Cohomology of group number 395 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 2.
  • Its center has rank 2.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 2 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    t12  +  t11  +  2·t10  +  t9  +  3·t8  +  t7  +  5·t6  +  t5  +  3·t4  +  t3  +  2·t2  +  t  +  1
    (t  −  1)2 · (t2  +  1)2 · (t4  +  1)2
  • The a-invariants are -∞,-∞,-2. 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 21 minimal generators of maximal degree 11:

  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. a_2_4, a nilpotent element of degree 2
  5. a_3_5, a nilpotent element of degree 3
  6. a_4_5, a nilpotent element of degree 4
  7. a_4_6, a nilpotent element of degree 4
  8. a_5_6, a nilpotent element of degree 5
  9. a_5_7, a nilpotent element of degree 5
  10. a_6_6, a nilpotent element of degree 6
  11. a_6_7, a nilpotent element of degree 6
  12. a_6_8, a nilpotent element of degree 6
  13. a_6_9, a nilpotent element of degree 6
  14. a_7_10, a nilpotent element of degree 7
  15. a_7_11, a nilpotent element of degree 7
  16. c_8_10, a Duflot regular element of degree 8
  17. c_8_11, a Duflot regular element of degree 8
  18. a_9_11, a nilpotent element of degree 9
  19. a_9_12, a nilpotent element of degree 9
  20. a_9_13, a nilpotent element of degree 9
  21. a_11_17, a nilpotent element of degree 11

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

Ring relations

There are 171 minimal relations of maximal degree 22:

  1. a_1_0·a_1_1
  2. a_1_0·a_1_2
  3. a_2_4·a_1_1 + a_1_23 + a_1_1·a_1_22 + a_1_12·a_1_2 + a_1_03
  4. a_2_4·a_1_0
  5. a_2_4·a_1_2 + a_1_23 + a_1_13
  6. a_2_42
  7. a_1_1·a_3_5 + a_1_1·a_1_23 + a_1_14
  8. a_1_2·a_3_5 + a_1_12·a_1_22 + a_1_13·a_1_2 + a_1_14
  9. a_1_13·a_1_22 + a_1_14·a_1_2
  10. a_1_12·a_1_23 + a_1_14·a_1_2 + a_1_15
  11. a_2_4·a_3_5
  12. a_4_5·a_1_0
  13. a_4_6·a_1_0 + a_1_02·a_3_5
  14. a_4_6·a_1_2 + a_4_5·a_1_1 + a_1_02·a_3_5 + a_1_14·a_1_2
  15. a_2_4·a_4_5 + a_4_6·a_1_12 + a_4_5·a_1_22
  16. a_2_4·a_4_6 + a_4_5·a_1_22 + a_4_5·a_1_1·a_1_2 + a_4_5·a_1_12
  17. a_1_1·a_5_6 + a_2_4·a_4_5 + a_4_5·a_1_22 + a_4_5·a_1_12
  18. a_1_2·a_5_6 + a_4_5·a_1_1·a_1_2 + a_4_5·a_1_12
  19. a_1_1·a_5_7
  20. a_3_52 + a_1_0·a_5_7 + a_1_0·a_5_6
  21. a_1_2·a_5_7
  22. a_4_5·a_3_5 + a_4_5·a_1_23 + a_4_5·a_1_13
  23. a_4_5·a_3_5 + a_2_4·a_5_6
  24. a_4_6·a_3_5 + a_4_5·a_3_5 + a_1_02·a_5_7 + a_1_02·a_5_6 + a_4_5·a_1_1·a_1_22
