David J. Green

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Cohomology of group number 343 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

  ( − 2) · (t7  +  3/2·t6  +  t5  +  t3  +  t2  +  t  +  1/2)

  (t  +  1)2 · (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 9:

1. a_1_0, a nilpotent element of degree 1
2. a_1_1, a nilpotent element of degree 1
3. a_1_2, a nilpotent element of degree 1
4. b_2_4, an 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. c_4_7, a Duflot regular element of degree 4
8. a_5_7, a nilpotent element of degree 5
9. a_5_8, a nilpotent element of degree 5
10. a_5_9, a nilpotent element of degree 5
11. a_5_10, a nilpotent element of degree 5
12. a_5_11, a nilpotent element of degree 5
13. b_6_16, an element of degree 6
14. a_7_19, a nilpotent element of degree 7
15. a_7_20, a nilpotent element of degree 7
16. c_8_21, a Duflot regular element of degree 8
17. a_9_27, a nilpotent element of degree 9
18. a_9_28, a nilpotent element of degree 9

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 18:

1. a_1_0·a_1_1
2. a_1_0·a_1_2
3. a_1_0³
4. a_1_1³
5. b_2_4·a_1_1 + a_1_2³ + a_1_1·a_1_2² + a_1_1²·a_1_2
6. a_1_2⁴
7. a_1_1·a_3_5
8. a_1_2·a_3_5 + a_1_1²·a_1_2²
9. a_1_1·a_3_6
10. a_1_2·a_3_6 + b_2_4·a_1_2² + b_2_4·a_1_0² + a_1_1·a_1_2³ + a_1_1²·a_1_2²
11. b_2_4·a_1_2³
12. a_1_0²·a_3_5
13. a_1_0²·a_3_6
14. a_3_5² + c_4_7·a_1_0²
15. a_1_1·a_5_7 + c_4_7·a_1_1·a_1_2
16. a_3_6² + a_3_5² + a_1_0·a_5_7 + b_2_4·a_1_0·a_3_6 + b_2_4·a_1_0·a_3_5
       + b_2_4²·a_1_0²
17. a_1_2·a_5_7 + b_2_4²·a_1_2² + c_4_7·a_1_2²
18. a_3_6² + a_1_0·a_5_8 + b_2_4·a_1_0·a_3_6 + b_2_4·a_1_0·a_3_5
19. a_3_5² + a_1_0·a_5_9
20. a_1_1·a_5_10 + a_1_1·a_5_9 + a_1_1·a_5_8 + c_4_7·a_1_1·a_1_2
21. a_3_5² + a_1_0·a_5_10 + b_2_4·a_1_0·a_3_6
22. a_1_2·a_5_10 + a_1_2·a_5_9 + a_1_1·a_5_9 + a_1_1·a_5_8 + b_2_4²·a_1_2²
       + b_2_4²·a_1_0²
23. a_1_1·a_5_11 + a_1_1·a_5_8 + c_4_7·a_1_1·a_1_2
24. a_3_6² + a_3_5·a_3_6 + a_3_5² + a_1_0·a_5_11 + b_2_4·a_1_0·a_3_6 + b_2_4·a_1_0·a_3_5
       + b_2_4²·a_1_0²
25. a_1_2·a_5_11 + a_1_1·a_5_9 + a_1_1·a_5_8 + b_2_4·a_1_0·a_3_5
26. a_1_0²·a_5_7
27. a_1_2²·a_5_8 + a_1_1·a_1_2·a_5_9 + a_1_1²·a_5_9 + c_4_7·a_1_2³
       + c_4_7·a_1_1·a_1_2² + c_4_7·a_1_1²·a_1_2
28. b_2_4·a_5_10 + b_2_4·a_5_9 + b_2_4²·a_3_6 + a_1_2²·a_5_9 + a_1_1²·a_5_9 + a_1_1²·a_5_8
       + c_4_7·a_1_2³
29. a_1_1²·a_5_8 + a_1_0²·a_5_11 + c_4_7·a_1_1²·a_1_2
30. b_6_16·a_1_1 + a_1_1²·a_5_9 + a_1_1²·a_5_8 + c_4_7·a_1_1·a_1_2²
31. b_6_16·a_1_0 + b_2_4²·a_3_5
32. b_6_16·a_1_2 + b_2_4·a_5_8 + b_2_4·a_5_7 + b_2_4³·a_1_0 + a_1_2²·a_5_9 + a_1_2²·a_5_8
       + b_2_4·c_4_7·a_1_0 + c_4_7·a_1_2³ + c_4_7·a_1_1²·a_1_2
33. a_3_5·a_5_8 + a_3_5·a_5_7 + b_2_4²·a_1_0·a_3_5 + c_4_7·a_1_0·a_3_5
34. a_3_6·a_5_8 + a_3_6·a_5_7 + b_2_4·a_1_2·a_5_8 + b_2_4²·a_1_0·a_3_6
       + b_2_4²·a_1_0·a_3_5 + b_2_4³·a_1_2² + a_1_1·a_1_2²·a_5_9 + c_4_7·a_1_0·a_3_6
       + b_2_4·c_4_7·a_1_2² + c_4_7·a_1_1·a_1_2³
35. a_3_6·a_5_8 + a_3_6·a_5_7 + b_2_4·a_1_2·a_5_8 + b_2_4²·a_1_0·a_3_6
       + b_2_4²·a_1_0·a_3_5 + b_2_4³·a_1_2² + a_1_2³·a_5_9 + a_1_1²·a_1_2·a_5_9
       + c_4_7·a_1_0·a_3_6 + b_2_4·c_4_7·a_1_2² + c_4_7·a_1_1²·a_1_2²
36. a_3_5·a_5_9 + a_1_1²·a_1_2·a_5_9 + c_4_7·a_1_0·a_3_5
37. a_3_6·a_5_9 + b_2_4·a_1_2·a_5_9 + b_2_4·a_1_0·a_5_7 + b_2_4²·a_1_0·a_3_5
       + b_2_4³·a_1_0² + c_4_7·a_1_0·a_3_6 + c_4_7·a_1_1·a_1_2³ + c_4_7·a_1_1²·a_1_2²
38. a_3_6·a_5_10 + a_3_6·a_5_9 + b_2_4·a_1_0·a_5_7 + b_2_4²·a_1_0·a_3_6
       + b_2_4²·a_1_0·a_3_5 + b_2_4³·a_1_0² + b_2_4·c_4_7·a_1_0²
39. a_3_6·a_5_11 + a_3_6·a_5_8 + a_3_5·a_5_10 + a_3_5·a_5_7 + b_2_4²·a_1_0·a_3_6
       + b_2_4²·a_1_0·a_3_5 + b_2_4³·a_1_2² + b_2_4³·a_1_0² + a_1_1²·a_1_2·a_5_9
       + c_4_7·a_1_0·a_3_6 + b_2_4·c_4_7·a_1_2² + c_4_7·a_1_1·a_1_2³
40. a_3_5·a_5_11 + a_3_5·a_5_7 + c_4_7·a_1_0·a_3_6 + c_4_7·a_1_1²·a_1_2²
41. a_3_5·a_5_10 + b_2_4·a_1_0·a_5_11 + b_2_4·a_1_0·a_5_7 + a_1_1²·a_1_2·a_5_9
       + c_4_7·a_1_0·a_3_5
42. a_3_6·a_5_8 + a_3_6·a_5_7 + a_1_1·a_7_19 + b_2_4·a_1_2·a_5_8 + b_2_4²·a_1_0·a_3_6
       + b_2_4²·a_1_0·a_3_5 + b_2_4³·a_1_2² + c_4_7·a_1_0·a_3_6 + b_2_4·c_4_7·a_1_2²
