David J. Green

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Cohomology of group number 1483 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 4 minimal generators and exponent 4.
• It is non-abelian.
• It has p-Rank 4.
• Its center has rank 3.
• It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 4.

Structure of the cohomology ring

General information

• The cohomology ring is of dimension 4 and depth 3.
• The depth coincides with the Duflot bound.
• The Poincaré series is

  ( − 1) · (t3  +  t  +  1) · (t4  −  3·t3  −  t  −  1)

  (t  +  1)2 · (t  −  1)4 · (t2  +  1)3

• The a-invariants are -∞,-∞,-∞,-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 16 minimal generators of maximal degree 5:

1. a_1_0, a nilpotent element of degree 1
2. a_1_2, a nilpotent element of degree 1
3. b_1_1, an element of degree 1
4. b_1_3, an element of degree 1
5. a_3_7, a nilpotent element of degree 3
6. b_3_5, an element of degree 3
7. b_3_6, an element of degree 3
8. b_3_8, an element of degree 3
9. b_3_9, an element of degree 3
10. b_3_10, an element of degree 3
11. b_3_11, an element of degree 3
12. c_4_19, a Duflot regular element of degree 4
13. c_4_20, a Duflot regular element of degree 4
14. c_4_21, a Duflot regular element of degree 4
15. b_5_31, an element of degree 5
16. b_5_32, an element of degree 5

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128

Ring relations

There are 62 minimal relations of maximal degree 10:

1. a_1_2·b_1_1 + a_1_0²
2. b_1_1·b_1_3 + b_1_1² + a_1_0·b_1_1 + a_1_0·a_1_2
3. a_1_0·b_1_3 + a_1_2² + a_1_0·a_1_2
4. a_1_0³
5. a_1_2³
6. a_1_2²·b_1_3 + a_1_0·a_1_2²
7. b_1_1·b_3_6 + b_1_1·b_3_5 + a_1_0·b_3_5
8. a_1_2·b_3_5 + a_1_0·b_3_6
9. b_1_3·b_3_5 + b_1_1·b_3_5 + a_1_2·b_3_6 + a_1_2·b_3_5
10. b_1_3·b_3_9 + b_1_3·b_3_8 + b_1_3·b_3_5 + b_1_1·b_3_9 + b_1_1·b_3_8 + b_1_1·b_3_5
       + b_1_3·a_3_7 + b_1_1·a_3_7 + a_1_2·b_3_8 + a_1_2·b_3_5 + a_1_0·b_3_8 + a_1_0·b_3_5
11. b_1_3·b_3_9 + b_1_3·b_3_8 + b_1_3·b_3_5 + b_1_1·b_3_8 + b_1_1·b_3_5 + b_1_3·a_3_7
       + a_1_2·b_3_8 + a_1_2·b_3_5 + a_1_0·b_3_9 + a_1_0·a_3_7
12. b_1_3·b_3_9 + b_1_3·b_3_8 + b_1_3·b_3_5 + b_1_1·b_3_8 + b_1_1·b_3_5 + b_1_3·a_3_7
       + a_1_2·b_3_9 + a_1_2·a_3_7
13. b_1_3·b_3_10 + b_1_3·b_3_9 + b_1_3·b_3_8 + b_1_3·b_3_5 + b_1_1·b_3_10 + b_1_1·b_3_8
       + b_1_1·b_3_5 + a_1_0·b_3_8 + a_1_0·b_3_5 + a_1_2·a_3_7
14. b_1_1·a_3_7 + a_1_0·b_3_10 + a_1_0·a_3_7
15. a_1_2·b_3_10 + a_1_2·a_3_7 + a_1_0·a_3_7
16. b_1_3·b_3_10 + b_1_3·b_3_9 + b_1_3·b_3_8 + b_1_3·b_3_5 + b_1_1·b_3_11 + b_1_1·b_3_8
       + b_1_1·b_3_5 + b_1_1·a_3_7 + a_1_0·b_3_5 + a_1_2·a_3_7 + a_1_0·a_3_7
17. b_1_3·b_3_9 + b_1_3·b_3_8 + b_1_3·b_3_5 + b_1_1·b_3_8 + b_1_1·b_3_5 + b_1_3·a_3_7
       + b_1_1·a_3_7 + a_1_2·b_3_8 + a_1_0·b_3_11 + a_1_0·b_3_8 + a_1_2·a_3_7
18. b_1_3·b_3_9 + b_1_3·b_3_8 + b_1_1·b_3_8 + a_1_2·b_3_11 + a_1_2·b_3_8 + a_1_2·b_3_5
       + a_1_2·a_3_7 + a_1_0·a_3_7
19. a_1_2²·b_3_6 + a_1_0·a_1_2·b_3_6 + a_1_0²·b_3_8 + a_1_0²·b_3_6 + a_1_0²·a_3_7
20. a_1_0·a_1_2·b_3_6 + a_1_0²·b_3_9 + a_1_0²·b_3_6 + a_1_0²·a_3_7
21. a_1_2²·b_3_11 + a_1_0²·b_3_10 + a_1_0²·b_3_6
22. b_3_6·b_3_9 + b_3_6·b_3_8 + b_3_5·b_3_8 + b_3_5·b_3_6 + b_3_5² + a_3_7·b_3_6
23. b_3_6·b_3_10 + b_3_5·b_3_10 + b_3_5·b_3_9 + a_3_7·b_3_6 + a_3_7·b_3_5
24. b_3_9·b_3_10 + a_3_7·b_3_10 + a_3_7·b_3_8 + a_3_7·b_3_5 + a_1_2·b_1_3²·b_3_8
       + a_1_2·b_1_3²·b_3_6
25. b_3_5·b_3_11 + b_3_5·b_3_10 + b_3_5·b_3_6 + b_3_5² + a_3_7·b_3_6 + a_3_7·b_3_5
26. b_3_9·b_3_11 + b_3_8·b_3_11 + b_3_8·b_3_10 + b_3_8·b_3_9 + b_3_5·b_3_6 + b_3_5²
       + b_1_3³·b_3_8 + b_1_3³·b_3_6 + b_1_1³·b_3_8 + b_1_1³·b_3_5 + a_3_7·b_3_11
       + a_3_7·b_3_8 + a_3_7·b_3_6 + a_3_7·b_3_5 + a_1_2·b_1_3²·b_3_8 + a_1_2·b_1_3²·b_3_6
27. b_3_9² + b_3_8² + b_3_5² + b_1_1³·b_3_10 + b_1_1³·b_3_5 + a_3_7² + c_4_19·b_1_1²
28. b_3_9² + b_3_8·b_3_9 + b_3_5·b_3_9 + a_3_7·b_3_8 + a_3_7·b_3_5 + a_1_2·b_1_3²·b_3_8
       + a_1_2·b_1_3²·b_3_6 + a_3_7² + c_4_19·a_1_0·b_1_1
29. b_3_9² + b_1_3³·b_3_8 + b_1_3³·b_3_6 + b_1_1³·b_3_8 + b_1_1³·b_3_5
       + a_1_2·b_1_3²·b_3_8 + a_1_2·b_1_3²·b_3_6 + a_3_7² + c_4_19·a_1_0²
30. b_3_10·b_3_11 + b_3_10² + b_3_8·b_3_9 + b_1_3³·b_3_8 + b_1_3³·b_3_6 + b_1_1³·b_3_8
       + b_1_1³·b_3_5 + a_3_7·b_3_11 + c_4_19·a_1_0·a_1_2
31. b_3_10² + b_3_8² + b_3_5² + b_1_3³·b_3_8 + b_1_3³·b_3_6 + b_1_1³·b_3_10
       + a_1_2·b_1_3²·b_3_8 + a_1_2·b_1_3²·b_3_6 + a_3_7² + c_4_20·b_1_1²
32. b_3_10·b_3_11 + b_3_10² + b_3_9² + b_3_5·b_3_9 + b_1_3³·b_3_8 + b_1_3³·b_3_6