       + a_4_5·a_1_12·a_1_2 + a_4_5·a_1_13
  25. a_2_4·a_5_7
  26. a_6_6·a_1_0
  27. a_6_7·a_1_0 + a_1_02·a_5_6
  28. a_6_8·a_1_1 + a_6_6·a_1_2 + a_4_5·a_3_5 + a_1_02·a_5_6
  29. a_6_8·a_1_0 + a_4_6·a_3_5 + a_4_5·a_3_5 + a_1_02·a_5_6 + a_4_5·a_1_1·a_1_22
       + a_4_5·a_1_12·a_1_2 + a_4_5·a_1_13
  30. a_6_8·a_1_2 + a_6_7·a_1_1 + a_4_5·a_3_5 + a_4_5·a_1_12·a_1_2 + a_4_5·a_1_13
  31. a_6_9·a_1_1 + a_6_7·a_1_2 + a_6_6·a_1_2 + a_4_6·a_3_5 + a_4_5·a_1_12·a_1_2
  32. a_6_9·a_1_0 + a_4_6·a_3_5 + a_4_5·a_3_5 + a_1_02·a_5_6 + a_4_5·a_1_1·a_1_22
       + a_4_5·a_1_12·a_1_2 + a_4_5·a_1_13
  33. a_6_9·a_1_2 + a_6_6·a_1_1 + a_4_5·a_1_1·a_1_22 + a_4_5·a_1_12·a_1_2
  34. a_4_52
  35. a_4_5·a_4_6 + a_4_5·a_1_12·a_1_22
  36. a_4_62
  37. a_2_4·a_6_7 + a_6_6·a_1_22 + a_6_6·a_1_1·a_1_2 + a_6_6·a_1_12
       + a_4_5·a_1_12·a_1_22 + a_4_5·a_1_13·a_1_2 + a_4_5·a_1_14
  38. a_2_4·a_6_6 + a_6_7·a_1_1·a_1_2 + a_6_6·a_1_22 + a_6_6·a_1_1·a_1_2
       + a_4_5·a_1_1·a_1_23 + a_4_5·a_1_12·a_1_22 + a_4_5·a_1_13·a_1_2 + a_4_5·a_1_14
  39. a_2_4·a_6_8 + a_2_4·a_6_6 + a_6_6·a_1_22 + a_6_6·a_1_1·a_1_2 + a_6_6·a_1_12
       + a_4_5·a_1_13·a_1_2 + a_4_5·a_1_14
  40. a_2_4·a_6_9 + a_6_6·a_1_22 + a_6_6·a_1_1·a_1_2 + a_6_6·a_1_12 + a_4_5·a_1_1·a_1_23
       + a_4_5·a_1_13·a_1_2 + a_4_5·a_1_14
  41. a_1_1·a_7_10 + a_6_6·a_1_22 + a_6_6·a_1_1·a_1_2 + a_6_6·a_1_12
       + a_4_5·a_1_1·a_1_23 + a_4_5·a_1_14
  42. a_3_5·a_5_6 + a_1_0·a_7_10 + a_4_5·a_1_12·a_1_22 + a_4_5·a_1_13·a_1_2
       + a_4_5·a_1_14
  43. a_1_2·a_7_10 + a_2_4·a_6_6 + a_4_5·a_1_14
  44. a_1_1·a_7_11 + a_2_4·a_6_6 + a_6_6·a_1_12 + a_4_5·a_1_14
  45. a_3_5·a_5_7 + a_3_5·a_5_6 + a_1_0·a_7_11 + a_4_5·a_1_12·a_1_22 + a_4_5·a_1_13·a_1_2
       + a_4_5·a_1_14
  46. a_1_2·a_7_11 + a_2_4·a_6_6 + a_6_6·a_1_22 + a_6_6·a_1_12 + a_4_5·a_1_1·a_1_23
       + a_4_5·a_1_12·a_1_22 + a_4_5·a_1_13·a_1_2 + a_4_5·a_1_14
  47. a_4_5·a_5_6
  48. a_4_5·a_5_7
  49. a_6_6·a_1_1·a_1_22 + a_6_6·a_1_12·a_1_2
  50. a_6_6·a_3_5 + a_6_6·a_1_23 + a_6_6·a_1_13
  51. a_6_7·a_3_5 + a_6_6·a_3_5 + a_4_6·a_5_6 + a_6_6·a_1_13
  52. a_6_8·a_3_5 + a_4_6·a_5_7 + a_6_6·a_1_13
  53. a_6_9·a_3_5 + a_6_6·a_3_5 + a_4_6·a_5_7 + a_6_6·a_1_13
  54. a_4_6·a_5_6 + a_1_02·a_7_10
  55. a_2_4·a_7_10
  56. a_4_6·a_5_7 + a_4_6·a_5_6 + a_1_02·a_7_11
  57. a_6_6·a_3_5 + a_2_4·a_7_11 + a_6_6·a_1_13
  58. a_4_6·a_6_7 + a_4_5·a_6_8 + a_6_6·a_1_13·a_1_2 + a_6_6·a_1_14