43. a_3_5·a_5_10 + a_3_5·a_5_7 + a_1_0·a_7_19 + b_2_4·a_1_0·a_5_7 + b_2_4²·a_1_0·a_3_5
       + a_1_1²·a_1_2·a_5_9 + b_2_4·c_4_7·a_1_0² + c_4_7·a_1_1²·a_1_2²
44. a_1_2·a_7_19 + b_2_4·a_1_2·a_5_8 + b_2_4²·a_1_0·a_3_5 + c_4_7·a_1_1·a_1_2³
45. a_3_6·a_5_8 + a_3_6·a_5_7 + a_1_1·a_7_20 + b_2_4·a_1_2·a_5_8 + b_2_4²·a_1_0·a_3_6
       + b_2_4²·a_1_0·a_3_5 + b_2_4³·a_1_2² + c_4_7·a_1_0·a_3_6 + b_2_4·c_4_7·a_1_2²
       + c_4_7·a_1_1·a_1_2³
46. a_3_6·a_5_7 + a_3_5·a_5_7 + a_1_0·a_7_20 + b_2_4·a_1_0·a_5_7 + b_2_4²·a_1_0·a_3_6
       + b_2_4³·a_1_2² + c_4_7·a_1_0·a_3_5 + b_2_4·c_4_7·a_1_2² + b_2_4·c_4_7·a_1_0²
       + c_4_7·a_1_1·a_1_2³
47. a_3_6·a_5_9 + a_3_6·a_5_8 + a_3_6·a_5_7 + a_1_2·a_7_20 + b_2_4·a_1_2·a_5_8
       + b_2_4²·a_1_0·a_3_6 + b_2_4³·a_1_2² + c_4_7·a_1_1·a_1_2³
48. b_6_16·a_3_6 + b_2_4²·a_5_11 + b_2_4²·a_5_8 + b_2_4⁴·a_1_2 + b_2_4⁴·a_1_0
       + b_2_4²·c_4_7·a_1_2 + b_2_4²·c_4_7·a_1_0
49. b_6_16·a_3_5 + a_1_1²·a_1_2²·a_5_9 + b_2_4²·c_4_7·a_1_0
50. a_1_0²·a_7_19
51. a_1_0²·a_7_20 + b_2_4·a_1_2²·a_5_9
52. a_5_7·a_5_8 + a_5_7² + b_2_4²·a_1_2·a_5_8 + b_2_4²·a_1_0·a_5_7 + b_2_4⁴·a_1_2²
       + c_4_7·a_1_2·a_5_8 + c_4_7·a_1_0·a_5_7 + c_4_7²·a_1_2²
53. a_5_7·a_5_9 + b_2_4²·a_1_2·a_5_9 + c_4_7·a_1_2·a_5_9 + c_4_7·a_1_0·a_5_7
54. a_5_8·a_5_10 + a_5_8·a_5_9 + a_5_8² + a_5_7·a_5_10 + a_5_7·a_5_9 + a_5_7²
       + b_2_4²·a_1_2·a_5_8 + b_2_4³·a_1_0·a_3_6 + b_2_4³·a_1_0·a_3_5 + b_2_4⁴·a_1_2²
       + b_2_4⁴·a_1_0² + b_2_4·c_4_7·a_1_0·a_3_6 + c_4_7²·a_1_0²
55. a_5_10² + a_5_9² + a_5_8² + a_5_7² + b_2_4²·a_1_0·a_5_7 + b_2_4³·a_1_0·a_3_6
       + b_2_4³·a_1_0·a_3_5 + b_2_4²·c_4_7·a_1_2² + b_2_4²·c_4_7·a_1_0²
       + c_4_7²·a_1_0²
56. a_5_11² + a_5_8² + b_2_4⁴·a_1_2² + b_2_4⁴·a_1_0² + c_4_7·a_1_0·a_5_7
       + b_2_4·c_4_7·a_1_0·a_3_6 + b_2_4·c_4_7·a_1_0·a_3_5 + b_2_4²·c_4_7·a_1_0²
       + c_4_7²·a_1_2²
57. a_5_10·a_5_11 + a_5_9·a_5_10 + a_5_9² + a_5_8·a_5_11 + a_5_7·a_5_11 + a_5_7·a_5_10
       + a_5_7·a_5_9 + b_2_4²·a_1_2·a_5_9 + b_2_4²·a_1_2·a_5_8 + c_4_7·a_1_1·a_5_9
       + c_4_7·a_1_1·a_5_8 + b_2_4·c_4_7·a_1_0·a_3_6 + b_2_4·c_4_7·a_1_0·a_3_5
58. a_5_8·a_5_11 + a_5_8² + a_5_7·a_5_11 + a_5_7² + b_2_4²·a_1_0·a_5_11
       + b_2_4⁴·a_1_0² + c_4_7·a_1_0·a_5_11 + b_2_4²·c_4_7·a_1_2²
       + b_2_4²·c_4_7·a_1_0² + c_4_7²·a_1_0²
59. a_5_9·a_5_11 + a_5_9·a_5_10 + a_5_9² + a_5_7·a_5_11 + a_5_7·a_5_10 + a_5_7·a_5_9
       + a_5_7² + a_3_6·a_7_19 + b_2_4²·a_1_2·a_5_9 + b_2_4²·a_1_0·a_5_7 + b_2_4⁴·a_1_2²
       + b_2_4⁴·a_1_0² + c_4_7·a_1_0·a_5_7 + c_4_7²·a_1_2²
60. a_5_9·a_5_11 + a_5_9·a_5_10 + a_5_9² + a_5_8·a_5_11 + a_5_8² + a_5_7·a_5_11 + a_5_7²
       + a_3_5·a_7_19 + b_2_4²·a_1_2·a_5_9 + b_2_4²·a_1_0·a_5_11 + b_2_4²·a_1_0·a_5_7
       + c_4_7·a_1_0·a_5_7 + b_2_4·c_4_7·a_1_0·a_3_5 + b_2_4²·c_4_7·a_1_2²
       + b_2_4²·c_4_7·a_1_0²
61. a_5_9·a_5_11 + a_5_9·a_5_10 + a_5_9² + a_5_8·a_5_11 + a_5_8² + a_5_7·a_5_11 + a_5_7²
       + b_2_4·a_1_0·a_7_19 + b_2_4²·a_1_2·a_5_9 + b_2_4²·a_1_0·a_5_7 + b_2_4³·a_1_0·a_3_5
       + b_2_4·c_4_7·a_1_0·a_3_6 + b_2_4·c_4_7·a_1_0·a_3_5 + b_2_4²·c_4_7·a_1_2²
       + b_2_4²·c_4_7·a_1_0² + c_4_7²·a_1_0²
62. a_5_9·a_5_11 + a_5_9·a_5_10 + a_5_9² + a_5_7·a_5_11 + a_3_6·a_7_20 + b_2_4²·a_1_2·a_5_9
       + b_2_4²·a_1_2·a_5_8 + b_2_4²·a_1_0·a_5_7 + b_2_4³·a_1_0·a_3_5 + b_2_4⁴·a_1_2²
       + c_4_7·a_1_1·a_5_9 + c_4_7·a_1_1·a_5_8 + b_2_4·c_4_7·a_1_0·a_3_6
       + b_2_4·c_4_7·a_1_0·a_3_5 + b_2_4²·c_4_7·a_1_0²
63. a_5_9·a_5_11 + a_5_9·a_5_10 + a_5_9² + a_5_8·a_5_11 + a_5_8² + a_3_5·a_7_20
       + b_2_4²·a_1_2·a_5_9 + b_2_4⁴·a_1_2² + c_4_7·a_1_1·a_5_9 + c_4_7·a_1_1·a_5_8
       + c_4_7·a_1_0·a_5_7 + b_2_4·c_4_7·a_1_0·a_3_6 + b_2_4·c_4_7·a_1_0·a_3_5
       + b_2_4²·c_4_7·a_1_2² + c_4_7²·a_1_2²
64. a_5_9·a_5_11 + a_5_9·a_5_10 + a_5_9² + a_5_8·a_5_11 + a_5_8² + a_5_7·a_5_11
       + a_5_7·a_5_10 + a_5_7·a_5_9 + a_5_7² + b_2_4·a_1_0·a_7_20 + b_2_4²·a_1_2·a_5_9
       + b_2_4²·a_1_0·a_5_11 + b_2_4³·a_1_0·a_3_6 + b_2_4⁴·a_1_2² + c_4_7·a_1_1·a_5_9
       + c_4_7·a_1_1·a_5_8 + b_2_4·c_4_7·a_1_0·a_3_6 + b_2_4·c_4_7·a_1_0·a_3_5
       + b_2_4²·c_4_7·a_1_0² + c_4_7²·a_1_0²