       + b_1_1³·b_3_8 + b_1_1³·b_3_5 + a_3_7·b_3_11 + a_3_7·b_3_10 + a_3_7·b_3_8 + a_3_7·b_3_5
       + a_1_2·b_1_3²·b_3_8 + a_1_2·b_1_3²·b_3_6 + c_4_20·a_1_0·b_1_1
33. b_3_9² + b_1_3³·b_3_8 + b_1_3³·b_3_6 + b_1_1³·b_3_8 + b_1_1³·b_3_5
       + a_1_2·b_1_3²·b_3_8 + a_1_2·b_1_3²·b_3_6 + c_4_20·a_1_0² + c_4_19·a_1_2²
34. b_3_10·b_3_11 + b_3_10² + b_3_8·b_3_9 + b_1_3³·b_3_8 + b_1_3³·b_3_6 + b_1_1³·b_3_8
       + b_1_1³·b_3_5 + a_3_7·b_3_10 + a_3_7·b_3_9 + a_3_7·b_3_8 + a_3_7·b_3_6
       + a_1_2·b_1_3²·b_3_11 + a_1_2·b_1_3²·b_3_6 + c_4_19·a_1_2·b_1_3 + c_4_20·a_1_0·a_1_2
       + c_4_19·a_1_2²
35. b_3_11² + b_3_10² + b_3_9² + b_3_8² + b_3_6² + b_1_3³·b_3_11 + b_1_3³·b_3_6
       + a_1_2·b_1_3²·b_3_11 + a_1_2·b_1_3²·b_3_8 + c_4_19·b_1_3² + c_4_20·a_1_2²
36. b_3_9² + b_3_6² + b_1_3³·b_3_8 + b_1_1³·b_3_10 + a_1_2·b_1_3²·b_3_11
       + a_1_2·b_1_3²·b_3_8 + a_1_2·b_1_3²·b_3_6 + c_4_21·b_1_3² + c_4_20·b_1_3²
       + c_4_21·a_1_0² + c_4_19·a_1_2²
37. b_3_10·b_3_11 + b_3_10² + b_3_8·b_3_9 + b_3_5·b_3_6 + b_3_5² + b_1_3³·b_3_8
       + b_1_3³·b_3_6 + b_1_1³·b_3_8 + b_1_1³·b_3_5 + a_3_7·b_3_10 + a_3_7·b_3_9 + a_3_7·b_3_8
       + a_3_7·b_3_6 + a_1_2·b_1_3²·b_3_11 + c_4_21·a_1_2·b_1_3 + c_4_20·a_1_2·b_1_3
       + c_4_19·a_1_2·b_1_3 + c_4_21·a_1_0·a_1_2 + c_4_19·a_1_2²
38. b_3_11² + b_3_9² + b_1_3³·b_3_11 + b_1_3³·b_3_8 + b_1_3³·b_3_6 + b_1_1³·b_3_10
       + b_1_1³·b_3_5 + a_1_2·b_1_3²·b_3_6 + a_3_7² + c_4_21·b_1_3² + c_4_21·b_1_1²
       + c_4_20·b_1_3² + c_4_19·b_1_3² + c_4_21·a_1_2²
39. b_3_8·b_3_9 + b_3_8² + b_3_6·b_3_11 + b_3_6² + b_3_5·b_3_9 + b_3_5·b_3_8 + b_3_5·b_3_6
       + b_1_3·b_5_31 + b_1_3³·b_3_11 + b_1_3³·b_3_8 + b_1_3³·b_3_6 + b_1_1³·b_3_5
       + a_3_7·b_3_8 + a_3_7·b_3_6 + a_1_2·b_1_3²·b_3_8 + c_4_21·b_1_3² + c_4_19·a_1_2²
40. b_3_10·b_3_11 + b_3_10² + b_3_8² + b_3_5·b_3_10 + b_3_5·b_3_8 + b_3_5·b_3_6 + b_3_5²
       + b_1_3³·b_3_8 + b_1_3³·b_3_6 + b_1_1·b_5_31 + b_1_1³·b_3_10 + b_1_1³·b_3_5
       + a_3_7·b_3_10 + a_3_7·b_3_9 + a_3_7·b_3_6 + a_3_7·b_3_5 + a_1_2·b_1_3²·b_3_11
       + a_1_2·b_1_3²·b_3_8 + a_1_2·b_1_3²·b_3_6 + c_4_21·b_1_1² + c_4_21·a_1_2·b_1_3
       + c_4_21·a_1_0·b_1_1 + c_4_20·a_1_2·b_1_3 + c_4_19·a_1_2·b_1_3 + c_4_19·a_1_2²
41. b_3_11² + b_3_10·b_3_11 + b_3_10² + b_3_9² + b_3_5·b_3_9 + b_3_5·b_3_6 + b_3_5²
       + b_1_3³·b_3_11 + b_1_3³·b_3_8 + b_1_3³·b_3_6 + b_1_1³·b_3_10 + b_1_1³·b_3_5
       + a_3_7·b_3_10 + a_3_7·b_3_9 + a_3_7·b_3_6 + a_1_2·b_1_3²·b_3_11 + a_1_2·b_1_3²·b_3_6
       + a_1_0·b_5_31 + a_3_7² + c_4_21·b_1_3² + c_4_21·b_1_1² + c_4_20·b_1_3²
       + c_4_19·b_1_3² + c_4_21·a_1_2·b_1_3 + c_4_20·a_1_2·b_1_3 + c_4_19·a_1_2·b_1_3
       + c_4_19·a_1_2²
42. b_3_10·b_3_11 + b_3_10² + b_3_9² + b_3_8·b_3_9 + a_3_7·b_3_11 + a_3_7·b_3_6
       + a_1_2·b_5_31 + a_1_2·b_1_3²·b_3_11 + a_3_7² + c_4_21·a_1_2·b_1_3
43. b_3_10·b_3_11 + b_3_10² + b_3_8·b_3_11 + b_3_8·b_3_9 + b_3_6·b_3_8 + b_3_6²
       + b_3_5·b_3_10 + b_3_5·b_3_8 + b_3_5² + b_1_3·b_5_32 + b_1_3³·b_3_11 + b_1_3³·b_3_8
       + a_3_7·b_3_11 + a_3_7·b_3_10 + a_3_7·b_3_9 + a_3_7·b_3_6 + a_3_7·b_3_5