  59. a_4_6·a_6_8 + a_4_5·a_6_6 + a_6_6·a_1_13·a_1_2
  60. a_4_6·a_6_6 + a_4_5·a_6_9 + a_6_6·a_1_13·a_1_2
  61. a_4_6·a_6_9 + a_4_5·a_6_7 + a_4_5·a_6_6 + a_6_6·a_1_13·a_1_2
  62. a_5_6·a_5_7 + a_5_62 + a_3_5·a_7_10
  63. a_5_72 + a_5_62 + a_3_5·a_7_11 + a_6_6·a_1_13·a_1_2 + a_6_6·a_1_14
  64. a_5_72 + a_5_62 + c_8_10·a_1_02
  65. a_5_72 + c_8_11·a_1_02
  66. a_1_1·a_9_11 + a_4_6·a_6_7 + a_4_6·a_6_6 + a_6_6·a_1_14 + c_8_10·a_1_1·a_1_2
       + c_8_10·a_1_12
  67. a_1_0·a_9_11
  68. a_1_2·a_9_11 + a_4_5·a_6_7 + a_4_5·a_6_6 + a_6_6·a_1_13·a_1_2 + c_8_10·a_1_22
       + c_8_10·a_1_1·a_1_2
  69. a_1_1·a_9_12 + a_4_6·a_6_6 + a_6_6·a_1_13·a_1_2 + a_6_6·a_1_14 + c_8_10·a_1_12
  70. a_5_72 + a_5_62 + a_1_0·a_9_12
  71. a_1_2·a_9_12 + a_4_5·a_6_6 + a_6_6·a_1_13·a_1_2 + a_6_6·a_1_14 + c_8_10·a_1_1·a_1_2
  72. a_1_1·a_9_13 + a_4_6·a_6_7 + a_6_6·a_1_13·a_1_2 + a_6_6·a_1_14 + c_8_10·a_1_1·a_1_2
       + c_8_10·a_1_12
  73. a_5_72 + a_5_6·a_5_7 + a_1_0·a_9_13
  74. a_1_2·a_9_13 + a_4_5·a_6_7 + c_8_10·a_1_22 + c_8_10·a_1_1·a_1_2
  75. a_6_6·a_5_7
  76. a_6_6·a_5_6 + a_4_5·a_6_7·a_1_2 + a_4_5·a_6_6·a_1_2 + a_4_5·a_6_6·a_1_1
  77. a_6_8·a_5_6 + a_6_7·a_5_7 + a_4_5·a_6_6·a_1_2 + a_4_5·a_6_6·a_1_1
  78. a_6_9·a_5_7 + a_6_8·a_5_7
  79. a_6_9·a_5_6 + a_6_7·a_5_7 + a_6_6·a_5_6 + a_4_5·a_6_7·a_1_1
  80. a_4_5·a_7_10 + a_4_5·a_6_7·a_1_1 + a_4_5·a_6_6·a_1_2 + a_4_5·a_6_6·a_1_1
  81. a_6_7·a_5_7 + a_6_7·a_5_6 + a_6_6·a_5_6 + a_4_6·a_7_10 + a_4_5·a_6_7·a_1_1
  82. a_6_6·a_5_6 + a_4_5·a_7_11 + a_4_5·a_6_7·a_1_1
  83. a_6_8·a_5_7 + a_6_7·a_5_6 + a_6_6·a_5_6 + a_4_6·a_7_11
  84. a_6_8·a_5_7 + a_6_7·a_5_6 + a_4_5·a_6_7·a_1_1 + a_4_5·a_6_6·a_1_2 + c_8_10·a_1_03
  85. a_6_8·a_5_7 + c_8_11·a_1_03
  86. a_6_8·a_5_7 + a_6_7·a_5_6 + a_6_6·a_5_6 + a_2_4·a_9_11 + a_4_5·a_6_6·a_1_2
       + a_4_5·a_6_6·a_1_1 + c_8_10·a_1_1·a_1_22 + c_8_10·a_1_12·a_1_2 + c_8_10·a_1_13
  87. a_6_8·a_5_7 + a_6_7·a_5_6 + a_2_4·a_9_12 + a_4_5·a_6_6·a_1_1 + c_8_10·a_1_23
       + c_8_10·a_1_1·a_1_22 + c_8_10·a_1_12·a_1_2
  88. a_6_8·a_5_7 + a_6_7·a_5_7 + a_1_02·a_9_13
  89. a_6_8·a_5_7 + a_6_7·a_5_6 + a_6_6·a_5_6 + a_2_4·a_9_13 + a_4_5·a_6_7·a_1_1
       + c_8_10·a_1_1·a_1_22 + c_8_10·a_1_12·a_1_2 + c_8_10·a_1_13
  90. a_6_72 + a_6_6·a_6_7 + a_6_62 + a_4_5·a_6_7·a_1_1·a_1_2 + a_4_5·a_6_6·a_1_22
       + a_4_5·a_6_6·a_1_12
  91. a_6_82 + a_6_6·a_6_7 + a_4_5·a_6_7·a_1_1·a_1_2 + a_4_5·a_6_6·a_1_22
       + a_4_5·a_6_6·a_1_12
