Cohomology of group number 1325 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) · (t8  +  t7  −  3·t6  +  3·t5  −  3·t4  −  3·t3  −  t2  −  2·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 14 minimal generators of maximal degree 8:

  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. b_3_5, an element of degree 3
  6. b_3_6, an element of degree 3
  7. b_3_7, 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. c_4_17, a Duflot regular element of degree 4
  12. c_4_18, a Duflot regular element of degree 4
  13. c_4_19, a Duflot regular element of degree 4
  14. b_8_67, an element of degree 8

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128

Ring relations

There are 42 minimal relations of maximal degree 16:

  1. b_1_1·b_1_3 + b_1_12 + a_1_2·b_1_1 + a_1_02
  2. a_1_0·b_1_3 + a_1_22 + a_1_0·a_1_2
  3. a_1_2·b_1_3 + a_1_0·b_1_1 + a_1_0·a_1_2
  4. a_1_03
  5. a_1_0·a_1_22
  6. a_1_0·a_1_2·b_1_1 + a_1_23
  7. a_1_2·b_3_5 + a_1_0·b_3_6
  8. b_1_3·b_3_6 + b_1_3·b_3_5 + b_1_1·b_3_5 + a_1_2·b_3_6
  9. b_1_3·b_3_6 + b_1_3·b_3_5 + b_1_1·b_3_9 + b_1_1·b_3_8 + b_1_1·b_3_7 + b_1_1·b_3_6
       + b_1_1·b_3_5 + b_1_14 + a_1_2·b_3_7 + a_1_0·b_3_8
  10. a_1_2·b_3_7 + a_1_0·b_3_9 + a_1_0·b_3_8 + a_1_0·b_3_7
  11. b_1_3·b_3_9 + b_1_3·b_3_8 + b_1_1·b_3_7 + b_1_14 + a_1_2·b_3_9 + a_1_2·b_3_8 + a_1_2·b_3_7
  12. b_1_3·b_3_10 + b_1_3·b_3_9 + b_1_3·b_3_6 + b_1_3·b_3_5 + b_1_34 + b_1_1·b_3_8
       + b_1_1·b_3_7 + b_1_1·b_3_5 + b_1_14 + a_1_2·b_3_8 + a_1_2·b_3_7 + a_1_2·b_3_5
       + a_1_0·b_3_8
  13. b_1_1·b_3_10 + b_1_14 + a_1_2·b_3_8 + a_1_2·b_3_5 + a_1_0·b_3_7
  14. b_1_3·b_3_9 + b_1_3·b_3_8 + b_1_3·b_3_6 + b_1_3·b_3_5 + b_1_1·b_3_7 + b_1_1·b_3_5 + b_1_14
       + a_1_2·b_3_8 + a_1_0·b_3_10 + a_1_0·b_3_8 + a_1_0·b_3_5
  15. b_1_1·b_3_6 + a_1_2·b_3_10 + a_1_2·b_3_8 + a_1_2·b_3_7 + a_1_2·b_3_5 + a_1_0·b_3_8
       + a_1_0·b_3_7 + a_1_0·b_3_5
  16. a_1_0·a_1_2·b_3_6 + a_1_02·b_3_5
  17. a_1_02·b_3_7 + a_1_02·b_3_5
  18. a_1_0·a_1_2·b_3_8 + a_1_02·b_3_10 + a_1_02·b_3_8
  19. b_3_6·b_3_7 + b_3_5·b_3_9 + b_3_5·b_3_8 + b_3_5·b_3_7 + b_1_13·b_3_5
  20. b_3_102 + b_3_92 + b_3_82 + b_3_72 + b_3_62 + b_1_36 + b_1_16 + c_4_18·b_1_32
       + c_4_18·b_1_12 + c_4_17·a_1_22 + c_4_17·a_1_02
  21. b_3_8·b_3_10 + b_3_82 + b_3_7·b_3_9 + b_3_7·b_3_8 + b_3_6·b_3_9 + b_3_6·b_3_8
       + b_1_33·b_3_8 + b_1_13·b_3_5 + c_4_17·b_1_12 + c_4_18·a_1_0·b_1_1
       + c_4_17·a_1_2·b_1_1
  22. b_3_8·b_3_10 + b_3_8·b_3_9 + b_3_7·b_3_10 + b_3_7·b_3_9 + b_3_6·b_3_10 + b_3_6·b_3_9
       + b_1_33·b_3_8 + b_1_33·b_3_7 + b_1_33·b_3_5 + b_1_13·b_3_8 + c_4_17·b_1_12