65. a_5_8² + a_5_7² + b_2_4⁴·a_1_0² + c_8_21·a_1_1² + b_2_4²·c_4_7·a_1_2²
       + c_4_7²·a_1_0²
66. a_5_7² + b_2_4⁴·a_1_2² + c_8_21·a_1_0² + c_4_7²·a_1_2² + c_4_7²·a_1_0²
67. a_5_9·a_5_10 + a_5_9² + a_5_8² + a_5_7² + b_2_4²·a_1_2·a_5_9 + b_2_4²·a_1_0·a_5_7
       + b_2_4³·a_1_0·a_3_5 + c_8_21·a_1_1·a_1_2 + c_4_7·a_1_1·a_5_9 + c_4_7·a_1_1·a_5_8
       + b_2_4·c_4_7·a_1_0·a_3_6 + b_2_4²·c_4_7·a_1_2² + c_4_7²·a_1_0²
68. a_5_9² + a_5_8² + a_5_7² + b_2_4⁴·a_1_2² + b_2_4⁴·a_1_0² + c_8_21·a_1_2²
       + c_4_7²·a_1_2²
69. a_1_1·a_9_27 + c_4_7·a_1_1·a_5_8 + c_4_7²·a_1_1·a_1_2 + c_4_7²·a_1_1²
70. a_5_7·a_5_10 + a_5_7·a_5_9 + a_5_7² + a_1_0·a_9_27 + b_2_4²·a_1_0·a_5_11
       + b_2_4²·a_1_0·a_5_7 + b_2_4³·a_1_0·a_3_5 + c_4_7·a_1_1·a_5_9 + c_4_7·a_1_1·a_5_8
       + b_2_4·c_4_7·a_1_0·a_3_6 + b_2_4·c_4_7·a_1_0·a_3_5 + b_2_4²·c_4_7·a_1_2²
       + c_4_7²·a_1_2² + c_4_7²·a_1_0²
71. a_5_9·a_5_10 + a_5_9² + a_5_8·a_5_9 + a_5_7·a_5_9 + a_5_7·a_5_8 + a_5_7² + a_1_2·a_9_27
       + b_2_4²·a_1_2·a_5_9 + b_2_4²·a_1_2·a_5_8 + b_2_4²·a_1_0·a_5_7 + b_2_4⁴·a_1_2²
       + b_2_4⁴·a_1_0² + c_4_7·a_1_0·a_5_7 + b_2_4·c_4_7·a_1_0·a_3_6
       + b_2_4²·c_4_7·a_1_2² + c_4_7²·a_1_1·a_1_2 + c_4_7²·a_1_0²
72. a_5_9·a_5_10 + a_5_9² + a_1_1·a_9_28 + b_2_4²·a_1_2·a_5_9 + b_2_4²·a_1_0·a_5_7
       + b_2_4³·a_1_0·a_3_5 + b_2_4⁴·a_1_0² + c_4_7·a_1_1·a_5_9 + c_4_7·a_1_1·a_5_8
       + b_2_4·c_4_7·a_1_0·a_3_6 + c_4_7²·a_1_1·a_1_2 + c_4_7²·a_1_1²
73. a_5_9·a_5_11 + a_5_9·a_5_10 + a_5_9² + a_5_7·a_5_11 + a_5_7·a_5_10 + a_5_7·a_5_9
       + a_5_7² + a_1_0·a_9_28 + b_2_4²·a_1_2·a_5_9 + b_2_4²·a_1_0·a_5_7
       + b_2_4·c_4_7·a_1_0·a_3_5 + b_2_4²·c_4_7·a_1_2² + c_4_7²·a_1_2²
       + c_4_7²·a_1_0²
74. a_5_9·a_5_11 + a_5_9² + a_5_8·a_5_11 + a_5_8² + a_5_7·a_5_11 + a_5_7·a_5_8
       + a_1_2·a_9_28 + b_2_4²·a_1_2·a_5_9 + b_2_4²·a_1_2·a_5_8 + b_2_4²·a_1_0·a_5_11
       + b_2_4³·a_1_0·a_3_5 + b_2_4⁴·a_1_2² + b_2_4⁴·a_1_0² + c_4_7·a_1_0·a_5_7
       + b_2_4·c_4_7·a_1_0·a_3_5 + b_2_4²·c_4_7·a_1_2² + b_2_4²·c_4_7·a_1_0²
       + c_4_7²·a_1_1·a_1_2
75. b_6_16·a_5_11 + b_6_16·a_5_8 + b_2_4³·a_5_8 + b_2_4³·a_5_7 + b_2_4⁴·a_3_5
       + b_2_4⁵·a_1_0 + b_2_4²·a_1_2²·a_5_9 + b_2_4·c_4_7·a_5_8 + b_2_4·c_4_7·a_5_7
       + b_2_4²·c_4_7·a_3_6 + b_2_4²·c_4_7·a_3_5 + c_4_7·a_1_2²·a_5_9
       + c_4_7·a_1_1·a_1_2·a_5_9 + c_4_7·a_1_1²·a_5_9 + b_2_4·c_4_7²·a_1_0
       + c_4_7²·a_1_1·a_1_2²
76. b_6_16·a_5_10 + b_6_16·a_5_9 + b_6_16·a_5_8 + b_6_16·a_5_7 + b_2_4³·a_5_11
       + b_2_4³·a_5_8 + b_2_4⁴·a_3_5 + b_2_4⁵·a_1_2 + b_2_4⁵·a_1_0 + b_2_4²·c_4_7·a_3_5
       + b_2_4³·c_4_7·a_1_0 + c_4_7·a_1_0²·a_5_11
77. b_6_16·a_5_7 + b_2_4²·a_7_19 + b_2_4³·a_5_11 + b_2_4⁴·a_3_5 + b_2_4⁵·a_1_2
       + b_2_4²·a_1_2²·a_5_9 + b_2_4·c_4_7·a_5_8 + b_2_4·c_4_7·a_5_7 + b_2_4²·c_4_7·a_3_5
       + b_2_4³·c_4_7·a_1_2 + c_4_7·a_1_2²·a_5_9 + c_4_7·a_1_1·a_1_2·a_5_9
       + c_4_7·a_1_1²·a_5_9 + b_2_4·c_4_7²·a_1_0 + c_4_7²·a_1_1·a_1_2²
78. b_6_16·a_5_9 + b_6_16·a_5_7 + b_2_4·a_9_27 + b_2_4²·a_7_20 + b_2_4³·a_5_11
       + b_2_4³·a_5_9 + b_2_4³·a_5_8 + b_2_4³·a_5_7 + b_2_4⁴·a_3_6 + b_2_4⁴·a_3_5
       + b_2_4⁵·a_1_2 + b_2_4⁵·a_1_0 + b_2_4²·a_1_2²·a_5_9 + b_2_4·c_8_21·a_1_0
       + b_2_4²·c_4_7·a_3_6 + b_2_4²·c_4_7·a_3_5 + b_2_4³·c_4_7·a_1_0
       + c_8_21·a_1_1·a_1_2² + c_4_7·a_1_0²·a_5_11 + c_4_7²·a_1_2³
       + c_4_7²·a_1_1·a_1_2² + c_4_7²·a_1_1²·a_1_2
79. b_6_16·a_5_8 + b_6_16·a_5_7 + b_2_4⁴·a_3_5 + a_1_2²·a_9_27 + b_2_4²·c_4_7·a_3_5
       + b_2_4³·c_4_7·a_1_2 + c_8_21·a_1_1²·a_1_2 + c_4_7²·a_1_1·a_1_2²
80. b_6_16·a_5_8 + b_6_16·a_5_7 + b_2_4⁴·a_3_5 + a_1_0²·a_9_28 + b_2_4²·a_1_2²·a_5_9
       + b_2_4²·c_4_7·a_3_5 + b_2_4³·c_4_7·a_1_2 + c_8_21·a_1_1²·a_1_2
       + c_4_7·a_1_1·a_1_2·a_5_9 + c_4_7·a_1_1²·a_5_9 + c_4_7·a_1_0²·a_5_11
       + c_4_7²·a_1_1·a_1_2² + c_4_7²·a_1_1²·a_1_2
81. a_5_11·a_7_19 + a_5_10·a_7_19 + a_5_8·a_7_20 + a_5_8·a_7_19 + a_5_7·a_7_20
       + b_2_4²·a_1_0·a_7_19 + b_2_4³·a_1_2·a_5_9 + b_2_4³·a_1_0·a_5_7
       + b_2_4⁴·a_1_0·a_3_6 + b_2_4⁴·a_1_0·a_3_5 + c_4_7·a_1_0·a_7_19
       + b_2_4·c_4_7·a_1_2·a_5_9 + b_2_4·c_4_7·a_1_2·a_5_8 + b_2_4·c_4_7·a_1_0·a_5_11