  92. a_6_8·a_6_9 + a_6_62 + a_4_5·a_6_7·a_1_1·a_1_2 + a_4_5·a_6_6·a_1_1·a_1_2
       + a_4_5·a_6_6·a_1_12
  93. a_6_92 + a_6_6·a_6_7 + a_6_62 + a_4_5·a_6_7·a_1_1·a_1_2 + a_4_5·a_6_6·a_1_22
       + a_4_5·a_6_6·a_1_12
  94. a_6_7·a_6_8 + a_6_6·a_6_9 + a_6_6·a_6_8 + a_4_5·a_6_7·a_1_1·a_1_2
       + a_4_5·a_6_6·a_1_1·a_1_2 + a_4_5·a_6_6·a_1_12
  95. a_6_7·a_6_9 + a_6_6·a_6_8 + a_4_5·a_6_7·a_1_1·a_1_2 + a_4_5·a_6_6·a_1_12
  96. a_5_7·a_7_10 + a_5_6·a_7_11 + a_5_6·a_7_10 + a_4_5·a_6_7·a_1_1·a_1_2
       + a_4_5·a_6_6·a_1_22
  97. a_5_7·a_7_11 + a_5_7·a_7_10 + a_5_6·a_7_10 + a_4_5·a_6_7·a_1_1·a_1_2
       + a_4_5·a_6_6·a_1_22 + a_4_5·a_6_6·a_1_1·a_1_2 + c_8_10·a_1_0·a_3_5
  98. a_6_6·a_6_7 + a_4_5·a_6_7·a_1_1·a_1_2 + a_4_5·a_6_6·a_1_22 + a_4_5·a_6_6·a_1_12
       + c_8_11·a_1_14 + c_8_10·a_1_12·a_1_22 + c_8_10·a_1_14
  99. a_6_7·a_6_8 + a_4_5·a_6_6·a_1_12 + c_8_11·a_1_13·a_1_2 + c_8_10·a_1_1·a_1_23
       + c_8_10·a_1_13·a_1_2
  100. a_6_6·a_6_7 + a_6_62 + a_4_5·a_6_7·a_1_1·a_1_2 + a_4_5·a_6_6·a_1_22
       + a_4_5·a_6_6·a_1_12 + c_8_11·a_1_12·a_1_22 + c_8_10·a_1_14
  101. a_6_7·a_6_8 + a_6_6·a_6_8 + a_4_5·a_6_7·a_1_1·a_1_2 + c_8_11·a_1_1·a_1_23
       + c_8_10·a_1_13·a_1_2
  102. a_5_7·a_7_11 + a_5_7·a_7_10 + c_8_11·a_1_0·a_3_5
  103. a_3_5·a_9_11 + a_4_5·a_6_7·a_1_1·a_1_2 + a_4_5·a_6_6·a_1_12 + c_8_10·a_1_1·a_1_23
       + c_8_10·a_1_12·a_1_22 + c_8_10·a_1_13·a_1_2
  104. a_5_7·a_7_11 + a_5_7·a_7_10 + a_5_6·a_7_10 + a_3_5·a_9_12 + c_8_10·a_1_1·a_1_23
       + c_8_10·a_1_14
  105. a_5_7·a_7_11 + a_3_5·a_9_13 + a_4_5·a_6_6·a_1_22 + a_4_5·a_6_6·a_1_1·a_1_2
       + a_4_5·a_6_6·a_1_12 + c_8_10·a_1_1·a_1_23 + c_8_10·a_1_12·a_1_22
       + c_8_10·a_1_13·a_1_2
  106. a_1_1·a_11_17 + a_6_7·a_6_8 + a_6_62 + a_4_5·a_6_6·a_1_22 + a_4_5·a_6_6·a_1_12
  107. a_5_7·a_7_10 + a_1_0·a_11_17
  108. a_1_2·a_11_17 + a_6_6·a_6_8 + a_6_6·a_6_7 + a_6_62 + a_4_5·a_6_7·a_1_1·a_1_2
       + a_4_5·a_6_6·a_1_1·a_1_2
  109. a_6_9·a_7_10 + a_6_8·a_7_10 + a_6_6·a_7_10 + a_4_5·a_6_6·a_1_23 + a_4_5·a_6_6·a_1_13
  110. a_6_9·a_7_11 + a_6_8·a_7_11
  111. a_6_8·a_7_10 + a_6_7·a_7_11 + a_6_7·a_7_10 + a_6_6·a_7_11
  112. a_6_9·a_7_10 + a_6_8·a_7_11 + a_6_7·a_7_10 + a_4_5·a_6_6·a_1_13 + c_8_10·a_1_02·a_3_5
  113. a_6_6·a_7_11 + a_4_5·a_6_6·a_1_23 + c_8_11·a_1_15 + c_8_10·a_1_14·a_1_2
       + c_8_10·a_1_15
  114. a_6_9·a_7_10 + a_6_8·a_7_10 + a_6_6·a_7_11 + a_4_5·a_6_6·a_1_23 + c_8_11·a_1_14·a_1_2
       + c_8_10·a_1_15