       + c_4_18·a_1_2·b_1_1 + c_4_17·a_1_2·b_1_1 + c_4_17·a_1_0·b_1_1 + c_4_18·a_1_02
  23. b_3_102 + b_3_92 + b_3_8·b_3_10 + b_3_8·b_3_9 + b_3_7·b_3_10 + b_3_7·b_3_9
       + b_3_6·b_3_10 + b_3_6·b_3_9 + b_1_33·b_3_8 + b_1_33·b_3_7 + b_1_33·b_3_5 + b_1_36
       + b_1_13·b_3_8 + b_1_13·b_3_7 + b_1_13·b_3_5 + b_1_16 + c_4_18·a_1_2·b_1_1
       + c_4_17·a_1_2·b_1_1 + c_4_17·a_1_0·b_1_1 + c_4_18·a_1_22 + c_4_17·a_1_22
       + c_4_17·a_1_02
  24. b_3_92 + b_3_72 + b_3_62 + b_3_52 + b_1_33·b_3_8 + b_1_33·b_3_5 + b_1_36
       + b_1_13·b_3_8 + b_1_13·b_3_7 + c_4_19·b_1_32 + c_4_17·b_1_32 + c_4_17·a_1_02
  25. b_3_102 + b_3_92 + b_3_8·b_3_10 + b_3_8·b_3_9 + b_3_82 + b_3_7·b_3_10 + b_3_7·b_3_9
       + b_3_6·b_3_10 + b_3_6·b_3_9 + b_3_52 + b_1_33·b_3_5 + b_1_36 + b_1_13·b_3_7 + b_1_16
       + c_4_19·b_1_12 + c_4_17·b_1_32 + c_4_18·a_1_2·b_1_1 + c_4_17·a_1_2·b_1_1
       + c_4_17·a_1_0·b_1_1
  26. b_3_102 + b_3_8·b_3_10 + b_3_82 + b_3_7·b_3_9 + b_3_7·b_3_8 + b_3_72 + b_3_6·b_3_9
       + b_3_6·b_3_8 + b_3_5·b_3_6 + b_1_33·b_3_8 + b_1_36 + b_1_13·b_3_8 + b_1_13·b_3_5
       + b_1_16 + c_4_18·b_1_32 + c_4_19·a_1_0·b_1_1 + c_4_17·a_1_2·b_1_1
       + c_4_17·a_1_0·b_1_1 + c_4_18·a_1_0·a_1_2 + c_4_17·a_1_22
  27. b_3_9·b_3_10 + b_3_92 + b_3_8·b_3_9 + b_3_7·b_3_10 + b_3_7·b_3_9 + b_3_6·b_3_10
       + b_3_6·b_3_9 + b_3_6·b_3_8 + b_3_5·b_3_10 + b_3_5·b_3_6 + b_1_13·b_3_7 + b_1_13·b_3_5
       + c_4_17·b_1_32 + c_4_17·b_1_12 + c_4_19·a_1_2·b_1_1 + c_4_17·a_1_2·b_1_1
       + c_4_17·a_1_0·b_1_1 + c_4_18·a_1_0·a_1_2 + c_4_17·a_1_0·a_1_2
  28. b_3_102 + b_3_92 + b_3_72 + b_1_33·b_3_8 + b_1_33·b_3_7 + b_1_36 + b_1_13·b_3_5
       + c_4_18·b_1_32 + c_4_17·b_1_32 + c_4_17·b_1_12 + c_4_19·a_1_02 + c_4_17·a_1_02
  29. b_3_102 + b_3_92 + b_3_8·b_3_9 + b_3_7·b_3_10 + b_3_7·b_3_9 + b_3_6·b_3_10 + b_3_6·b_3_8
       + b_3_62 + b_1_33·b_3_8 + b_1_33·b_3_5 + b_1_36 + b_1_13·b_3_5 + b_1_16
       + c_4_17·b_1_32 + c_4_17·b_1_12 + c_4_18·a_1_2·b_1_1 + c_4_19·a_1_0·a_1_2
       + c_4_17·a_1_22 + c_4_17·a_1_0·a_1_2
  30. b_3_92 + b_3_8·b_3_10 + b_3_8·b_3_9 + b_3_7·b_3_10 + b_3_7·b_3_9 + b_3_6·b_3_10
       + b_3_6·b_3_9 + b_3_62 + b_1_33·b_3_5 + b_1_13·b_3_5 + b_1_16 + c_4_17·b_1_32
       + c_4_17·b_1_12 + c_4_18·a_1_2·b_1_1 + c_4_17·a_1_2·b_1_1 + c_4_17·a_1_0·b_1_1
       + c_4_19·a_1_22
  31. a_1_0·b_3_5·b_3_7
  32. b_3_5·b_3_7·b_3_8 + b_1_33·b_3_7·b_3_8 + b_1_36·b_3_7 + b_1_36·b_3_5 + b_1_39
       + b_1_13·b_3_7·b_3_8 + b_1_13·b_3_5·b_3_7 + b_1_16·b_3_7 + b_1_16·b_3_5
       + b_8_67·b_1_3 + c_4_19·b_1_32·b_3_8 + c_4_19·b_1_32·b_3_7 + c_4_19·b_1_32·b_3_5