       + b_2_4³·c_4_7·a_1_0² + c_4_7²·a_1_0·a_3_6
82. a_5_11·a_7_20 + a_5_11·a_7_19 + a_5_10·a_7_20 + a_5_10·a_7_19 + a_5_9·a_7_20
       + a_5_7·a_7_20 + b_2_4²·a_1_0·a_7_20 + b_2_4³·a_1_2·a_5_8 + b_2_4⁴·a_1_0·a_3_6
       + b_2_4⁴·a_1_0·a_3_5 + c_4_7·a_1_0·a_7_19 + b_2_4·c_4_7·a_1_0·a_5_11
       + b_2_4²·c_4_7·a_1_0·a_3_6 + c_4_7·a_1_1²·a_1_2·a_5_9 + b_2_4·c_4_7²·a_1_2²
       + b_2_4·c_4_7²·a_1_0² + c_4_7²·a_1_1·a_1_2³
83. a_5_11·a_7_19 + a_5_10·a_7_19 + a_5_9·a_7_19 + a_5_7·a_7_19 + b_2_4²·a_1_0·a_7_20
       + b_2_4³·a_1_2·a_5_8 + b_2_4³·a_1_0·a_5_7 + b_2_4⁴·a_1_0·a_3_6 + b_2_4⁴·a_1_0·a_3_5
       + b_2_4⁵·a_1_2² + c_4_7·a_1_0·a_7_20 + c_4_7·a_1_0·a_7_19 + b_2_4·c_4_7·a_1_2·a_5_8
       + b_2_4·c_4_7·a_1_0·a_5_11 + b_2_4³·c_4_7·a_1_0² + c_4_7²·a_1_0·a_3_6
       + c_4_7²·a_1_1·a_1_2³
84. b_6_16² + b_2_4⁴·c_4_7 + c_8_21·a_1_1²·a_1_2²
85. b_6_16² + a_5_8·a_7_19 + a_5_7·a_7_19 + b_2_4²·a_1_0·a_7_19 + b_2_4³·a_1_2·a_5_8
       + b_2_4⁵·a_1_2² + b_2_4⁴·c_4_7 + c_4_7·a_1_0·a_7_19 + b_2_4·c_4_7·a_1_2·a_5_8
       + b_2_4³·c_4_7·a_1_2² + b_2_4³·c_4_7·a_1_0² + c_8_21·a_1_1·a_1_2³
       + c_4_7·a_1_1·a_1_2²·a_5_9 + b_2_4·c_4_7²·a_1_2² + c_4_7²·a_1_1·a_1_2³
86. a_5_10·a_7_19 + a_5_9·a_7_19 + a_5_8·a_7_19 + a_5_7·a_7_20 + b_2_4²·a_1_0·a_7_19
       + b_2_4³·a_1_2·a_5_9 + c_8_21·a_1_0·a_3_6 + c_4_7·a_1_0·a_7_19
       + b_2_4·c_4_7·a_1_2·a_5_9 + b_2_4·c_4_7·a_1_0·a_5_11 + b_2_4·c_4_7·a_1_0·a_5_7
       + b_2_4²·c_4_7·a_1_0·a_3_6 + b_2_4³·c_4_7·a_1_2² + c_4_7²·a_1_0·a_3_6
       + c_4_7²·a_1_1·a_1_2³ + c_4_7²·a_1_1²·a_1_2²
87. a_5_11·a_7_20 + a_5_11·a_7_19 + a_5_10·a_7_19 + a_5_8·a_7_19 + a_5_7·a_7_20
       + b_2_4²·a_1_0·a_7_20 + b_2_4²·a_1_0·a_7_19 + b_2_4⁴·a_1_0·a_3_6 + b_2_4⁵·a_1_0²
       + c_8_21·a_1_0·a_3_5 + c_4_7·a_1_0·a_7_19 + b_2_4·c_4_7·a_1_0·a_5_11
       + b_2_4·c_4_7·a_1_0·a_5_7 + b_2_4²·c_4_7·a_1_0·a_3_6 + b_2_4³·c_4_7·a_1_2²
       + b_2_4³·c_4_7·a_1_0² + c_4_7²·a_1_1²·a_1_2²
88. a_5_11·a_7_20 + a_5_11·a_7_19 + a_5_9·a_7_19 + a_5_7·a_7_20 + b_2_4²·a_1_0·a_7_19
       + b_2_4³·a_1_0·a_5_11 + b_2_4³·a_1_0·a_5_7 + b_2_4⁴·a_1_0·a_3_5 + b_2_4⁵·a_1_0²
       + c_4_7·a_1_0·a_7_19 + b_2_4·c_8_21·a_1_0² + b_2_4·c_4_7·a_1_0·a_5_11
       + b_2_4·c_4_7·a_1_0·a_5_7 + b_2_4²·c_4_7·a_1_0·a_3_5 + b_2_4³·c_4_7·a_1_2²
       + b_2_4³·c_4_7·a_1_0² + c_4_7·a_1_1·a_1_2²·a_5_9 + b_2_4·c_4_7²·a_1_2²
       + c_4_7²·a_1_1²·a_1_2²
89. a_5_11·a_7_20 + a_5_10·a_7_19 + a_5_9·a_7_20 + a_5_8·a_7_19 + a_5_7·a_7_20
       + b_2_4²·a_1_0·a_7_20 + b_2_4²·a_1_0·a_7_19 + b_2_4⁴·a_1_0·a_3_6 + b_2_4⁵·a_1_2²
       + c_4_7·a_1_0·a_7_19 + b_2_4·c_8_21·a_1_2² + b_2_4·c_4_7·a_1_2·a_5_9
       + b_2_4·c_4_7·a_1_0·a_5_7 + b_2_4³·c_4_7·a_1_2² + c_4_7²·a_1_0·a_3_6
       + b_2_4·c_4_7²·a_1_2² + b_2_4·c_4_7²·a_1_0²
90. a_5_11·a_7_20 + a_5_11·a_7_19 + a_5_10·a_7_19 + a_5_9·a_7_19 + a_5_8·a_7_19 + a_3_6·a_9_27
       + b_2_4²·a_1_0·a_7_20 + b_2_4²·a_1_0·a_7_19 + b_2_4³·a_1_2·a_5_9 + b_2_4⁵·a_1_2²
       + c_4_7·a_1_0·a_7_19 + b_2_4·c_4_7·a_1_2·a_5_8 + b_2_4·c_4_7·a_1_0·a_5_11
       + b_2_4·c_4_7·a_1_0·a_5_7 + b_2_4²·c_4_7·a_1_0·a_3_6 + b_2_4³·c_4_7·a_1_0²
       + c_4_7·a_1_1·a_1_2²·a_5_9 + c_4_7·a_1_1²·a_1_2·a_5_9 + c_4_7²·a_1_0·a_3_6
       + c_4_7²·a_1_1²·a_1_2²
91. a_5_11·a_7_20 + a_5_11·a_7_19 + a_5_9·a_7_19 + a_5_7·a_7_20 + a_5_7·a_7_19 + a_3_5·a_9_27
       + b_2_4³·a_1_2·a_5_8 + b_2_4³·a_1_0·a_5_7 + b_2_4⁵·a_1_0²
       + b_2_4·c_4_7·a_1_2·a_5_8 + b_2_4·c_4_7·a_1_0·a_5_11 + b_2_4·c_4_7·a_1_0·a_5_7
       + b_2_4²·c_4_7·a_1_0·a_3_6 + b_2_4²·c_4_7·a_1_0·a_3_5 + b_2_4³·c_4_7·a_1_2²
       + b_2_4³·c_4_7·a_1_0² + c_4_7·a_1_1·a_1_2²·a_5_9 + b_2_4·c_4_7²·a_1_2²
       + b_2_4·c_4_7²·a_1_0² + c_4_7²·a_1_1·a_1_2³ + c_4_7²·a_1_1²·a_1_2²
92. a_5_9·a_7_19 + b_2_4·a_1_2·a_9_27 + b_2_4²·a_1_0·a_7_19 + b_2_4³·a_1_2·a_5_9
       + b_2_4³·a_1_0·a_5_11 + b_2_4³·a_1_0·a_5_7 + b_2_4⁴·a_1_0·a_3_5 + c_4_7·a_1_0·a_7_19
       + b_2_4·c_4_7·a_1_2·a_5_9 + b_2_4·c_4_7·a_1_2·a_5_8 + b_2_4²·c_4_7·a_1_0·a_3_5
       + b_2_4³·c_4_7·a_1_2² + b_2_4³·c_4_7·a_1_0² + c_4_7·a_1_1·a_1_2²·a_5_9
       + c_4_7·a_1_1²·a_1_2·a_5_9 + b_2_4·c_4_7²·a_1_2² + c_4_7²·a_1_1·a_1_2³