  115. a_6_9·a_7_10 + a_6_8·a_7_11 + a_6_6·a_7_11 + a_4_5·a_6_6·a_1_13 + c_8_11·a_1_02·a_3_5
  116. a_4_5·a_9_11 + a_4_5·a_6_6·a_1_23 + a_4_5·a_6_6·a_1_13 + a_4_5·c_8_10·a_1_2
       + a_4_5·c_8_10·a_1_1
  117. a_6_9·a_7_10 + a_6_8·a_7_11 + a_6_7·a_7_10 + a_4_6·a_9_11 + a_4_5·a_6_6·a_1_13
       + a_4_6·c_8_10·a_1_1 + a_4_5·c_8_10·a_1_1 + c_8_10·a_1_14·a_1_2
  118. a_4_5·a_9_12 + a_4_5·a_6_6·a_1_23 + a_4_5·c_8_10·a_1_1
  119. a_6_9·a_7_10 + a_6_8·a_7_11 + a_6_7·a_7_10 + a_4_6·a_9_12 + a_4_5·a_6_6·a_1_23
       + a_4_5·a_6_6·a_1_13 + a_4_6·c_8_10·a_1_1
  120. a_4_5·a_9_13 + a_4_5·c_8_10·a_1_2 + a_4_5·c_8_10·a_1_1
  121. a_6_9·a_7_10 + a_6_7·a_7_10 + a_4_6·a_9_13 + a_4_5·a_6_6·a_1_13 + a_4_6·c_8_10·a_1_1
       + a_4_5·c_8_10·a_1_1 + c_8_10·a_1_14·a_1_2
  122. a_6_9·a_7_10 + a_6_6·a_7_11 + a_1_02·a_11_17 + a_4_5·a_6_6·a_1_23
       + a_4_5·a_6_6·a_1_13
  123. a_2_4·a_11_17 + a_4_5·a_6_6·a_1_23
  124. a_7_112 + a_7_10·a_7_11 + c_8_10·a_1_0·a_5_7
  125. a_7_10·a_7_11 + c_8_10·a_1_0·a_5_6
  126. a_7_112 + a_7_102 + a_4_5·a_6_6·a_1_14 + c_8_11·a_1_0·a_5_7 + c_8_11·a_1_0·a_5_6
  127. a_5_7·a_9_11
  128. a_5_6·a_9_11 + a_4_6·c_8_10·a_1_12 + a_4_5·c_8_10·a_1_1·a_1_2
  129. a_7_112 + a_7_10·a_7_11 + a_5_7·a_9_12
  130. a_7_10·a_7_11 + a_5_6·a_9_12 + a_4_5·a_6_6·a_1_14 + a_4_6·c_8_10·a_1_12
       + a_4_5·c_8_10·a_1_12
  131. a_7_112 + a_7_102 + a_5_7·a_9_13 + a_4_5·a_6_6·a_1_14
  132. a_7_10·a_7_11 + a_7_102 + a_5_6·a_9_13 + a_4_5·a_6_6·a_1_14 + a_4_6·c_8_10·a_1_12
       + a_4_5·c_8_10·a_1_1·a_1_2
  133. a_7_10·a_7_11 + a_7_102 + a_3_5·a_11_17
  134. a_6_8·a_9_11 + a_6_7·c_8_10·a_1_1 + a_6_6·c_8_10·a_1_2 + c_8_10·a_1_02·a_5_6
       + a_4_5·c_8_11·a_1_1·a_1_22 + a_4_5·c_8_10·a_1_12·a_1_2
  135. a_6_9·a_9_11 + a_6_7·c_8_10·a_1_2 + a_6_6·c_8_10·a_1_2 + a_6_6·c_8_10·a_1_1
       + c_8_10·a_1_02·a_5_7 + c_8_10·a_1_02·a_5_6 + a_4_5·c_8_11·a_1_13
       + a_4_5·c_8_10·a_1_23 + a_4_5·c_8_10·a_1_1·a_1_22 + a_4_5·c_8_10·a_1_12·a_1_2
       + a_4_5·c_8_10·a_1_13
  136. a_6_6·a_9_11 + a_6_6·c_8_10·a_1_2 + a_6_6·c_8_10·a_1_1 + a_4_5·c_8_11·a_1_12·a_1_2
       + a_4_5·c_8_10·a_1_23 + a_4_5·c_8_10·a_1_12·a_1_2
  137. a_6_7·a_9_11 + a_6_7·c_8_10·a_1_2 + a_6_7·c_8_10·a_1_1 + a_4_5·c_8_11·a_1_23
       + a_4_5·c_8_10·a_1_12·a_1_2
  138. a_6_8·a_9_12 + a_6_6·c_8_10·a_1_2 + c_8_10·a_1_02·a_5_7 + c_8_10·a_1_02·a_5_6
       + a_4_5·c_8_11·a_1_1·a_1_22 + a_4_5·c_8_11·a_1_13 + a_4_5·c_8_10·a_1_23
       + a_4_5·c_8_10·a_1_1·a_1_22 + a_4_5·c_8_10·a_1_13