       + c_4_19·b_1_35 + c_4_19·b_1_12·b_3_5 + c_4_19·b_1_15 + c_4_18·b_1_32·b_3_8
       + c_4_18·b_1_35 + c_4_18·b_1_12·b_3_8 + c_4_18·b_1_12·b_3_7 + c_4_18·b_1_15
       + c_4_17·b_1_12·b_3_8 + c_4_17·b_1_12·b_3_5 + c_4_17·b_1_15 + c_4_19·a_1_22·b_3_10
       + c_4_19·a_1_0·a_1_2·b_3_10 + c_4_19·a_1_02·b_3_9 + c_4_19·a_1_02·b_3_6
       + c_4_18·a_1_22·b_3_8 + c_4_18·a_1_02·b_3_10 + c_4_18·a_1_02·b_3_5
       + c_4_17·a_1_22·b_3_6 + c_4_17·a_1_02·b_3_10 + c_4_17·a_1_02·b_3_9
       + c_4_17·a_1_02·b_3_8
  33. b_3_5·b_3_7·b_3_10 + b_3_5·b_3_7·b_3_8 + b_1_33·b_3_5·b_3_7 + b_1_13·b_3_5·b_3_7
       + b_1_19 + b_8_67·b_1_1 + c_4_19·b_1_12·b_3_8 + c_4_19·b_1_12·b_3_7
       + c_4_18·b_1_12·b_3_7 + c_4_17·b_1_12·b_3_8 + c_4_17·b_1_12·b_3_5 + c_4_17·b_1_15
       + c_4_19·a_1_0·a_1_2·b_3_10 + c_4_19·a_1_02·b_3_10 + c_4_19·a_1_02·b_3_6
       + c_4_18·a_1_22·b_3_6 + c_4_18·a_1_0·a_1_2·b_3_10 + c_4_18·a_1_02·b_3_9
       + c_4_18·a_1_02·b_3_8 + c_4_18·a_1_02·b_3_6 + c_4_17·a_1_22·b_3_10
       + c_4_17·a_1_22·b_3_6 + c_4_17·a_1_0·a_1_2·b_3_10
  34. b_8_67·a_1_0 + c_4_19·a_1_02·b_3_10 + c_4_19·a_1_02·b_3_9 + c_4_19·a_1_02·b_3_8
       + c_4_19·a_1_02·b_3_6 + c_4_18·a_1_22·b_3_10 + c_4_18·a_1_22·b_3_6
       + c_4_18·a_1_0·a_1_2·b_3_10 + c_4_18·a_1_02·b_3_9 + c_4_18·a_1_02·b_3_6
       + c_4_17·a_1_22·b_3_10 + c_4_17·a_1_22·b_3_6 + c_4_17·a_1_0·a_1_2·b_3_10
       + c_4_17·a_1_02·b_3_9 + c_4_17·a_1_02·b_3_6 + c_4_17·a_1_02·b_3_5
  35. b_8_67·a_1_2 + c_4_19·a_1_22·b_3_6 + c_4_19·a_1_0·a_1_2·b_3_10 + c_4_19·a_1_02·b_3_5
       + c_4_18·a_1_22·b_3_6 + c_4_18·a_1_02·b_3_10 + c_4_18·a_1_02·b_3_9
       + c_4_18·a_1_02·b_3_8 + c_4_18·a_1_02·b_3_6 + c_4_17·a_1_22·b_3_8
       + c_4_17·a_1_22·b_3_6 + c_4_17·a_1_0·a_1_2·b_3_10 + c_4_17·a_1_02·b_3_8
       + c_4_17·a_1_02·b_3_6 + c_4_17·a_1_02·b_3_5
  36. b_1_38·b_3_7 + b_1_18·b_3_7 + b_8_67·b_3_10 + b_8_67·b_1_33
       + c_4_19·b_1_3·b_3_7·b_3_8 + c_4_19·b_1_3·b_3_5·b_3_8 + c_4_19·b_1_3·b_3_5·b_3_7
       + c_4_19·b_1_34·b_3_8 + c_4_19·b_1_34·b_3_7 + c_4_19·b_1_34·b_3_5 + c_4_19·b_1_37
       + c_4_19·b_1_1·b_3_7·b_3_8 + c_4_19·b_1_1·b_3_5·b_3_8 + c_4_19·b_1_1·b_3_5·b_3_7
       + c_4_19·b_1_14·b_3_8 + c_4_19·b_1_14·b_3_7 + c_4_19·b_1_14·b_3_5 + c_4_19·b_1_17
       + c_4_18·b_1_3·b_3_5·b_3_7 + c_4_18·b_1_34·b_3_8 + c_4_18·b_1_34·b_3_7
       + c_4_18·b_1_1·b_3_5·b_3_7 + c_4_18·b_1_14·b_3_8 + c_4_18·b_1_14·b_3_7
       + c_4_17·b_1_34·b_3_7 + c_4_17·b_1_34·b_3_5 + c_4_17·b_1_37 + c_4_17·b_1_14·b_3_7
       + c_4_17·b_1_14·b_3_5 + c_4_17·b_1_17 + c_4_18·a_1_0·b_3_7·b_3_10
       + c_4_18·a_1_0·b_3_6·b_3_8 + c_4_18·a_1_0·b_3_5·b_3_9 + c_4_17·a_1_0·b_3_6·b_3_8