       + a_1_2·b_1_3²·b_3_11 + a_1_2·b_1_3²·b_3_6 + c_4_21·b_1_3² + c_4_21·a_1_2·b_1_3
       + c_4_21·a_1_0·b_1_1 + c_4_20·a_1_2·b_1_3 + c_4_19·a_1_2²
44. b_3_10·b_3_11 + b_3_10² + b_3_9² + b_3_8·b_3_10 + b_3_6² + b_3_5·b_3_10 + b_3_5·b_3_9
       + b_1_3³·b_3_8 + b_1_1·b_5_32 + b_1_1³·b_3_8 + a_3_7·b_3_10 + a_3_7·b_3_6 + a_3_7·b_3_5
       + a_1_2·b_1_3²·b_3_6 + c_4_21·b_1_3² + c_4_21·b_1_1² + c_4_20·b_1_3²
       + c_4_21·a_1_0·b_1_1 + c_4_19·a_1_2·b_1_3 + c_4_19·a_1_2²
45. b_3_11² + b_3_10² + b_3_9² + b_3_8·b_3_9 + b_3_8² + b_3_5·b_3_9 + b_3_5·b_3_6
       + b_3_5² + b_1_3³·b_3_11 + b_1_3³·b_3_8 + b_1_3³·b_3_6 + b_1_1³·b_3_10 + a_3_7·b_3_11
       + a_3_7·b_3_8 + a_3_7·b_3_6 + a_1_2·b_1_3²·b_3_11 + a_1_2·b_1_3²·b_3_8 + a_1_0·b_5_32
       + c_4_21·b_1_3² + c_4_20·b_1_3² + c_4_19·b_1_3² + c_4_21·a_1_2·b_1_3
       + c_4_21·a_1_0·b_1_1 + c_4_20·a_1_2·b_1_3 + c_4_19·a_1_2·b_1_3 + c_4_19·a_1_2²
46. b_3_10·b_3_11 + b_3_8·b_3_9 + b_3_8² + b_3_5·b_3_6 + b_3_5² + b_1_1³·b_3_5
       + a_3_7·b_3_11 + a_3_7·b_3_9 + a_1_2·b_5_32 + a_1_2·b_1_3²·b_3_11 + a_1_2·b_1_3²·b_3_8
       + c_4_21·b_1_1² + c_4_21·a_1_2·b_1_3
47. a_1_2²·b_5_31 + a_1_0·a_1_2·b_5_32 + a_1_0·a_1_2·b_5_31 + a_1_0²·b_5_31
       + c_4_20·a_1_0²·a_1_2 + c_4_19·a_1_0·a_1_2² + c_4_19·a_1_0²·a_1_2
48. b_3_6·b_5_31 + b_1_3²·b_3_6·b_3_8 + b_1_1³·b_5_31 + b_1_1⁵·b_3_8
       + a_1_2·b_1_3·b_3_6·b_3_8 + a_1_2·b_1_3⁴·b_3_11 + a_1_2·b_1_3⁴·b_3_6
       + c_4_21·b_1_3·b_3_11 + c_4_21·b_1_1·b_3_8 + c_4_20·b_1_3·b_3_11 + c_4_20·b_1_3·b_3_6
       + c_4_20·b_1_1·b_3_8 + c_4_19·b_1_1·b_3_5 + c_4_19·b_1_1⁴ + c_4_21·a_1_2·b_3_11
       + c_4_21·a_1_2·b_3_8 + c_4_21·a_1_0·b_3_10 + c_4_21·a_1_0·b_3_6 + c_4_20·a_1_2·b_3_11
       + c_4_20·a_1_2·b_3_8 + c_4_20·a_1_0·b_3_10 + c_4_20·a_1_0·b_3_6 + c_4_19·a_1_2·b_1_3³
       + c_4_19·a_1_0·b_3_6 + c_4_19·a_1_0·b_3_5 + c_4_21·a_1_0·a_3_7 + c_4_20·a_1_0·a_3_7
49. b_3_11·b_5_32 + b_3_10·b_5_31 + b_3_8·b_5_31 + b_1_3⁵·b_3_11 + b_1_3⁵·b_3_8
       + b_1_3⁵·b_3_6 + b_1_1²·b_3_5·b_3_8 + b_1_1³·b_5_31 + b_1_1⁵·b_3_10 + b_1_1⁵·b_3_8
       + b_1_1⁵·b_3_5 + a_1_2·b_1_3·b_3_6·b_3_8 + a_1_2·b_1_3⁴·b_3_11 + c_4_21·b_1_3·b_3_8
       + c_4_21·b_1_3⁴ + c_4_21·b_1_1·b_3_10 + c_4_21·b_1_1·b_3_8 + c_4_21·b_1_1·b_3_5
       + c_4_21·b_1_1⁴ + c_4_20·b_1_3·b_3_11 + c_4_20·b_1_3⁴ + c_4_20·b_1_1·b_3_5
       + c_4_19·b_1_3·b_3_8 + c_4_19·b_1_3⁴ + c_4_19·b_1_1·b_3_10 + c_4_19·b_1_1·b_3_8
       + c_4_19·b_1_1·b_3_5 + c_4_19·b_1_1⁴ + c_4_21·a_1_2·b_3_11 + c_4_21·a_1_2·b_3_8
       + c_4_21·a_1_0·b_3_10 + c_4_21·a_1_0·b_3_9 + c_4_21·a_1_0·b_3_8 + c_4_21·a_1_0·b_3_5
       + c_4_20·a_1_2·b_3_6 + c_4_20·a_1_0·b_3_10 + c_4_20·a_1_0·b_3_9 + c_4_20·a_1_0·b_3_8
       + c_4_19·a_1_2·b_3_8 + c_4_19·a_1_2·b_1_3³ + c_4_19·a_1_0·b_3_10 + c_4_19·a_1_0·b_3_9
       + c_4_19·a_1_0·b_3_5 + c_4_21·a_1_2·a_3_7 + c_4_20·a_1_2·a_3_7 + c_4_20·a_1_0·a_3_7
       + c_4_19·a_1_2·a_3_7
50. b_3_11·b_5_31 + b_3_9·b_5_31 + b_3_8·b_5_31 + b_1_3²·b_3_6·b_3_8 + b_1_3³·b_5_32
       + b_1_3³·b_5_31 + b_1_3⁵·b_3_11 + b_1_3⁵·b_3_8 + b_1_1³·b_5_31 + b_1_1⁵·b_3_10