  139. a_6_9·a_9_12 + a_6_7·c_8_10·a_1_2 + a_6_6·c_8_10·a_1_2 + c_8_10·a_1_02·a_5_6
       + a_4_5·c_8_11·a_1_1·a_1_22 + a_4_5·c_8_10·a_1_23 + a_4_5·c_8_10·a_1_1·a_1_22
       + a_4_5·c_8_10·a_1_13
  140. a_6_6·a_9_12 + a_6_6·c_8_10·a_1_1 + a_4_5·c_8_11·a_1_23 + a_4_5·c_8_10·a_1_12·a_1_2
  141. a_6_7·a_9_12 + a_6_7·c_8_10·a_1_1 + c_8_10·a_1_02·a_5_6 + a_4_5·c_8_11·a_1_23
       + a_4_5·c_8_11·a_1_12·a_1_2 + a_4_5·c_8_10·a_1_23
  142. a_6_8·a_9_13 + a_6_7·c_8_10·a_1_1 + a_6_6·c_8_10·a_1_2 + c_8_11·a_1_02·a_5_7
       + c_8_11·a_1_02·a_5_6 + c_8_10·a_1_02·a_5_6 + a_4_5·c_8_11·a_1_13
       + a_4_5·c_8_10·a_1_1·a_1_22 + a_4_5·c_8_10·a_1_12·a_1_2
  143. a_6_9·a_9_13 + a_6_7·c_8_10·a_1_2 + a_6_6·c_8_10·a_1_2 + a_6_6·c_8_10·a_1_1
       + c_8_11·a_1_02·a_5_7 + c_8_11·a_1_02·a_5_6 + c_8_10·a_1_02·a_5_7
       + c_8_10·a_1_02·a_5_6 + a_4_5·c_8_11·a_1_1·a_1_22 + a_4_5·c_8_11·a_1_13
       + a_4_5·c_8_10·a_1_23 + a_4_5·c_8_10·a_1_1·a_1_22 + a_4_5·c_8_10·a_1_12·a_1_2
  144. a_6_6·a_9_13 + a_6_6·c_8_10·a_1_2 + a_6_6·c_8_10·a_1_1 + a_4_5·c_8_11·a_1_23
       + a_4_5·c_8_11·a_1_12·a_1_2 + a_4_5·c_8_10·a_1_23
  145. a_6_7·a_9_13 + a_6_7·c_8_10·a_1_2 + a_6_7·c_8_10·a_1_1 + c_8_11·a_1_02·a_5_7
       + c_8_11·a_1_02·a_5_6 + c_8_10·a_1_02·a_5_7 + a_4_5·c_8_11·a_1_12·a_1_2
       + a_4_5·c_8_10·a_1_23 + a_4_5·c_8_10·a_1_12·a_1_2
  146. a_4_5·a_11_17 + a_4_5·c_8_11·a_1_1·a_1_22 + a_4_5·c_8_11·a_1_12·a_1_2
       + a_4_5·c_8_11·a_1_13 + a_4_5·c_8_10·a_1_23 + a_4_5·c_8_10·a_1_1·a_1_22
       + a_4_5·c_8_10·a_1_12·a_1_2
  147. a_4_6·a_11_17 + c_8_11·a_1_02·a_5_7 + c_8_11·a_1_02·a_5_6 + c_8_10·a_1_02·a_5_7
       + a_4_5·c_8_11·a_1_23 + a_4_5·c_8_11·a_1_13 + a_4_5·c_8_10·a_1_1·a_1_22
       + a_4_5·c_8_10·a_1_12·a_1_2 + a_4_5·c_8_10·a_1_13
  148. a_7_11·a_9_11 + a_6_6·c_8_10·a_1_22 + a_4_5·c_8_11·a_1_1·a_1_23
       + a_4_5·c_8_11·a_1_13·a_1_2 + a_4_5·c_8_11·a_1_14 + a_4_5·c_8_10·a_1_13·a_1_2
       + a_4_5·c_8_10·a_1_14
  149. a_7_10·a_9_11 + a_6_7·c_8_10·a_1_1·a_1_2 + a_6_6·c_8_10·a_1_12
       + a_4_5·c_8_11·a_1_1·a_1_23 + a_4_5·c_8_11·a_1_12·a_1_22
       + a_4_5·c_8_11·a_1_13·a_1_2 + a_4_5·c_8_10·a_1_1·a_1_23
       + a_4_5·c_8_10·a_1_12·a_1_22 + a_4_5·c_8_10·a_1_13·a_1_2
  150. a_7_11·a_9_12 + c_8_10·a_1_0·a_7_11 + a_6_7·c_8_10·a_1_1·a_1_2 + a_6_6·c_8_10·a_1_22
       + a_6_6·c_8_10·a_1_1·a_1_2 + a_6_6·c_8_10·a_1_12 + a_4_5·c_8_11·a_1_12·a_1_22