       + c_4_17·a_1_0·b_3_5·b_3_8 + c_4_192·b_1_33 + c_4_192·b_1_13 + c_4_182·b_1_33
       + c_4_182·b_1_13 + c_4_17·c_4_19·a_1_22·b_1_1 + c_4_172·a_1_22·b_1_1
       + c_4_18·c_4_19·a_1_23 + c_4_182·a_1_02·a_1_2 + c_4_17·c_4_18·a_1_23
       + c_4_17·c_4_18·a_1_02·a_1_2
  37. b_1_35·b_3_7·b_3_8 + b_1_311 + b_1_15·b_3_5·b_3_7 + b_1_18·b_3_8 + b_1_18·b_3_5
       + b_8_67·b_3_5 + c_4_19·b_1_3·b_3_7·b_3_8 + c_4_19·b_1_3·b_3_5·b_3_8
       + c_4_19·b_1_3·b_3_5·b_3_7 + c_4_19·b_1_34·b_3_8 + c_4_19·b_1_34·b_3_7
       + c_4_19·b_1_14·b_3_8 + c_4_18·b_1_3·b_3_7·b_3_8 + c_4_18·b_1_3·b_3_5·b_3_8
       + c_4_18·b_1_34·b_3_7 + c_4_18·b_1_34·b_3_5 + c_4_18·b_1_1·b_3_5·b_3_8
       + c_4_18·b_1_1·b_3_5·b_3_7 + c_4_18·b_1_14·b_3_7 + c_4_18·b_1_14·b_3_5
       + c_4_18·b_1_17 + c_4_17·b_1_3·b_3_7·b_3_8 + c_4_17·b_1_1·b_3_5·b_3_8
       + c_4_17·b_1_14·b_3_8 + c_4_17·b_1_14·b_3_7 + c_4_17·b_1_14·b_3_5
       + c_4_19·a_1_0·b_3_7·b_3_10 + c_4_19·a_1_0·b_3_6·b_3_8 + c_4_19·a_1_0·b_3_5·b_3_10
       + c_4_18·a_1_0·b_3_7·b_3_10 + c_4_18·a_1_0·b_3_5·b_3_10 + c_4_17·a_1_0·b_3_7·b_3_10
       + c_4_17·a_1_0·b_3_6·b_3_8 + c_4_17·a_1_0·b_3_5·b_3_10 + c_4_17·a_1_0·b_3_5·b_3_8
       + c_4_192·b_1_33 + c_4_192·b_1_13 + c_4_18·c_4_19·b_1_33
       + c_4_18·c_4_19·b_1_13 + c_4_17·c_4_19·b_1_33 + c_4_17·c_4_18·b_1_13
       + c_4_172·b_1_13 + c_4_192·a_1_22·b_1_1 + c_4_182·a_1_22·b_1_1
       + c_4_17·c_4_19·a_1_22·b_1_1 + c_4_17·c_4_18·a_1_22·b_1_1 + c_4_192·a_1_23
       + c_4_192·a_1_02·a_1_2 + c_4_18·c_4_19·a_1_23 + c_4_18·c_4_19·a_1_02·a_1_2
       + c_4_182·a_1_23 + c_4_182·a_1_02·a_1_2 + c_4_172·a_1_23
       + c_4_172·a_1_02·a_1_2
  38. b_1_35·b_3_7·b_3_8 + b_1_311 + b_1_15·b_3_7·b_3_8 + b_1_111 + b_8_67·b_3_6
       + c_4_19·b_1_3·b_3_7·b_3_8 + c_4_19·b_1_3·b_3_5·b_3_8 + c_4_19·b_1_3·b_3_5·b_3_7
       + c_4_19·b_1_34·b_3_8 + c_4_19·b_1_34·b_3_7 + c_4_19·b_1_1·b_3_7·b_3_8
       + c_4_19·b_1_1·b_3_5·b_3_8 + c_4_19·b_1_1·b_3_5·b_3_7 + c_4_19·b_1_14·b_3_8
       + c_4_19·b_1_14·b_3_7 + c_4_18·b_1_3·b_3_7·b_3_8 + c_4_18·b_1_3·b_3_5·b_3_8
       + c_4_18·b_1_34·b_3_7 + c_4_18·b_1_34·b_3_5 + c_4_18·b_1_1·b_3_7·b_3_8
       + c_4_18·b_1_1·b_3_5·b_3_8 + c_4_18·b_1_14·b_3_7 + c_4_18·b_1_14·b_3_5
       + c_4_17·b_1_3·b_3_7·b_3_8 + c_4_17·b_1_1·b_3_7·b_3_8 + c_4_19·a_1_0·b_3_5·b_3_9
       + c_4_18·a_1_0·b_3_7·b_3_10 + c_4_18·a_1_0·b_3_5·b_3_10 + c_4_18·a_1_0·b_3_5·b_3_8
       + c_4_17·a_1_0·b_3_7·b_3_10 + c_4_17·a_1_0·b_3_6·b_3_8 + c_4_17·a_1_0·b_3_5·b_3_10
       + c_4_17·a_1_0·b_3_5·b_3_9 + c_4_17·a_1_0·b_3_5·b_3_8 + c_4_192·b_1_33