       + a_3_7·b_5_31 + a_1_2·b_1_3·b_3_6·b_3_8 + c_4_21·b_1_3·b_3_6 + c_4_21·b_1_3⁴
       + c_4_21·b_1_1⁴ + c_4_20·b_1_3·b_3_11 + c_4_20·b_1_3·b_3_6 + c_4_20·b_1_3⁴
       + c_4_20·b_1_1·b_3_8 + c_4_20·b_1_1·b_3_5 + c_4_20·b_1_1⁴ + c_4_19·b_1_3·b_3_6
       + c_4_19·b_1_3⁴ + c_4_19·b_1_1·b_3_10 + c_4_19·b_1_1·b_3_8 + c_4_19·b_1_1·b_3_5
       + c_4_21·a_1_2·b_3_11 + c_4_21·a_1_2·b_3_6 + c_4_21·a_1_2·b_1_3³
       + c_4_21·a_1_0·b_3_10 + c_4_21·a_1_0·b_3_9 + c_4_21·a_1_0·b_3_6 + c_4_20·a_1_2·b_3_11
       + c_4_20·a_1_2·b_3_8 + c_4_20·a_1_2·b_1_3³ + c_4_20·a_1_0·b_3_10 + c_4_19·a_1_2·b_3_6
       + c_4_19·a_1_2·b_1_3³ + c_4_19·a_1_0·b_3_9 + c_4_19·a_1_0·b_3_6 + c_4_19·a_1_0·b_3_5
       + c_4_20·a_1_0·a_3_7 + c_4_19·a_1_2·a_3_7
51. a_3_7·b_5_31 + a_1_2·b_1_3²·b_5_32 + a_1_2·b_1_3²·b_5_31 + a_1_2·b_1_3⁴·b_3_11
       + a_1_2·b_1_3⁴·b_3_8 + c_4_21·a_1_2·b_3_11 + c_4_21·a_1_2·b_3_6
       + c_4_21·a_1_2·b_1_3³ + c_4_21·a_1_0·b_3_9 + c_4_21·a_1_0·b_3_6
       + c_4_20·a_1_2·b_1_3³ + c_4_20·a_1_0·b_3_5 + c_4_19·a_1_2·b_3_6
       + c_4_19·a_1_2·b_1_3³ + c_4_19·a_1_0·b_3_10 + c_4_19·a_1_0·b_3_5 + c_4_21·a_1_2·a_3_7
52. b_3_10·b_5_31 + b_3_9·b_5_31 + b_3_8·b_5_31 + b_3_5·b_5_31 + b_1_1³·b_5_32
       + b_1_1³·b_5_31 + c_4_21·b_1_1·b_3_10 + c_4_21·b_1_1·b_3_8 + c_4_21·b_1_1·b_3_5
       + c_4_20·b_1_1·b_3_5 + c_4_19·b_1_1·b_3_10 + c_4_19·b_1_1·b_3_8 + c_4_19·b_1_1·b_3_5
       + c_4_21·a_1_0·b_3_8 + c_4_21·a_1_0·b_3_5 + c_4_19·a_1_0·b_3_6 + c_4_19·a_1_0·b_3_5
       + c_4_21·a_1_2·a_3_7
53. b_3_10·b_5_32 + b_3_10·b_5_31 + b_3_9·b_5_31 + b_3_8·b_5_31 + b_1_1²·b_3_5·b_3_8
       + b_1_1³·b_5_31 + a_3_7·b_5_31 + a_1_2·b_1_3²·b_5_31 + a_1_2·b_1_3⁴·b_3_11
       + a_1_2·b_1_3⁴·b_3_8 + a_1_2·b_1_3⁴·b_3_6 + c_4_21·b_1_1·b_3_10 + c_4_21·b_1_1·b_3_5
       + c_4_20·b_1_1·b_3_10 + c_4_20·b_1_1·b_3_5 + c_4_20·b_1_1⁴ + c_4_19·b_1_1·b_3_10
       + c_4_19·b_1_1·b_3_5 + c_4_21·a_1_2·b_3_6 + c_4_21·a_1_2·b_1_3³ + c_4_21·a_1_0·b_3_9
       + c_4_21·a_1_0·b_3_8 + c_4_21·a_1_0·b_3_6 + c_4_21·a_1_0·b_3_5 + c_4_20·a_1_2·b_3_8
       + c_4_20·a_1_0·b_3_8 + c_4_20·a_1_0·b_3_6 + c_4_19·a_1_2·b_3_8 + c_4_19·a_1_2·b_3_6
       + c_4_19·a_1_0·b_3_5 + c_4_21·a_1_2·a_3_7 + c_4_20·a_1_2·a_3_7 + c_4_20·a_1_0·a_3_7
54. b_3_5·b_5_31 + b_1_1²·b_3_5·b_3_8 + b_1_1³·b_5_31 + b_1_1⁵·b_3_8
       + a_1_2·b_1_3·b_3_6·b_3_8 + a_1_0²·a_1_2·b_5_32 + c_4_21·b_1_1·b_3_10
       + c_4_21·b_1_1·b_3_8 + c_4_20·b_1_1·b_3_10 + c_4_20·b_1_1·b_3_8 + c_4_20·b_1_1·b_3_5
       + c_4_19·b_1_1·b_3_5 + c_4_19·b_1_1⁴ + c_4_21·a_1_2·b_3_11 + c_4_21·a_1_0·b_3_10
       + c_4_20·a_1_2·b_3_11 + c_4_20·a_1_2·b_3_6 + c_4_20·a_1_0·b_3_10 + c_4_20·a_1_0·b_3_6
       + c_4_19·a_1_0·b_3_5 + c_4_21·a_1_0·a_3_7 + c_4_20·a_1_0·a_3_7
55. b_3_5·b_5_31 + b_1_1²·b_3_5·b_3_8 + b_1_1³·b_5_31 + b_1_1⁵·b_3_8 + a_3_7·b_5_32
       + a_3_7·b_5_31 + a_1_2·b_1_3·b_3_6·b_3_8 + c_4_21·b_1_1·b_3_10 + c_4_21·b_1_1·b_3_8
       + c_4_20·b_1_1·b_3_10 + c_4_20·b_1_1·b_3_8 + c_4_20·b_1_1·b_3_5 + c_4_19·b_1_1·b_3_5