       + a_4_5·c_8_11·a_1_13·a_1_2 + a_4_5·c_8_10·a_1_12·a_1_22 + a_4_5·c_8_10·a_1_14
  151. a_7_10·a_9_12 + c_8_10·a_1_0·a_7_10 + a_6_6·c_8_10·a_1_22 + a_6_6·c_8_10·a_1_1·a_1_2
       + a_6_6·c_8_10·a_1_12 + a_4_5·c_8_11·a_1_12·a_1_22 + a_4_5·c_8_11·a_1_13·a_1_2
       + a_4_5·c_8_11·a_1_14 + a_4_5·c_8_10·a_1_12·a_1_22 + a_4_5·c_8_10·a_1_13·a_1_2
       + a_4_5·c_8_10·a_1_14
  152. a_7_11·a_9_13 + c_8_10·a_1_0·a_7_11 + c_8_10·a_1_0·a_7_10 + a_6_6·c_8_10·a_1_22
       + a_4_5·c_8_11·a_1_1·a_1_23 + a_4_5·c_8_11·a_1_12·a_1_22 + a_4_5·c_8_11·a_1_14
       + a_4_5·c_8_10·a_1_1·a_1_23
  153. a_7_10·a_9_13 + c_8_11·a_1_0·a_7_11 + c_8_10·a_1_0·a_7_11 + c_8_10·a_1_0·a_7_10
       + a_6_7·c_8_10·a_1_1·a_1_2 + a_6_6·c_8_10·a_1_12 + a_4_5·c_8_11·a_1_1·a_1_23
       + a_4_5·c_8_11·a_1_14
  154. a_5_7·a_11_17 + c_8_11·a_1_0·a_7_10
  155. a_5_6·a_11_17 + c_8_11·a_1_0·a_7_11 + c_8_11·a_1_0·a_7_10 + c_8_10·a_1_0·a_7_11
       + c_8_10·a_1_0·a_7_10 + a_4_5·c_8_11·a_1_1·a_1_23 + a_4_5·c_8_11·a_1_12·a_1_22
       + a_4_5·c_8_11·a_1_13·a_1_2 + a_4_5·c_8_10·a_1_1·a_1_23 + a_4_5·c_8_10·a_1_14
  156. a_6_8·a_11_17 + c_8_11·a_1_02·a_7_10 + a_6_6·c_8_11·a_1_23 + a_6_6·c_8_10·a_1_23
       + a_6_6·c_8_10·a_1_13
  157. a_6_9·a_11_17 + c_8_11·a_1_02·a_7_10 + a_6_6·c_8_11·a_1_23 + a_6_6·c_8_11·a_1_13
       + a_6_6·c_8_10·a_1_13
  158. a_6_6·a_11_17 + a_6_6·c_8_11·a_1_13 + a_6_6·c_8_10·a_1_23
  159. a_6_7·a_11_17 + c_8_11·a_1_02·a_7_11 + c_8_11·a_1_02·a_7_10 + c_8_10·a_1_02·a_7_11
       + c_8_10·a_1_02·a_7_10 + a_6_6·c_8_11·a_1_23 + a_6_6·c_8_11·a_1_13
       + a_6_6·c_8_10·a_1_13
  160. a_9_112 + c_8_102·a_1_22 + c_8_102·a_1_12
  161. a_9_11·a_9_12 + a_4_5·a_6_8·c_8_10 + a_4_5·a_6_6·c_8_10 + a_6_6·c_8_10·a_1_13·a_1_2
       + c_8_102·a_1_1·a_1_2 + c_8_102·a_1_12
  162. a_9_122 + c_8_102·a_1_12 + c_8_102·a_1_02
  163. a_9_11·a_9_13 + a_4_5·a_6_9·c_8_10 + a_4_5·a_6_6·c_8_10 + a_6_6·c_8_10·a_1_13·a_1_2
       + c_8_102·a_1_22 + c_8_102·a_1_12
  164. a_9_132 + c_8_10·c_8_11·a_1_02 + c_8_102·a_1_22 + c_8_102·a_1_12
  165. a_9_12·a_9_13 + c_8_10·a_1_0·a_9_13 + a_4_5·a_6_9·c_8_10 + a_4_5·a_6_8·c_8_10
       + a_4_5·a_6_6·c_8_10 + a_6_6·c_8_10·a_1_13·a_1_2 + c_8_102·a_1_1·a_1_2
       + c_8_102·a_1_12
  166. a_7_11·a_11_17 + c_8_10·a_1_0·a_9_13 + a_6_6·c_8_11·a_1_14
       + a_6_6·c_8_10·a_1_13·a_1_2 + c_8_10·c_8_11·a_1_02
  167. a_7_10·a_11_17 + c_8_11·a_1_0·a_9_13 + c_8_10·a_1_0·a_9_13
  168. a_9_11·a_11_17 + a_4_5·a_6_6·c_8_11·a_1_22 + a_4_5·a_6_6·c_8_11·a_1_1·a_1_2