93. a_5_10·a_7_19 + a_5_9·a_7_20 + a_5_8·a_7_19 + a_3_6·a_9_28 + b_2_4³·a_1_2·a_5_9
       + b_2_4⁵·a_1_2² + b_2_4·c_4_7·a_1_0·a_5_7 + b_2_4²·c_4_7·a_1_0·a_3_6
       + b_2_4²·c_4_7·a_1_0·a_3_5 + b_2_4³·c_4_7·a_1_2² + c_4_7·a_1_1²·a_1_2·a_5_9
       + c_4_7²·a_1_0·a_3_6 + b_2_4·c_4_7²·a_1_2² + c_4_7²·a_1_1·a_1_2³
       + c_4_7²·a_1_1²·a_1_2²
94. b_6_16² + a_5_11·a_7_19 + a_5_8·a_7_19 + a_3_5·a_9_28 + b_2_4²·a_1_0·a_7_19
       + b_2_4⁵·a_1_2² + b_2_4⁴·c_4_7 + b_2_4·c_4_7·a_1_0·a_5_11
       + c_4_7·a_1_1·a_1_2²·a_5_9 + b_2_4·c_4_7²·a_1_2² + c_4_7²·a_1_1²·a_1_2²
95. a_5_10·a_7_19 + a_5_9·a_7_19 + a_5_8·a_7_19 + a_5_7·a_7_19 + b_2_4·a_1_0·a_9_28
       + b_2_4²·a_1_0·a_7_19 + b_2_4³·a_1_2·a_5_8 + b_2_4⁵·a_1_0² + c_4_7·a_1_0·a_7_19
       + b_2_4·c_4_7·a_1_2·a_5_8 + b_2_4·c_4_7·a_1_0·a_5_7 + b_2_4²·c_4_7·a_1_0·a_3_5
       + b_2_4³·c_4_7·a_1_0² + c_4_7·a_1_1·a_1_2²·a_5_9 + b_2_4·c_4_7²·a_1_2²
       + b_2_4·c_4_7²·a_1_0² + c_4_7²·a_1_1·a_1_2³
96. b_6_16·a_7_19 + b_2_4³·a_7_19 + b_2_4⁴·a_5_11 + b_2_4⁵·a_3_5 + b_2_4⁶·a_1_2
       + b_2_4³·a_1_2²·a_5_9 + b_2_4²·c_4_7·a_5_8 + b_2_4³·c_4_7·a_3_6
       + b_2_4⁴·c_4_7·a_1_0 + b_2_4·c_4_7·a_1_2²·a_5_9 + b_2_4²·c_4_7²·a_1_2
97. b_6_16·a_7_20 + b_2_4²·a_9_28 + b_2_4²·a_9_27 + b_2_4⁴·a_5_9 + b_2_4⁵·a_3_5
       + b_2_4⁶·a_1_2 + b_2_4⁶·a_1_0 + b_2_4²·c_8_21·a_1_2 + b_2_4²·c_8_21·a_1_0
       + b_2_4²·c_4_7·a_5_11 + b_2_4²·c_4_7·a_5_8 + b_2_4³·c_4_7·a_3_5
       + b_2_4⁴·c_4_7·a_1_2 + b_2_4⁴·c_4_7·a_1_0 + b_2_4·c_4_7·a_1_2²·a_5_9
       + c_4_7·a_1_1²·a_1_2²·a_5_9 + b_2_4²·c_4_7²·a_1_0
98. a_7_20² + b_2_4⁴·a_1_0·a_5_7 + b_2_4⁵·a_1_0·a_3_6 + b_2_4⁵·a_1_0·a_3_5
       + c_8_21·a_1_0·a_5_7 + b_2_4·c_8_21·a_1_0·a_3_6 + b_2_4·c_8_21·a_1_0·a_3_5
       + b_2_4⁴·c_4_7·a_1_2² + b_2_4⁴·c_4_7·a_1_0² + c_4_7²·a_1_0·a_5_7
       + b_2_4·c_4_7²·a_1_0·a_3_6 + b_2_4·c_4_7²·a_1_0·a_3_5 + b_2_4²·c_4_7²·a_1_2²
       + c_4_7³·a_1_0²
99. a_7_19² + b_2_4⁶·a_1_2² + b_2_4²·c_8_21·a_1_0² + b_2_4²·c_4_7·a_1_0·a_5_7
       + b_2_4³·c_4_7·a_1_0·a_3_6 + b_2_4³·c_4_7·a_1_0·a_3_5 + c_4_7·c_8_21·a_1_0²
       + b_2_4²·c_4_7²·a_1_2² + b_2_4²·c_4_7²·a_1_0²
100. a_7_19² + a_5_8·a_9_27 + a_5_7·a_9_27 + b_2_4³·a_1_0·a_7_20 + b_2_4⁴·a_1_0·a_5_11
        + b_2_4⁵·a_1_0·a_3_6 + b_2_4⁵·a_1_0·a_3_5 + b_2_4⁶·a_1_2²
        + b_2_4·c_4_7·a_1_0·a_7_20 + b_2_4·c_4_7·a_1_0·a_7_19 + b_2_4²·c_4_7·a_1_2·a_5_9
        + b_2_4²·c_4_7·a_1_0·a_5_7 + b_2_4³·c_4_7·a_1_0·a_3_6 + c_4_7·c_8_21·a_1_1²
        + c_4_7²·a_1_1·a_5_8 + b_2_4·c_4_7²·a_1_0·a_3_6 + b_2_4·c_4_7²·a_1_0·a_3_5
        + b_2_4²·c_4_7²·a_1_0² + c_4_7³·a_1_1·a_1_2
101. a_7_20² + a_5_11·a_9_27 + b_2_4²·a_1_2·a_9_27 + b_2_4⁴·a_1_2·a_5_8
        + b_2_4⁴·a_1_0·a_5_11 + b_2_4⁵·a_1_0·a_3_6 + b_2_4⁶·a_1_2² + c_8_21·a_1_0·a_5_11
        + c_8_21·a_1_0·a_5_7 + b_2_4·c_4_7·a_1_0·a_7_20 + b_2_4²·c_4_7·a_1_0·a_5_7
        + b_2_4³·c_4_7·a_1_0·a_3_6 + b_2_4³·c_4_7·a_1_0·a_3_5 + b_2_4⁴·c_4_7·a_1_0²
        + c_4_7·c_8_21·a_1_1² + c_4_7²·a_1_1·a_5_8 + c_4_7²·a_1_0·a_5_7
        + b_2_4·c_4_7²·a_1_0·a_3_6 + b_2_4²·c_4_7²·a_1_2² + b_2_4²·c_4_7²·a_1_0²
        + c_4_7³·a_1_1·a_1_2 + c_4_7³·a_1_0²
102. a_5_10·a_9_27 + a_5_9·a_9_27 + b_2_4²·a_1_2·a_9_27 + b_2_4³·a_1_0·a_7_20
        + b_2_4⁴·a_1_2·a_5_9 + b_2_4⁶·a_1_2² + b_2_4⁶·a_1_0² + b_2_4·c_8_21·a_1_0·a_3_6
        + b_2_4²·c_8_21·a_1_0² + b_2_4²·c_4_7·a_1_0·a_5_11 + b_2_4²·c_4_7·a_1_0·a_5_7
        + b_2_4⁴·c_4_7·a_1_2² + b_2_4⁴·c_4_7·a_1_0² + c_4_7·c_8_21·a_1_1²
        + c_4_7²·a_1_1·a_5_8 + c_4_7³·a_1_1·a_1_2
103. a_7_19² + a_5_9·a_9_27 + b_2_4⁴·a_1_2·a_5_8 + b_2_4⁴·a_1_0·a_5_7
        + b_2_4⁵·a_1_0·a_3_5 + c_8_21·a_1_2·a_5_8 + c_8_21·a_1_1·a_5_9 + c_8_21·a_1_1·a_5_8
        + c_4_7·a_1_2·a_9_27 + b_2_4·c_4_7·a_1_0·a_7_20 + b_2_4·c_4_7·a_1_0·a_7_19
        + b_2_4²·c_8_21·a_1_2² + b_2_4²·c_4_7·a_1_2·a_5_8 + b_2_4²·c_4_7·a_1_0·a_5_7
        + b_2_4³·c_4_7·a_1_0·a_3_5 + c_4_7·c_8_21·a_1_2² + c_4_7·c_8_21·a_1_1·a_1_2
        + c_4_7·c_8_21·a_1_1² + c_4_7²·a_1_1·a_5_9 + b_2_4·c_4_7²·a_1_0·a_3_6
        + b_2_4·c_4_7²·a_1_0·a_3_5 + b_2_4²·c_4_7²·a_1_0² + c_4_7³·a_1_1·a_1_2
104. a_7_19·a_7_20 + a_5_7·a_9_28 + b_2_4³·a_1_0·a_7_20 + b_2_4³·a_1_0·a_7_19