       + c_4_19·b_1_1⁴ + c_4_21·a_1_2·b_3_6 + c_4_21·a_1_0·b_3_9 + c_4_21·a_1_0·b_3_6
       + c_4_20·a_1_0·b_3_8 + c_4_19·a_1_2·b_3_8 + c_4_19·a_1_2·b_3_6 + c_4_19·a_1_0·b_3_10
       + c_4_19·a_1_0·b_3_8 + c_4_19·a_1_0·b_3_6 + c_4_19·a_1_0·b_3_5 + c_4_21·a_1_0·a_3_7
       + c_4_19·a_1_0·a_3_7
56. b_3_9·b_5_31 + b_3_8·b_5_31 + b_3_5·b_5_32 + b_3_5·b_5_31 + b_1_1²·b_3_5·b_3_8
       + b_1_1⁵·b_3_8 + b_1_1⁵·b_3_5 + a_1_2·b_1_3⁴·b_3_11 + a_1_2·b_1_3⁴·b_3_8
       + a_1_2·b_1_3⁴·b_3_6 + c_4_21·b_1_1·b_3_8 + c_4_21·b_1_1⁴ + c_4_20·b_1_1⁴
       + c_4_19·b_1_1·b_3_8 + c_4_19·b_1_1⁴ + c_4_21·a_1_2·b_3_11 + c_4_21·a_1_2·b_3_8
       + c_4_21·a_1_2·b_3_6 + c_4_21·a_1_2·b_1_3³ + c_4_21·a_1_0·b_3_6 + c_4_21·a_1_0·b_3_5
       + c_4_20·a_1_2·b_3_6 + c_4_20·a_1_2·b_1_3³ + c_4_19·a_1_2·b_3_6
       + c_4_19·a_1_2·b_1_3³ + c_4_19·a_1_0·b_3_10 + c_4_21·a_1_2·a_3_7 + c_4_21·a_1_0·a_3_7
57. b_3_11·b_5_31 + b_3_10·b_5_31 + b_3_8·b_5_31 + b_3_6·b_5_32 + b_3_5·b_5_31
       + b_1_3²·b_3_6·b_3_8 + b_1_3⁵·b_3_11 + b_1_3⁵·b_3_8 + b_1_3⁵·b_3_6 + b_1_1⁵·b_3_10
       + a_1_2·b_1_3·b_3_6·b_3_8 + a_1_2·b_1_3⁴·b_3_8 + a_1_2·b_1_3⁴·b_3_6
       + c_4_21·b_1_3·b_3_8 + c_4_21·b_1_3·b_3_6 + c_4_21·b_1_3⁴ + c_4_21·b_1_1·b_3_5
       + c_4_20·b_1_3·b_3_11 + c_4_20·b_1_3⁴ + c_4_20·b_1_1·b_3_10 + c_4_19·b_1_3·b_3_6
       + c_4_19·b_1_3⁴ + c_4_19·b_1_1·b_3_8 + c_4_19·b_1_1·b_3_5 + c_4_21·a_1_2·b_3_11
       + c_4_21·a_1_2·b_1_3³ + c_4_21·a_1_0·b_3_8 + c_4_20·a_1_2·b_3_8
       + c_4_20·a_1_2·b_1_3³ + c_4_19·a_1_0·b_3_10 + c_4_19·a_1_0·b_3_9 + c_4_19·a_1_0·b_3_8
       + c_4_19·a_1_0·b_3_6 + c_4_21·a_1_0·a_3_7 + c_4_19·a_1_2·a_3_7
58. b_3_10·b_5_31 + b_3_8·b_5_32 + b_3_5·b_5_31 + b_1_3²·b_3_6·b_3_8 + b_1_3³·b_5_31
       + b_1_3⁵·b_3_11 + b_1_3⁵·b_3_8 + b_1_3⁵·b_3_6 + b_1_1²·b_3_5·b_3_8 + b_1_1⁵·b_3_10
       + a_1_2·b_1_3⁴·b_3_11 + a_1_2·b_1_3⁴·b_3_8 + a_1_2·b_1_3⁴·b_3_6 + c_4_21·b_1_3⁴
       + c_4_20·b_1_3·b_3_8 + c_4_20·b_1_1·b_3_10 + c_4_19·b_1_1⁴ + c_4_21·a_1_2·b_3_6
       + c_4_21·a_1_0·b_3_10 + c_4_21·a_1_0·b_3_5 + c_4_20·a_1_2·b_3_11 + c_4_20·a_1_2·b_3_8
       + c_4_20·a_1_2·b_3_6 + c_4_20·a_1_0·b_3_10 + c_4_20·a_1_0·b_3_8 + c_4_20·a_1_0·b_3_6
       + c_4_20·a_1_0·b_3_5 + c_4_19·a_1_2·b_3_6 + c_4_19·a_1_2·b_1_3³ + c_4_19·a_1_0·b_3_10
       + c_4_19·a_1_0·b_3_8 + c_4_19·a_1_0·b_3_6 + c_4_19·a_1_0·b_3_5 + c_4_20·a_1_0·a_3_7
59. b_3_9·b_5_32 + b_1_3²·b_3_6·b_3_8 + b_1_3³·b_5_31 + b_1_3⁵·b_3_11 + b_1_3⁵·b_3_8
       + b_1_3⁵·b_3_6 + b_1_1²·b_3_5·b_3_8 + b_1_1³·b_5_31 + b_1_1⁵·b_3_10 + b_1_1⁵·b_3_8
       + b_1_1⁵·b_3_5 + a_3_7·b_5_31 + a_1_2·b_1_3²·b_5_31 + a_1_2·b_1_3⁴·b_3_11
       + a_1_2·b_1_3⁴·b_3_8 + a_1_2·b_1_3⁴·b_3_6 + c_4_21·b_1_3⁴ + c_4_21·b_1_1⁴
       + c_4_20·b_1_3·b_3_8 + c_4_20·b_1_1·b_3_8 + c_4_21·a_1_2·b_3_11 + c_4_21·a_1_2·b_3_8
       + c_4_21·a_1_2·b_3_6 + c_4_21·a_1_0·b_3_10 + c_4_21·a_1_0·b_3_8 + c_4_21·a_1_0·b_3_5
       + c_4_20·a_1_2·b_3_11 + c_4_20·a_1_2·b_1_3³ + c_4_20·a_1_0·b_3_10
       + c_4_20·a_1_0·b_3_5 + c_4_19·a_1_2·b_3_8 + c_4_19·a_1_2·b_3_6 + c_4_19·a_1_0·b_3_9
