1. a_1_0 → 0, an element of degree 1
2. a_1_1 → 0, an element of degree 1
3. a_2_0 → 0, an element of degree 2
4. b_2_1 → c_1_3², an element of degree 2
5. b_2_2 → c_1_3² + c_1_2·c_1_3, an element of degree 2
6. b_2_3 → c_1_2², an element of degree 2
7. a_3_2 → 0, an element of degree 3
8. a_3_3 → 0, an element of degree 3
9. a_3_5 → 0, an element of degree 3
10. a_3_4 → 0, an element of degree 3
11. b_3_6 → c_1_2·c_1_3² + c_1_2²·c_1_3, an element of degree 3
12. b_3_7 → c_1_3³ + c_1_2²·c_1_3, an element of degree 3
13. a_4_3 → 0, an element of degree 4
14. a_4_4 → 0, an element of degree 4
15. b_4_10 → c_1_3⁴ + c_1_2·c_1_3³ + c_1_1·c_1_3³ + c_1_1·c_1_2·c_1_3² + c_1_1²·c_1_3²
       + c_1_1²·c_1_2·c_1_3 + c_1_0²·c_1_2·c_1_3, an element of degree 4
16. b_4_11 → c_1_2·c_1_3³ + c_1_2²·c_1_3² + c_1_1·c_1_3³ + c_1_1·c_1_2²·c_1_3
       + c_1_1²·c_1_3² + c_1_1²·c_1_2² + c_1_0²·c_1_2·c_1_3 + c_1_0²·c_1_2², an element of degree 4
17. c_4_12 → c_1_3⁴ + c_1_2·c_1_3³ + c_1_2²·c_1_3² + c_1_1·c_1_3³ + c_1_1·c_1_2·c_1_3²
       + c_1_1²·c_1_2·c_1_3 + c_1_1⁴ + c_1_0²·c_1_2·c_1_3 + c_1_0²·c_1_2², an element of degree 4
18. c_4_13 → c_1_3⁴ + c_1_2·c_1_3³ + c_1_2²·c_1_3² + c_1_1²·c_1_3² + c_1_1⁴
       + c_1_0²·c_1_3² + c_1_0⁴, an element of degree 4
19. a_5_15 → 0, an element of degree 5
20. a_5_14 → 0, an element of degree 5
21. b_5_19 → c_1_2·c_1_3⁴ + c_1_2³·c_1_3² + c_1_1·c_1_3⁴ + c_1_1·c_1_2²·c_1_3²
       + c_1_1²·c_1_3³ + c_1_1²·c_1_2²·c_1_3 + c_1_0²·c_1_2·c_1_3²
       + c_1_0²·c_1_2²·c_1_3, an element of degree 5
22. a_6_9 → 0, an element of degree 6

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128