        + b_2_4⁴·a_1_0·a_5_7 + b_2_4⁵·a_1_0·a_3_6 + b_2_4⁵·a_1_0·a_3_5 + b_2_4⁶·a_1_2²
        + b_2_4·c_4_7·a_1_0·a_7_20 + b_2_4²·c_8_21·a_1_2² + b_2_4²·c_8_21·a_1_0²
        + b_2_4²·c_4_7·a_1_0·a_5_11 + b_2_4²·c_4_7·a_1_0·a_5_7 + b_2_4⁴·c_4_7·a_1_2²
        + b_2_4⁴·c_4_7·a_1_0² + c_4_7·c_8_21·a_1_2² + c_4_7·c_8_21·a_1_1·a_1_2
        + c_4_7²·a_1_2·a_5_8 + c_4_7²·a_1_1·a_5_9 + c_4_7²·a_1_1·a_5_8 + c_4_7²·a_1_0·a_5_7
        + b_2_4·c_4_7²·a_1_0·a_3_6 + b_2_4·c_4_7²·a_1_0·a_3_5 + b_2_4²·c_4_7²·a_1_2²
        + b_2_4²·c_4_7²·a_1_0² + c_4_7³·a_1_1·a_1_2 + c_4_7³·a_1_0²
105. a_7_19·a_7_20 + a_5_9·a_9_28 + a_5_9·a_9_27 + b_2_4³·a_1_0·a_7_20
        + b_2_4³·a_1_0·a_7_19 + b_2_4⁴·a_1_2·a_5_8 + b_2_4⁵·a_1_0·a_3_6 + c_8_21·a_1_2·a_5_9
        + c_8_21·a_1_2·a_5_8 + c_8_21·a_1_1·a_5_8 + c_8_21·a_1_0·a_5_11 + c_8_21·a_1_0·a_5_7
        + b_2_4·c_8_21·a_1_0·a_3_6 + b_2_4²·c_8_21·a_1_0² + b_2_4²·c_4_7·a_1_2·a_5_9
        + b_2_4²·c_4_7·a_1_0·a_5_7 + b_2_4³·c_4_7·a_1_0·a_3_6 + b_2_4⁴·c_4_7·a_1_0²
        + c_4_7·c_8_21·a_1_2² + c_4_7·c_8_21·a_1_1·a_1_2 + c_4_7·c_8_21·a_1_1²
        + c_4_7²·a_1_2·a_5_9 + c_4_7²·a_1_2·a_5_8 + c_4_7²·a_1_0·a_5_7
        + b_2_4²·c_4_7²·a_1_2² + c_4_7³·a_1_2² + c_4_7³·a_1_0²
106. a_7_20² + a_5_9·a_9_27 + a_5_8·a_9_28 + a_5_7·a_9_27 + b_2_4³·a_1_0·a_7_20
        + b_2_4⁴·a_1_2·a_5_9 + b_2_4⁶·a_1_2² + b_2_4⁶·a_1_0² + c_8_21·a_1_1·a_5_9
        + c_8_21·a_1_0·a_5_11 + c_8_21·a_1_0·a_5_7 + b_2_4·c_8_21·a_1_0·a_3_6
        + b_2_4·c_4_7·a_1_0·a_7_19 + b_2_4²·c_8_21·a_1_2² + b_2_4²·c_8_21·a_1_0²
        + b_2_4²·c_4_7·a_1_2·a_5_9 + b_2_4²·c_4_7·a_1_2·a_5_8 + b_2_4²·c_4_7·a_1_0·a_5_11
        + b_2_4³·c_4_7·a_1_0·a_3_6 + b_2_4⁴·c_4_7·a_1_2² + c_4_7·c_8_21·a_1_2²
        + c_4_7·c_8_21·a_1_1·a_1_2 + c_4_7·c_8_21·a_1_1² + c_4_7²·a_1_0·a_5_7
        + b_2_4·c_4_7²·a_1_0·a_3_6 + b_2_4·c_4_7²·a_1_0·a_3_5 + b_2_4²·c_4_7²·a_1_0²
        + c_4_7³·a_1_2² + c_4_7³·a_1_1·a_1_2 + c_4_7³·a_1_0²
107. a_7_19² + a_5_9·a_9_27 + a_5_7·a_9_27 + b_2_4²·a_1_2·a_9_27 + b_2_4²·a_1_0·a_9_28
        + b_2_4³·a_1_0·a_7_20 + b_2_4³·a_1_0·a_7_19 + b_2_4⁴·a_1_2·a_5_8
        + b_2_4⁴·a_1_0·a_5_11 + b_2_4⁴·a_1_0·a_5_7 + b_2_4⁵·a_1_0·a_3_6
        + b_2_4⁵·a_1_0·a_3_5 + b_2_4⁶·a_1_0² + c_8_21·a_1_2·a_5_8 + c_8_21·a_1_1·a_5_9
        + c_8_21·a_1_1·a_5_8 + c_8_21·a_1_0·a_5_7 + b_2_4·c_8_21·a_1_0·a_3_6
        + b_2_4·c_4_7·a_1_0·a_7_19 + b_2_4²·c_8_21·a_1_2² + b_2_4²·c_4_7·a_1_2·a_5_8
        + b_2_4²·c_4_7·a_1_0·a_5_11 + b_2_4²·c_4_7·a_1_0·a_5_7 + b_2_4³·c_4_7·a_1_0·a_3_5
        + c_4_7·c_8_21·a_1_2² + c_4_7·c_8_21·a_1_1·a_1_2 + c_4_7·c_8_21·a_1_1²
        + c_4_7²·a_1_1·a_5_9 + c_4_7³·a_1_1·a_1_2
108. a_7_20² + a_7_19² + a_5_11·a_9_28 + a_5_9·a_9_27 + a_5_7·a_9_27 + b_2_4⁵·a_1_0·a_3_6
        + b_2_4⁵·a_1_0·a_3_5 + b_2_4⁶·a_1_2² + c_8_21·a_1_2·a_5_8 + c_8_21·a_1_1·a_5_8
        + c_8_21·a_1_0·a_5_11 + c_8_21·a_1_0·a_5_7 + b_2_4·c_8_21·a_1_0·a_3_6
        + b_2_4·c_4_7·a_1_0·a_7_20 + b_2_4·c_4_7·a_1_0·a_7_19 + b_2_4²·c_8_21·a_1_2²
        + b_2_4²·c_4_7·a_1_2·a_5_9 + b_2_4²·c_4_7·a_1_0·a_5_11 + b_2_4²·c_4_7·a_1_0·a_5_7
        + b_2_4³·c_4_7·a_1_0·a_3_6 + b_2_4⁴·c_4_7·a_1_2² + b_2_4⁴·c_4_7·a_1_0²
        + c_4_7·c_8_21·a_1_2² + c_4_7·c_8_21·a_1_1² + b_2_4·c_4_7²·a_1_0·a_3_6
        + b_2_4·c_4_7²·a_1_0·a_3_5 + b_2_4²·c_4_7²·a_1_2²
109. a_7_20² + a_7_19·a_7_20 + a_5_10·a_9_28 + a_5_9·a_9_27 + b_2_4²·a_1_2·a_9_27
        + b_2_4³·a_1_0·a_7_19 + b_2_4⁴·a_1_2·a_5_8 + b_2_4⁴·a_1_0·a_5_11
        + b_2_4⁴·a_1_0·a_5_7 + b_2_4⁵·a_1_0·a_3_6 + b_2_4⁵·a_1_0·a_3_5 + b_2_4⁶·a_1_0²
        + c_8_21·a_1_2·a_5_9 + c_8_21·a_1_2·a_5_8 + c_8_21·a_1_1·a_5_9 + c_8_21·a_1_1·a_5_8
        + c_8_21·a_1_0·a_5_11 + b_2_4·c_4_7·a_1_0·a_7_20 + b_2_4·c_4_7·a_1_0·a_7_19
        + b_2_4²·c_8_21·a_1_2² + b_2_4²·c_8_21·a_1_0² + b_2_4²·c_4_7·a_1_2·a_5_9
        + b_2_4²·c_4_7·a_1_2·a_5_8 + b_2_4³·c_4_7·a_1_0·a_3_6 + c_4_7·c_8_21·a_1_2²
        + c_4_7·c_8_21·a_1_1² + c_4_7²·a_1_2·a_5_9 + c_4_7²·a_1_2·a_5_8
        + c_4_7²·a_1_1·a_5_9 + b_2_4·c_4_7²·a_1_0·a_3_5 + b_2_4²·c_4_7²·a_1_0²
        + c_4_7³·a_1_2² + c_4_7³·a_1_1·a_1_2