       + c_4_19·a_1_0·b_3_8 + c_4_19·a_1_0·b_3_6 + c_4_21·a_1_0·a_3_7 + c_4_20·a_1_0·a_3_7
       + c_4_19·a_1_0·a_3_7
60. b_5_31² + b_1_3⁵·b_5_31 + b_1_3⁷·b_3_6 + a_1_2·b_1_3⁴·b_5_31 + a_1_2·b_1_3⁶·b_3_11
       + a_1_2·b_1_3⁶·b_3_8 + c_4_21·b_1_3³·b_3_11 + c_4_21·b_1_3³·b_3_6 + c_4_21·b_1_3⁶
       + c_4_21·b_1_1³·b_3_8 + c_4_20·b_1_3³·b_3_11 + c_4_20·b_1_3³·b_3_6
       + c_4_20·b_1_1³·b_3_10 + c_4_20·b_1_1⁶ + c_4_19·b_1_3³·b_3_6 + c_4_19·b_1_3⁶
       + c_4_19·b_1_1³·b_3_5 + c_4_19·b_1_1⁶ + c_4_21·a_1_2·b_1_3²·b_3_11
       + c_4_21·a_1_2·b_1_3²·b_3_8 + c_4_21·a_1_2·b_1_3⁵ + c_4_20·a_1_2·b_1_3²·b_3_11
       + c_4_20·a_1_2·b_1_3²·b_3_8 + c_4_19·a_1_2·b_1_3²·b_3_11 + c_4_19·a_1_2·b_1_3⁵
       + c_4_21²·b_1_3² + c_4_20·c_4_21·b_1_1² + c_4_20²·b_1_1²
       + c_4_19·c_4_21·b_1_3² + c_4_19·c_4_21·b_1_1² + c_4_19·c_4_20·b_1_3²
       + c_4_19·c_4_20·b_1_1² + c_4_19²·b_1_1² + c_4_20·c_4_21·a_1_2²
       + c_4_20·c_4_21·a_1_0² + c_4_20²·a_1_2² + c_4_20²·a_1_0²
       + c_4_19·c_4_21·a_1_2² + c_4_19·c_4_20·a_1_2² + c_4_19²·a_1_0²
61. b_5_32² + b_1_3⁵·b_5_32 + b_1_3⁵·b_5_31 + b_1_3⁷·b_3_11 + b_1_3⁷·b_3_8
       + b_1_3⁷·b_3_6 + b_1_1⁷·b_3_10 + b_1_1⁷·b_3_5 + a_1_2·b_1_3⁴·b_5_32
       + a_1_2·b_1_3⁶·b_3_11 + a_1_2·b_1_3⁶·b_3_6 + c_4_20·b_1_1³·b_3_10
       + c_4_20·b_1_1³·b_3_5 + c_4_20·b_1_1⁶ + c_4_19·b_1_3³·b_3_8 + c_4_19·b_1_3³·b_3_6
       + c_4_19·b_1_3⁶ + c_4_19·b_1_1³·b_3_10 + c_4_21·a_1_2·b_1_3⁵
       + c_4_19·a_1_2·b_1_3²·b_3_8 + c_4_19·a_1_2·b_1_3²·b_3_6 + c_4_21²·b_1_1²
       + c_4_20²·b_1_3² + c_4_20²·b_1_1² + c_4_19·c_4_20·b_1_1² + c_4_19²·b_1_1²
       + c_4_21²·a_1_2² + c_4_21²·a_1_0² + c_4_20²·a_1_2² + c_4_20²·a_1_0²
62. b_5_32² + b_5_31·b_5_32 + b_1_3²·b_3_6·b_5_32 + b_1_3⁴·b_3_6·b_3_8
       + b_1_1²·b_3_5·b_5_32 + b_1_1⁴·b_3_5·b_3_8 + a_1_2·b_1_3·b_3_6·b_5_32
       + a_1_2·b_1_3⁴·b_5_31 + a_1_2·b_1_3⁶·b_3_8 + c_4_21·b_1_3·b_5_32
       + c_4_21·b_1_3³·b_3_11 + c_4_21·b_1_3³·b_3_6 + c_4_21·b_1_1·b_5_31
       + c_4_21·b_1_1³·b_3_10 + c_4_21·b_1_1³·b_3_5 + c_4_20·b_3_5·b_3_9
       + c_4_20·b_3_5·b_3_8 + c_4_20·b_1_3·b_5_31 + c_4_20·b_1_3³·b_3_11
       + c_4_20·b_1_1·b_5_31 + c_4_20·b_1_1³·b_3_10 + c_4_20·b_1_1³·b_3_8
       + c_4_20·b_1_1³·b_3_5 + c_4_19·b_3_6·b_3_8 + c_4_19·b_3_5·b_3_9 + c_4_19·b_3_5·b_3_8
       + c_4_19·b_1_1·b_5_32 + c_4_19·b_1_1·b_5_31 + c_4_21·a_1_2·b_5_31
       + c_4_21·a_1_2·b_1_3²·b_3_8 + c_4_21·a_1_2·b_1_3⁵ + c_4_21·a_1_0·b_5_31
       + c_4_20·a_1_2·b_5_31 + c_4_20·a_1_2·b_1_3²·b_3_8 + c_4_20·a_1_2·b_1_3²·b_3_6
       + c_4_19·a_1_2·b_1_3⁵ + c_4_19·a_1_0·b_5_32 + c_4_20·c_4_21·b_1_3²
       + c_4_20²·b_1_3² + c_4_19·c_4_20·b_1_1² + c_4_21²·a_1_2·b_1_3
       + c_4_21²·a_1_0·b_1_1 + c_4_20·c_4_21·a_1_2·b_1_3 + c_4_20·c_4_21·a_1_0·b_1_1
       + c_4_19·c_4_21·a_1_2·b_1_3 + c_4_19·c_4_21·a_1_0·b_1_1 + c_4_19·c_4_20·a_1_2·b_1_3
       + c_4_19²·a_1_0·b_1_1 + c_4_21²·a_1_0² + c_4_20·c_4_21·a_1_2²
       + c_4_20·c_4_21·a_1_0·a_1_2 + c_4_20·c_4_21·a_1_0² + c_4_19·c_4_21·a_1_0·a_1_2
       + c_4_19·c_4_20·a_1_2² + c_4_19·c_4_20·a_1_0²