       + c_4_192·b_1_13 + c_4_18·c_4_19·b_1_33 + c_4_18·c_4_19·b_1_13
       + c_4_17·c_4_19·b_1_33 + c_4_17·c_4_19·b_1_13 + c_4_18·c_4_19·a_1_22·b_1_1
       + c_4_17·c_4_19·a_1_22·b_1_1 + c_4_17·c_4_18·a_1_22·b_1_1 + c_4_18·c_4_19·a_1_23
       + c_4_18·c_4_19·a_1_02·a_1_2 + c_4_182·a_1_23 + c_4_17·c_4_19·a_1_02·a_1_2
       + c_4_17·c_4_18·a_1_23 + c_4_17·c_4_18·a_1_02·a_1_2 + c_4_172·a_1_02·a_1_2
  39. b_1_311 + b_1_15·b_3_5·b_3_8 + b_1_15·b_3_5·b_3_7 + b_1_18·b_3_8 + b_1_18·b_3_7
       + b_8_67·b_3_7 + b_8_67·b_1_33 + b_8_67·b_1_13 + c_4_19·b_1_3·b_3_7·b_3_8
       + c_4_19·b_1_3·b_3_5·b_3_7 + c_4_19·b_1_34·b_3_7 + c_4_19·b_1_37
       + c_4_19·b_1_1·b_3_5·b_3_7 + c_4_19·b_1_14·b_3_8 + c_4_19·b_1_14·b_3_7
       + c_4_19·b_1_17 + c_4_18·b_1_3·b_3_7·b_3_8 + c_4_18·b_1_3·b_3_5·b_3_8
       + c_4_18·b_1_34·b_3_7 + c_4_18·b_1_34·b_3_5 + c_4_18·b_1_1·b_3_7·b_3_8
       + c_4_18·b_1_14·b_3_8 + c_4_18·b_1_14·b_3_7 + c_4_18·b_1_14·b_3_5
       + c_4_17·b_1_3·b_3_5·b_3_8 + c_4_17·b_1_34·b_3_8 + c_4_17·b_1_1·b_3_7·b_3_8
       + c_4_17·b_1_1·b_3_5·b_3_7 + c_4_17·b_1_14·b_3_8 + c_4_17·b_1_14·b_3_7
       + c_4_17·b_1_14·b_3_5 + c_4_17·b_1_17 + c_4_19·a_1_0·b_3_5·b_3_8
       + c_4_18·a_1_0·b_3_7·b_3_10 + c_4_18·a_1_0·b_3_5·b_3_10 + c_4_18·a_1_0·b_3_5·b_3_8
       + c_4_17·a_1_0·b_3_6·b_3_8 + c_4_17·a_1_0·b_3_5·b_3_8 + c_4_18·c_4_19·b_1_33
       + c_4_182·b_1_13 + c_4_17·c_4_19·b_1_33 + c_4_17·c_4_18·b_1_13
       + c_4_192·a_1_22·b_1_1 + c_4_182·a_1_22·b_1_1 + c_4_17·c_4_18·a_1_22·b_1_1
       + c_4_192·a_1_23 + c_4_18·c_4_19·a_1_02·a_1_2 + c_4_182·a_1_23
       + c_4_17·c_4_19·a_1_23 + c_4_17·c_4_18·a_1_23 + c_4_17·c_4_18·a_1_02·a_1_2
  40. b_1_38·b_3_7 + b_1_15·b_3_7·b_3_8 + b_1_15·b_3_5·b_3_8 + b_1_15·b_3_5·b_3_7
       + b_1_18·b_3_8 + b_1_18·b_3_5 + b_8_67·b_3_8 + c_4_19·b_1_3·b_3_7·b_3_8
       + c_4_19·b_1_3·b_3_5·b_3_8 + c_4_19·b_1_3·b_3_5·b_3_7 + c_4_19·b_1_34·b_3_8
       + c_4_19·b_1_34·b_3_7 + c_4_19·b_1_34·b_3_5 + c_4_19·b_1_37
       + c_4_19·b_1_1·b_3_5·b_3_8 + c_4_19·b_1_1·b_3_5·b_3_7 + c_4_19·b_1_14·b_3_7
       + c_4_18·b_1_3·b_3_5·b_3_7 + c_4_18·b_1_34·b_3_8 + c_4_18·b_1_34·b_3_7
       + c_4_18·b_1_1·b_3_7·b_3_8 + c_4_18·b_1_14·b_3_8 + c_4_18·b_1_14·b_3_5
       + c_4_17·b_1_34·b_3_7 + c_4_17·b_1_34·b_3_5 + c_4_17·b_1_37
       + c_4_17·b_1_1·b_3_5·b_3_8 + c_4_17·b_1_14·b_3_7 + c_4_17·b_1_14·b_3_5
       + c_4_18·a_1_0·b_3_7·b_3_10 + c_4_18·a_1_0·b_3_6·b_3_8 + c_4_18·a_1_0·b_3_5·b_3_9
       + c_4_17·a_1_0·b_3_7·b_3_10 + c_4_17·a_1_0·b_3_5·b_3_9 + c_4_192·b_1_33
       + c_4_192·b_1_13 + c_4_18·c_4_19·b_1_13 + c_4_182·b_1_33 + c_4_182·b_1_13