110. a_7_20² + a_7_19·a_7_20 + a_7_19² + b_2_4³·a_1_0·a_7_20 + b_2_4³·a_1_0·a_7_19
        + b_2_4⁴·a_1_2·a_5_9 + b_2_4⁴·a_1_0·a_5_7 + b_2_4⁶·a_1_2² + c_8_21·a_1_0·a_5_11
        + c_4_7·a_1_0·a_9_28 + b_2_4·c_4_7·a_1_0·a_7_20 + b_2_4²·c_4_7·a_1_2·a_5_9
        + b_2_4²·c_4_7·a_1_2·a_5_8 + b_2_4²·c_4_7·a_1_0·a_5_11 + b_2_4²·c_4_7·a_1_0·a_5_7
        + b_2_4³·c_4_7·a_1_0·a_3_6 + b_2_4³·c_4_7·a_1_0·a_3_5 + b_2_4⁴·c_4_7·a_1_0²
        + b_2_4²·c_4_7²·a_1_0²
111. b_6_16·a_9_27 + b_2_4³·a_9_28 + b_2_4⁴·a_7_20 + b_2_4⁵·a_5_7 + b_2_4⁶·a_3_6
        + b_2_4⁷·a_1_2 + b_2_4⁷·a_1_0 + b_2_4²·a_1_0²·a_9_28 + b_2_4²·c_8_21·a_3_5
        + b_2_4³·c_8_21·a_1_2 + b_2_4³·c_4_7·a_5_9 + b_2_4³·c_4_7·a_5_8
        + b_2_4⁵·c_4_7·a_1_2 + b_2_4³·c_4_7²·a_1_2 + c_4_7·c_8_21·a_1_1²·a_1_2
        + c_4_7²·a_1_1·a_1_2·a_5_9 + c_4_7²·a_1_0²·a_5_11
112. b_6_16·a_9_28 + b_6_16·a_9_27 + b_2_4³·a_9_27 + b_2_4⁴·a_7_20 + b_2_4⁴·a_7_19
        + b_2_4⁵·a_5_9 + b_2_4⁶·a_3_6 + b_2_4⁶·a_3_5 + b_2_4⁴·a_1_2²·a_5_9
        + b_2_4·c_8_21·a_5_8 + b_2_4·c_8_21·a_5_7 + b_2_4²·c_8_21·a_3_5 + b_2_4²·c_4_7·a_7_20
        + b_2_4³·c_4_7·a_5_8 + b_2_4³·c_4_7·a_5_7 + b_2_4⁴·c_4_7·a_3_6 + b_2_4⁵·c_4_7·a_1_2
        + b_2_4⁵·c_4_7·a_1_0 + c_8_21·a_1_2²·a_5_9 + c_8_21·a_1_1·a_1_2·a_5_9
        + c_8_21·a_1_0²·a_5_11 + c_4_7·a_1_0²·a_9_28 + b_2_4·c_4_7·c_8_21·a_1_0
        + b_2_4·c_4_7²·a_5_8 + b_2_4·c_4_7²·a_5_7 + b_2_4²·c_4_7²·a_3_6
        + b_2_4³·c_4_7²·a_1_0 + c_4_7²·a_1_2²·a_5_9 + c_4_7²·a_1_0²·a_5_11
        + b_2_4·c_4_7³·a_1_0 + c_4_7³·a_1_1²·a_1_2
113. a_7_20·a_9_28 + b_2_4³·a_1_2·a_9_27 + b_2_4⁴·a_1_0·a_7_19 + b_2_4⁵·a_1_0·a_5_7
        + c_8_21·a_1_0·a_7_19 + b_2_4·c_8_21·a_1_2·a_5_9 + b_2_4·c_8_21·a_1_2·a_5_8
        + b_2_4·c_8_21·a_1_0·a_5_11 + b_2_4²·c_4_7·a_1_0·a_7_20 + b_2_4³·c_8_21·a_1_2²
        + b_2_4³·c_8_21·a_1_0² + b_2_4⁴·c_4_7·a_1_0·a_3_6 + b_2_4⁵·c_4_7·a_1_2²
        + b_2_4⁵·c_4_7·a_1_0² + c_8_21·a_1_1·a_1_2²·a_5_9 + c_4_7²·a_1_0·a_7_20
        + c_4_7²·a_1_0·a_7_19 + b_2_4·c_4_7²·a_1_2·a_5_9 + b_2_4²·c_4_7²·a_1_0·a_3_5
        + b_2_4³·c_4_7²·a_1_2² + c_4_7·c_8_21·a_1_1²·a_1_2² + c_4_7³·a_1_0·a_3_6
        + b_2_4·c_4_7³·a_1_2² + c_4_7³·a_1_1·a_1_2³ + c_4_7³·a_1_1²·a_1_2²
114. a_7_20·a_9_27 + a_7_19·a_9_28 + a_7_19·a_9_27 + b_2_4³·a_1_2·a_9_27
        + b_2_4⁴·a_1_0·a_7_20 + b_2_4⁴·a_1_0·a_7_19 + b_2_4⁵·a_1_2·a_5_8
        + b_2_4⁵·a_1_0·a_5_11 + b_2_4⁷·a_1_2² + c_8_21·a_1_0·a_7_20 + c_8_21·a_1_0·a_7_19
        + b_2_4·c_8_21·a_1_0·a_5_7 + b_2_4·c_4_7·a_1_2·a_9_27 + b_2_4²·c_8_21·a_1_0·a_3_5
        + b_2_4²·c_4_7·a_1_0·a_7_19 + b_2_4³·c_8_21·a_1_0² + b_2_4³·c_4_7·a_1_0·a_5_7
        + b_2_4⁴·c_4_7·a_1_0·a_3_6 + b_2_4⁴·c_4_7·a_1_0·a_3_5 + b_2_4⁵·c_4_7·a_1_2²
        + b_2_4⁵·c_4_7·a_1_0² + c_4_7·c_8_21·a_1_0·a_3_6 + c_4_7·c_8_21·a_1_0·a_3_5
        + b_2_4·c_4_7·c_8_21·a_1_2² + b_2_4·c_4_7·c_8_21·a_1_0²
        + b_2_4·c_4_7²·a_1_0·a_5_11 + b_2_4²·c_4_7²·a_1_0·a_3_6 + b_2_4³·c_4_7²·a_1_2²
        + b_2_4³·c_4_7²·a_1_0² + c_4_7·c_8_21·a_1_1·a_1_2³
        + c_4_7²·a_1_1²·a_1_2·a_5_9 + b_2_4·c_4_7³·a_1_2² + b_2_4·c_4_7³·a_1_0²
        + c_4_7³·a_1_1·a_1_2³
115. a_7_19·a_9_27 + b_2_4³·a_1_0·a_9_28 + b_2_4⁴·a_1_0·a_7_20 + b_2_4⁵·a_1_2·a_5_8
        + b_2_4⁵·a_1_0·a_5_11 + b_2_4⁶·a_1_0·a_3_6 + b_2_4⁶·a_1_0·a_3_5 + b_2_4⁷·a_1_2²
        + b_2_4⁷·a_1_0² + c_8_21·a_1_0·a_7_19 + b_2_4·c_8_21·a_1_0·a_5_11
        + b_2_4·c_8_21·a_1_0·a_5_7 + b_2_4·c_4_7·a_1_2·a_9_27 + b_2_4²·c_8_21·a_1_0·a_3_6
        + b_2_4²·c_8_21·a_1_0·a_3_5 + b_2_4²·c_4_7·a_1_0·a_7_20
        + b_2_4²·c_4_7·a_1_0·a_7_19 + b_2_4³·c_4_7·a_1_2·a_5_9 + b_2_4⁴·c_4_7·a_1_0·a_3_6
        + b_2_4⁴·c_4_7·a_1_0·a_3_5 + b_2_4⁵·c_4_7·a_1_2² + b_2_4⁵·c_4_7·a_1_0²
        + b_2_4·c_4_7²·a_1_0·a_5_7 + b_2_4³·c_4_7²·a_1_2² + c_4_7·c_8_21·a_1_1·a_1_2³
        + c_4_7·c_8_21·a_1_1²·a_1_2² + b_2_4·c_4_7³·a_1_0² + c_4_7³·a_1_1·a_1_2³
        + c_4_7³·a_1_1²·a_1_2²
116. a_7_20·a_9_27 + a_7_19·a_9_27 + b_2_4³·a_1_2·a_9_27 + b_2_4⁴·a_1_0·a_7_20
        + b_2_4⁵·a_1_2·a_5_9 + b_2_4⁵·a_1_2·a_5_8 + b_2_4⁵·a_1_0·a_5_7 + b_2_4⁶·a_1_0·a_3_5