       + a_4_5·a_6_6·c_8_11·a_1_12 + a_4_5·a_6_6·c_8_10·a_1_22
       + c_8_10·c_8_11·a_1_1·a_1_23 + c_8_10·c_8_11·a_1_14 + c_8_102·a_1_12·a_1_22
       + c_8_102·a_1_13·a_1_2 + c_8_102·a_1_14
  169. a_9_13·a_11_17 + c_8_11·a_1_0·a_11_17 + a_4_5·a_6_7·c_8_11·a_1_1·a_1_2
       + a_4_5·a_6_6·c_8_11·a_1_22 + a_4_5·a_6_6·c_8_11·a_1_1·a_1_2
       + a_4_5·a_6_6·c_8_10·a_1_1·a_1_2 + a_4_5·a_6_6·c_8_10·a_1_12 + c_8_112·a_1_0·a_3_5
       + c_8_10·c_8_11·a_1_0·a_3_5 + c_8_10·c_8_11·a_1_1·a_1_23 + c_8_10·c_8_11·a_1_14
       + c_8_102·a_1_12·a_1_22 + c_8_102·a_1_13·a_1_2 + c_8_102·a_1_14
  170. a_9_12·a_11_17 + c_8_10·a_1_0·a_11_17 + a_4_5·a_6_7·c_8_11·a_1_1·a_1_2
       + a_4_5·a_6_6·c_8_11·a_1_12 + a_4_5·a_6_6·c_8_10·a_1_1·a_1_2
       + a_4_5·a_6_6·c_8_10·a_1_12 + c_8_10·c_8_11·a_1_12·a_1_22
       + c_8_10·c_8_11·a_1_13·a_1_2 + c_8_10·c_8_11·a_1_14 + c_8_102·a_1_1·a_1_23
       + c_8_102·a_1_12·a_1_22 + c_8_102·a_1_13·a_1_2
  171. a_11_172 + a_4_5·a_6_6·c_8_10·a_1_14 + c_8_112·a_1_0·a_5_7 + c_8_112·a_1_0·a_5_6
       + c_8_10·c_8_11·a_1_0·a_5_7 + c_8_10·c_8_11·a_1_0·a_5_6

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 22.
  • The completion test was perfect: It applied in the last degree in which a generator or relation was found.
  • The following is a filter regular homogeneous system of parameters:
    1. c_8_10, a Duflot regular element of degree 8
    2. c_8_11, a Duflot regular element of degree 8
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 14].
  • The filter degree type of any filter regular HSOP is [-1, -2, -2].

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. a_2_4 → 0, an element of degree 2
  5. a_3_5 → 0, an element of degree 3
  6. a_4_5 → 0, an element of degree 4
  7. a_4_6 → 0, an element of degree 4
  8. a_5_6 → 0, an element of degree 5
  9. a_5_7 → 0, an element of degree 5
  10. a_6_6 → 0, an element of degree 6
  11. a_6_7 → 0, an element of degree 6
  12. a_6_8 → 0, an element of degree 6
  13. a_6_9 → 0, an element of degree 6
  14. a_7_10 → 0, an element of degree 7
  15. a_7_11 → 0, an element of degree 7
  16. c_8_10 → c_1_18, an element of degree 8
  17. c_8_11 → c_1_08, an element of degree 8
  18. a_9_11 → 0, an element of degree 9
  19. a_9_12 → 0, an element of degree 9
  20. a_9_13 → 0, an element of degree 9
  21. a_11_17 → 0, an element of degree 11

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