       + c_4_17·c_4_18·b_1_13 + c_4_192·a_1_22·b_1_1 + c_4_18·c_4_19·a_1_22·b_1_1
       + c_4_182·a_1_22·b_1_1 + c_4_17·c_4_19·a_1_22·b_1_1 + c_4_172·a_1_22·b_1_1
       + c_4_192·a_1_23 + c_4_192·a_1_02·a_1_2 + c_4_18·c_4_19·a_1_23
       + c_4_18·c_4_19·a_1_02·a_1_2 + c_4_182·a_1_23 + c_4_17·c_4_19·a_1_23
       + c_4_17·c_4_19·a_1_02·a_1_2 + c_4_17·c_4_18·a_1_02·a_1_2
  41. b_1_38·b_3_7 + b_1_15·b_3_7·b_3_8 + b_1_18·b_3_7 + b_1_18·b_3_5 + b_1_111
       + b_8_67·b_3_9 + b_8_67·b_1_13 + c_4_19·b_1_3·b_3_7·b_3_8 + c_4_19·b_1_3·b_3_5·b_3_8
       + c_4_19·b_1_3·b_3_5·b_3_7 + c_4_19·b_1_34·b_3_8 + c_4_19·b_1_34·b_3_7
       + c_4_19·b_1_34·b_3_5 + c_4_19·b_1_37 + c_4_19·b_1_1·b_3_7·b_3_8
       + c_4_19·b_1_1·b_3_5·b_3_8 + c_4_19·b_1_1·b_3_5·b_3_7 + c_4_19·b_1_14·b_3_8
       + c_4_19·b_1_14·b_3_7 + c_4_18·b_1_3·b_3_5·b_3_7 + c_4_18·b_1_34·b_3_8
       + c_4_18·b_1_34·b_3_7 + c_4_18·b_1_1·b_3_7·b_3_8 + c_4_18·b_1_1·b_3_5·b_3_8
       + c_4_18·b_1_14·b_3_5 + c_4_17·b_1_34·b_3_7 + c_4_17·b_1_34·b_3_5 + c_4_17·b_1_37
       + c_4_17·b_1_1·b_3_7·b_3_8 + c_4_17·b_1_1·b_3_5·b_3_7 + c_4_17·b_1_17
       + c_4_19·a_1_0·b_3_7·b_3_10 + c_4_19·a_1_0·b_3_6·b_3_8 + c_4_19·a_1_0·b_3_5·b_3_10
       + c_4_18·a_1_0·b_3_7·b_3_10 + c_4_18·a_1_0·b_3_5·b_3_10 + c_4_17·a_1_0·b_3_7·b_3_10
       + c_4_17·a_1_0·b_3_6·b_3_8 + c_4_192·b_1_33 + c_4_192·b_1_13 + c_4_182·b_1_33
       + c_4_17·c_4_19·b_1_13 + c_4_18·c_4_19·a_1_22·b_1_1 + c_4_17·c_4_19·a_1_22·b_1_1
       + c_4_17·c_4_18·a_1_22·b_1_1 + c_4_192·a_1_02·a_1_2 + c_4_18·c_4_19·a_1_23
       + c_4_18·c_4_19·a_1_02·a_1_2 + c_4_182·a_1_23 + c_4_17·c_4_19·a_1_23
       + c_4_17·c_4_19·a_1_02·a_1_2 + c_4_172·a_1_23 + c_4_172·a_1_02·a_1_2
  42. b_1_310·b_3_5·b_3_8 + b_8_67·b_1_32·b_3_7·b_3_8 + b_8_67·b_1_32·b_3_5·b_3_8
       + b_8_67·b_1_35·b_3_7 + b_8_67·b_1_35·b_3_5 + b_8_67·b_1_38
       + b_8_67·b_1_12·b_3_5·b_3_8 + b_8_67·b_1_15·b_3_8 + b_8_67·b_1_15·b_3_5 + b_8_672
       + c_4_19·b_1_39·b_3_8 + c_4_19·b_1_19·b_3_8 + c_4_19·b_1_112
       + c_4_19·b_8_67·b_1_3·b_3_7 + c_4_19·b_8_67·b_1_3·b_3_5 + c_4_19·b_8_67·b_1_34
       + c_4_19·b_8_67·b_1_1·b_3_5 + c_4_19·b_8_67·b_1_14 + c_4_18·b_1_36·b_3_5·b_3_8
       + c_4_18·b_1_39·b_3_8 + c_4_18·b_1_39·b_3_5 + c_4_18·b_1_112
       + c_4_18·b_8_67·b_1_3·b_3_8 + c_4_18·b_8_67·b_1_3·b_3_7 + c_4_18·b_8_67·b_1_3·b_3_5
       + c_4_18·b_8_67·b_1_34 + c_4_18·b_8_67·b_1_1·b_3_8 + c_4_18·b_8_67·b_1_14
       + c_4_17·b_1_36·b_3_5·b_3_8 + c_4_17·b_1_39·b_3_8 + c_4_17·b_1_39·b_3_5
       + c_4_17·b_1_19·b_3_8 + c_4_17·b_1_19·b_3_7 + c_4_17·b_1_19·b_3_5 + c_4_17·b_1_112