        + c_8_21·a_1_0·a_7_20 + c_8_21·a_1_0·a_7_19 + b_2_4·c_8_21·a_1_2·a_5_8
        + b_2_4·c_8_21·a_1_0·a_5_7 + b_2_4·c_4_7·a_1_2·a_9_27 + b_2_4·c_4_7·a_1_0·a_9_28
        + b_2_4²·c_8_21·a_1_0·a_3_6 + b_2_4³·c_8_21·a_1_0² + b_2_4³·c_4_7·a_1_2·a_5_8
        + b_2_4³·c_4_7·a_1_0·a_5_11 + b_2_4³·c_4_7·a_1_0·a_5_7 + b_2_4⁵·c_4_7·a_1_2²
        + b_2_4⁵·c_4_7·a_1_0² + c_8_21·a_1_1·a_1_2²·a_5_9 + b_2_4·c_4_7·c_8_21·a_1_2²
        + b_2_4·c_4_7²·a_1_0·a_5_11 + b_2_4³·c_4_7²·a_1_2² + c_4_7·c_8_21·a_1_1·a_1_2³
        + c_4_7²·a_1_1²·a_1_2·a_5_9 + b_2_4·c_4_7³·a_1_0²
117. a_9_27² + b_2_4⁸·a_1_0² + b_2_4²·c_8_21·a_1_0·a_5_7 + b_2_4³·c_8_21·a_1_0·a_3_6
        + b_2_4³·c_8_21·a_1_0·a_3_5 + b_2_4⁴·c_8_21·a_1_2² + b_2_4⁴·c_8_21·a_1_0²
        + b_2_4⁴·c_4_7·a_1_0·a_5_7 + b_2_4⁵·c_4_7·a_1_0·a_3_6 + b_2_4⁵·c_4_7·a_1_0·a_3_5
        + c_8_21²·a_1_0² + b_2_4²·c_4_7·c_8_21·a_1_2² + b_2_4²·c_4_7·c_8_21·a_1_0²
        + b_2_4⁴·c_4_7²·a_1_2² + c_4_7²·c_8_21·a_1_1² + b_2_4²·c_4_7³·a_1_0²
        + c_4_7⁴·a_1_1²
118. a_9_28² + b_2_4²·c_8_21·a_1_0·a_5_7 + b_2_4³·c_8_21·a_1_0·a_3_6
        + b_2_4³·c_8_21·a_1_0·a_3_5 + b_2_4⁴·c_8_21·a_1_0² + b_2_4⁶·c_4_7·a_1_2²
        + b_2_4⁶·c_4_7·a_1_0² + c_8_21²·a_1_2² + c_8_21²·a_1_1²
        + c_4_7·c_8_21·a_1_0·a_5_7 + b_2_4·c_4_7·c_8_21·a_1_0·a_3_6
        + b_2_4·c_4_7·c_8_21·a_1_0·a_3_5 + b_2_4²·c_4_7·c_8_21·a_1_2²
        + b_2_4²·c_4_7·c_8_21·a_1_0² + b_2_4⁴·c_4_7²·a_1_2² + b_2_4²·c_4_7³·a_1_2²
        + b_2_4²·c_4_7³·a_1_0² + c_4_7⁴·a_1_2² + c_4_7⁴·a_1_1²
119. a_9_27·a_9_28 + b_2_4⁵·a_1_0·a_7_20 + b_2_4⁶·a_1_0·a_5_7 + b_2_4⁷·a_1_0·a_3_6
        + b_2_4⁷·a_1_0·a_3_5 + b_2_4⁸·a_1_0² + c_8_21·a_1_2·a_9_27 + c_8_21·a_1_0·a_9_28
        + b_2_4·c_8_21·a_1_0·a_7_19 + b_2_4²·c_8_21·a_1_0·a_5_11
        + b_2_4²·c_8_21·a_1_0·a_5_7 + b_2_4²·c_4_7·a_1_2·a_9_27
        + b_2_4³·c_8_21·a_1_0·a_3_6 + b_2_4³·c_8_21·a_1_0·a_3_5
        + b_2_4³·c_4_7·a_1_0·a_7_19 + b_2_4⁴·c_8_21·a_1_2² + b_2_4⁴·c_4_7·a_1_2·a_5_9
        + b_2_4⁴·c_4_7·a_1_0·a_5_11 + b_2_4⁵·c_4_7·a_1_0·a_3_5 + c_4_7·c_8_21·a_1_1·a_5_8
        + c_4_7²·a_1_2·a_9_27 + b_2_4·c_4_7·c_8_21·a_1_0·a_3_6 + b_2_4·c_4_7²·a_1_0·a_7_20
        + b_2_4²·c_4_7·c_8_21·a_1_0² + b_2_4²·c_4_7²·a_1_2·a_5_9
        + b_2_4²·c_4_7²·a_1_0·a_5_7 + b_2_4³·c_4_7²·a_1_0·a_3_5
        + b_2_4⁴·c_4_7²·a_1_2² + b_2_4⁴·c_4_7²·a_1_0² + c_4_7²·c_8_21·a_1_1·a_1_2
        + c_4_7²·c_8_21·a_1_1² + c_4_7³·a_1_1·a_5_8 + b_2_4·c_4_7³·a_1_0·a_3_6
        + b_2_4²·c_4_7³·a_1_2² + b_2_4²·c_4_7³·a_1_0² + c_4_7⁴·a_1_1·a_1_2
        + c_4_7⁴·a_1_1²

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 18.
• 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_7, a Duflot regular element of degree 4
  2. c_8_21, a Duflot regular element of degree 8
  3. b_2_4, 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. a_1_2 → 0, an element of degree 1
4. b_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. c_4_7 → c_1_0⁴, an element of degree 4
8. a_5_7 → 0, an element of degree 5
9. a_5_8 → 0, an element of degree 5
10. a_5_9 → 0, an element of degree 5
11. a_5_10 → 0, an element of degree 5
12. a_5_11 → 0, an element of degree 5
13. b_6_16 → 0, an element of degree 6
14. a_7_19 → 0, an element of degree 7
15. a_7_20 → 0, an element of degree 7
16. c_8_21 → c_1_1⁸, an element of degree 8
17. a_9_27 → 0, an element of degree 9
18. a_9_28 → 0, an element of degree 9

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_1_2 → 0, an element of degree 1
4. b_2_4 → c_1_2², 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. c_4_7 → c_1_0⁴, an element of degree 4
8. a_5_7 → 0, an element of degree 5
9. a_5_8 → 0, an element of degree 5
10. a_5_9 → 0, an element of degree 5
11. a_5_10 → 0, an element of degree 5
12. a_5_11 → 0, an element of degree 5
13. b_6_16 → c_1_0²·c_1_2⁴, an element of degree 6
14. a_7_19 → 0, an element of degree 7
15. a_7_20 → 0, an element of degree 7
16. c_8_21 → c_1_2⁸ + c_1_1⁴·c_1_2⁴ + c_1_1⁸ + c_1_0⁴·c_1_2⁴, an element of degree 8
17. a_9_27 → 0, an element of degree 9
18. a_9_28 → 0, an element of degree 9

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128