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Data used for Benson′s test

• Benson′s completion test succeeded in degree 10.
• 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_19, a Duflot regular element of degree 4
  2. c_4_20, a Duflot regular element of degree 4
  3. c_4_21, a Duflot regular element of degree 4
  4. b_1_3², an element of degree 2
• The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 8, 10].
• The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].

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Restriction maps

Restriction map to the greatest central el. ab. subgp., which is of rank 3

1. a_1_0 → 0, an element of degree 1
2. a_1_2 → 0, an element of degree 1
3. b_1_1 → 0, an element of degree 1
4. b_1_3 → 0, an element of degree 1
5. a_3_7 → 0, an element of degree 3
6. b_3_5 → 0, an element of degree 3
7. b_3_6 → 0, an element of degree 3
8. b_3_8 → 0, an element of degree 3
9. b_3_9 → 0, an element of degree 3
10. b_3_10 → 0, an element of degree 3
11. b_3_11 → 0, an element of degree 3
12. c_4_19 → c_1_2⁴ + c_1_1⁴ + c_1_0⁴, an element of degree 4
13. c_4_20 → c_1_0⁴, an element of degree 4
14. c_4_21 → c_1_2⁴ + c_1_0⁴, an element of degree 4
15. b_5_31 → 0, an element of degree 5
16. b_5_32 → 0, an element of degree 5

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

1. a_1_0 → 0, an element of degree 1
2. a_1_2 → 0, an element of degree 1
3. b_1_1 → 0, an element of degree 1
4. b_1_3 → c_1_3, an element of degree 1
5. a_3_7 → 0, an element of degree 3
6. b_3_5 → 0, an element of degree 3
7. b_3_6 → c_1_2·c_1_3² + c_1_2²·c_1_3, an element of degree 3
8. b_3_8 → c_1_2·c_1_3², an element of degree 3
9. b_3_9 → c_1_2·c_1_3², an element of degree 3
10. b_3_10 → 0, an element of degree 3
11. b_3_11 → c_1_2·c_1_3² + c_1_1·c_1_3² + c_1_1²·c_1_3 + c_1_0·c_1_3² + c_1_0²·c_1_3, an element of degree 3
12. c_4_19 → c_1_2²·c_1_3² + c_1_2⁴ + c_1_1·c_1_3³ + c_1_1⁴ + c_1_0·c_1_3³ + c_1_0⁴, an element of degree 4
13. c_4_20 → c_1_2·c_1_3³ + c_1_2²·c_1_3² + c_1_0²·c_1_3² + c_1_0⁴, an element of degree 4
14. c_4_21 → c_1_2²·c_1_3² + c_1_2⁴ + c_1_0²·c_1_3² + c_1_0⁴, an element of degree 4
15. b_5_31 → c_1_2·c_1_3⁴ + c_1_2³·c_1_3² + c_1_1·c_1_3⁴ + c_1_1·c_1_2·c_1_3³
       + c_1_1·c_1_2²·c_1_3² + c_1_1²·c_1_3³ + c_1_1²·c_1_2·c_1_3²
       + c_1_1²·c_1_2²·c_1_3 + c_1_0·c_1_3⁴ + c_1_0·c_1_2·c_1_3³
       + c_1_0·c_1_2²·c_1_3² + c_1_0²·c_1_2·c_1_3² + c_1_0²·c_1_2²·c_1_3
       + c_1_0⁴·c_1_3, an element of degree 5
16. b_5_32 → c_1_2²·c_1_3³ + c_1_2³·c_1_3² + c_1_1·c_1_3⁴ + c_1_1·c_1_2·c_1_3³
       + c_1_1²·c_1_3³ + c_1_1²·c_1_2·c_1_3² + c_1_0·c_1_3⁴ + c_1_0·c_1_2·c_1_3³
       + c_1_0²·c_1_2·c_1_3² + c_1_0⁴·c_1_3, an element of degree 5

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

1. a_1_0 → 0, an element of degree 1
2. a_1_2 → 0, an element of degree 1
3. b_1_1 → c_1_3, an element of degree 1
4. b_1_3 → c_1_3, an element of degree 1
5. a_3_7 → 0, an element of degree 3
6. b_3_5 → c_1_2·c_1_3² + c_1_2²·c_1_3, an element of degree 3
7. b_3_6 → c_1_2·c_1_3² + c_1_2²·c_1_3, an element of degree 3
8. b_3_8 → c_1_1·c_1_3² + c_1_1²·c_1_3 + c_1_0·c_1_3² + c_1_0²·c_1_3, an element of degree 3
9. b_3_9 → 0, an element of degree 3
10. b_3_10 → c_1_2·c_1_3² + c_1_2²·c_1_3 + c_1_1·c_1_3² + c_1_1²·c_1_3, an element of degree 3
11. b_3_11 → c_1_2·c_1_3² + c_1_2²·c_1_3 + c_1_1·c_1_3² + c_1_1²·c_1_3, an element of degree 3
12. c_4_19 → c_1_2²·c_1_3² + c_1_2⁴ + c_1_1·c_1_3³ + c_1_1⁴ + c_1_0²·c_1_3² + c_1_0⁴, an element of degree 4
13. c_4_20 → c_1_0·c_1_3³ + c_1_0⁴, an element of degree 4
14. c_4_21 → c_1_2·c_1_3³ + c_1_2⁴ + c_1_0²·c_1_3² + c_1_0⁴, an element of degree 4
15. b_5_31 → c_1_1²·c_1_3³ + c_1_1⁴·c_1_3 + c_1_0·c_1_3⁴ + c_1_0·c_1_2·c_1_3³
       + c_1_0·c_1_2²·c_1_3² + c_1_0²·c_1_3³ + c_1_0²·c_1_2·c_1_3²
       + c_1_0²·c_1_2²·c_1_3, an element of degree 5
16. b_5_32 → c_1_1²·c_1_3³ + c_1_1⁴·c_1_3 + c_1_0·c_1_3⁴ + c_1_0·c_1_2·c_1_3³
       + c_1_0·c_1_2²·c_1_3² + c_1_0·c_1_1·c_1_3³ + c_1_0·c_1_1²·c_1_3²
       + c_1_0²·c_1_2·c_1_3² + c_1_0²·c_1_2²·c_1_3 + c_1_0²·c_1_1·c_1_3²
       + c_1_0²·c_1_1²·c_1_3 + c_1_0⁴·c_1_3, an element of degree 5

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 128