       + c_4_17·b_8_67·b_1_3·b_3_8 + c_4_17·b_8_67·b_1_3·b_3_5 + c_4_17·b_8_67·b_1_1·b_3_8
       + c_4_17·b_8_67·b_1_1·b_3_7 + c_4_17·b_8_67·b_1_14 + c_4_192·b_1_32·b_3_5·b_3_8
       + c_4_192·b_1_35·b_3_8 + c_4_192·b_1_35·b_3_7 + c_4_192·b_1_35·b_3_5
       + c_4_192·b_1_38 + c_4_192·b_1_12·b_3_7·b_3_8 + c_4_192·b_1_12·b_3_5·b_3_8
       + c_4_192·b_1_18 + c_4_18·c_4_19·b_1_32·b_3_7·b_3_8
       + c_4_18·c_4_19·b_1_32·b_3_5·b_3_7 + c_4_18·c_4_19·b_1_35·b_3_8
       + c_4_18·c_4_19·b_1_38 + c_4_18·c_4_19·b_1_12·b_3_7·b_3_8
       + c_4_18·c_4_19·b_1_15·b_3_7 + c_4_18·c_4_19·b_1_15·b_3_5
       + c_4_182·b_1_32·b_3_5·b_3_7 + c_4_182·b_1_35·b_3_8 + c_4_182·b_1_35·b_3_5
       + c_4_182·b_1_38 + c_4_182·b_1_12·b_3_7·b_3_8 + c_4_182·b_1_12·b_3_5·b_3_8
       + c_4_182·b_1_15·b_3_8 + c_4_182·b_1_15·b_3_7 + c_4_182·b_1_15·b_3_5
       + c_4_182·b_1_18 + c_4_17·c_4_19·b_1_32·b_3_7·b_3_8
       + c_4_17·c_4_19·b_1_32·b_3_5·b_3_8 + c_4_17·c_4_19·b_1_35·b_3_8
       + c_4_17·c_4_19·b_1_35·b_3_7 + c_4_17·c_4_19·b_1_38
       + c_4_17·c_4_19·b_1_12·b_3_5·b_3_8 + c_4_17·c_4_19·b_1_15·b_3_8
       + c_4_17·c_4_19·b_1_15·b_3_7 + c_4_17·c_4_19·b_1_18
       + c_4_17·c_4_18·b_1_32·b_3_5·b_3_7 + c_4_17·c_4_18·b_1_35·b_3_7
       + c_4_17·c_4_18·b_1_12·b_3_7·b_3_8 + c_4_17·c_4_18·b_1_12·b_3_5·b_3_8
       + c_4_17·c_4_18·b_1_12·b_3_5·b_3_7 + c_4_17·c_4_18·b_1_15·b_3_8
       + c_4_17·c_4_18·b_1_15·b_3_7 + c_4_17·c_4_18·b_1_15·b_3_5 + c_4_17·c_4_18·b_1_18
       + c_4_172·b_1_32·b_3_7·b_3_8 + c_4_172·b_1_12·b_3_7·b_3_8
       + c_4_172·b_1_12·b_3_5·b_3_7 + c_4_172·b_1_15·b_3_8 + c_4_172·b_1_15·b_3_5
       + c_4_172·b_1_18 + c_4_18·c_4_19·a_1_02·b_3_5·b_3_10
       + c_4_18·c_4_19·a_1_02·b_3_5·b_3_8 + c_4_182·a_1_02·b_3_5·b_3_10
       + c_4_182·a_1_02·b_3_5·b_3_8 + c_4_17·c_4_19·a_1_02·b_3_5·b_3_10
       + c_4_17·c_4_18·a_1_02·b_3_5·b_3_8 + c_4_193·b_1_34 + c_4_193·b_1_14
       + c_4_18·c_4_192·b_1_14 + c_4_182·c_4_19·b_1_34 + c_4_183·b_1_34
       + c_4_17·c_4_192·b_1_34 + c_4_17·c_4_192·b_1_14 + c_4_17·c_4_18·c_4_19·b_1_14
       + c_4_17·c_4_182·b_1_34 + c_4_17·c_4_182·b_1_14 + c_4_172·c_4_19·b_1_14
       + c_4_172·c_4_18·b_1_34 + c_4_172·c_4_18·b_1_14

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 16.
  • The completion test was perfect: It applied in the last degree in which a generator or relation was found.
  • The following is a filter regular homogeneous system of parameters:
    1. c_4_17, a Duflot regular element of degree 4
    2. c_4_18, a Duflot regular element of degree 4
    3. c_4_19, 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].

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 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. b_3_5 → 0, an element of degree 3
  6. b_3_6 → 0, an element of degree 3
  7. b_3_7 → 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. c_4_17 → c_1_24 + c_1_04, an element of degree 4
  12. c_4_18 → c_1_04, an element of degree 4
  13. c_4_19 → c_1_24 + c_1_14 + c_1_04, an element of degree 4
  14. b_8_67 → 0, an element of degree 8

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. b_3_5 → 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
  6. b_3_6 → 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
  7. b_3_7 → c_1_33 + c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
  8. b_3_8 → c_1_33 + 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
  9. b_3_9 → c_1_33 + 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
  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. c_4_17 → c_1_34 + c_1_22·c_1_32 + c_1_24 + c_1_1·c_1_33 + c_1_12·c_1_32
       + c_1_02·c_1_32 + c_1_04, an element of degree 4
  12. c_4_18 → c_1_02·c_1_32 + c_1_04, an element of degree 4
  13. c_4_19 → c_1_34 + c_1_2·c_1_33 + 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
  14. b_8_67 → c_1_1·c_1_2·c_1_36 + c_1_1·c_1_24·c_1_33 + c_1_12·c_1_22·c_1_34
       + c_1_12·c_1_24·c_1_32 + c_1_14·c_1_2·c_1_33 + c_1_14·c_1_22·c_1_32
       + c_1_0·c_1_2·c_1_36 + c_1_0·c_1_22·c_1_35 + c_1_0·c_1_1·c_1_2·c_1_35
       + c_1_0·c_1_1·c_1_22·c_1_34 + c_1_0·c_1_12·c_1_35
       + c_1_0·c_1_12·c_1_2·c_1_34 + c_1_0·c_1_12·c_1_22·c_1_33
       + c_1_0·c_1_14·c_1_33 + c_1_02·c_1_1·c_1_35 + c_1_02·c_1_1·c_1_2·c_1_34
       + c_1_02·c_1_1·c_1_22·c_1_33 + c_1_02·c_1_12·c_1_2·c_1_33
       + c_1_02·c_1_12·c_1_22·c_1_32 + c_1_02·c_1_14·c_1_32 + c_1_03·c_1_35
       + c_1_04·c_1_34 + c_1_04·c_1_2·c_1_33 + c_1_04·c_1_22·c_1_32
       + c_1_04·c_1_1·c_1_33 + c_1_04·c_1_12·c_1_32 + c_1_05·c_1_33
       + c_1_06·c_1_32, an element of degree 8

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. b_3_5 → c_1_33 + 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
  6. b_3_6 → 0, an element of degree 3
  7. b_3_7 → c_1_33 + c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
  8. b_3_8 → c_1_33 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  9. b_3_9 → c_1_33 + c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
  10. b_3_10 → c_1_33, an element of degree 3
  11. c_4_17 → c_1_2·c_1_33 + c_1_24 + c_1_1·c_1_33 + c_1_12·c_1_32 + c_1_0·c_1_33 + c_1_04, an element of degree 4
  12. c_4_18 → c_1_34 + c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_1·c_1_33 + c_1_12·c_1_32
       + c_1_02·c_1_32 + c_1_04, an element of degree 4
  13. c_4_19 → c_1_22·c_1_32 + c_1_24 + c_1_12·c_1_32 + c_1_14 + c_1_02·c_1_32 + c_1_04, an element of degree 4
  14. b_8_67 → c_1_2·c_1_37 + c_1_22·c_1_36 + c_1_23·c_1_35 + c_1_24·c_1_34
       + c_1_25·c_1_33 + c_1_26·c_1_32 + c_1_1·c_1_22·c_1_35 + c_1_1·c_1_24·c_1_33
       + c_1_12·c_1_36 + c_1_12·c_1_2·c_1_35 + c_1_12·c_1_24·c_1_32
       + c_1_14·c_1_34 + c_1_14·c_1_2·c_1_33 + c_1_14·c_1_22·c_1_32
       + c_1_0·c_1_2·c_1_36 + c_1_0·c_1_24·c_1_33 + c_1_0·c_1_1·c_1_2·c_1_35
       + c_1_0·c_1_1·c_1_22·c_1_34 + c_1_0·c_1_12·c_1_35
       + c_1_0·c_1_12·c_1_2·c_1_34 + c_1_0·c_1_12·c_1_22·c_1_33
       + c_1_0·c_1_14·c_1_33 + c_1_02·c_1_36 + c_1_02·c_1_22·c_1_34
       + c_1_02·c_1_24·c_1_32 + c_1_02·c_1_1·c_1_35 + c_1_02·c_1_1·c_1_2·c_1_34
       + c_1_02·c_1_1·c_1_22·c_1_33 + c_1_02·c_1_12·c_1_2·c_1_33
       + c_1_02·c_1_12·c_1_22·c_1_32 + c_1_02·c_1_14·c_1_32 + c_1_03·c_1_35
       + c_1_04·c_1_2·c_1_33 + c_1_04·c_1_22·c_1_32 + c_1_04·c_1_1·c_1_33
       + c_1_04·c_1_12·c_1_32 + c_1_05·c_1_33 + c_1_06·c_1_32, an element of degree 8

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 128


Simon A. King David J. Green
Fakultät für Mathematik und Informatik Fakultät für Mathematik und Informatik
Friedrich-Schiller-Universität Jena Friedrich-Schiller-Universität Jena
Ernst-Abbe-Platz 2 Ernst-Abbe-Platz 2
D-07743 Jena D-07743 Jena
Germany Germany
E-mail: simon dot king at uni hyphen jena dot de
Tel: +49 (0)3641 9-46184
Fax: +49 (0)3641 9-46162
Office: Zi. 3524, Ernst-Abbe-Platz 2
E-mail: david dot green at uni hyphen jena dot de
Tel: +49 3641 9-46166
Fax: +49 3641 9-46162
Office: Zi 3512, Ernst-Abbe-Platz 2

Last change: 25.08.2009