Cohomology of group number 126 of order 128

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

General information on the group

  • The group has 2 minimal generators and exponent 8.
  • It is non-abelian.
  • It has p-Rank 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
    ( − 1) · (t5  −  t4  +  2·t2  −  t  +  1)
    (t  −  1)3 · (t2  +  1) · (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 19 minimal generators of maximal degree 8:

  1. a_1_0, a nilpotent element of degree 1
  2. a_1_1, a nilpotent element of degree 1
  3. a_2_0, a nilpotent element of degree 2
  4. a_2_1, a nilpotent element of degree 2
  5. a_2_2, a nilpotent element of degree 2
  6. c_2_3, a Duflot regular element of degree 2
  7. a_3_3, a nilpotent element of degree 3
  8. a_3_4, a nilpotent element of degree 3
  9. a_3_5, a nilpotent element of degree 3
  10. a_3_6, a nilpotent element of degree 3
  11. a_3_7, a nilpotent element of degree 3
  12. a_4_8, a nilpotent element of degree 4
  13. b_4_9, an element of degree 4
  14. a_5_10, a nilpotent element of degree 5
  15. a_5_12, a nilpotent element of degree 5
  16. a_6_13, a nilpotent element of degree 6
  17. b_6_16, an element of degree 6
  18. a_7_20, a nilpotent element of degree 7
  19. c_8_26, a Duflot regular 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 116 minimal relations of maximal degree 14:

  1. a_1_02
  2. a_1_12
  3. a_1_0·a_1_1
  4. a_2_0·a_1_1
  5. a_2_0·a_1_0
  6. a_2_1·a_1_0
  7. a_2_2·a_1_1
  8. a_2_2·a_1_0 + a_2_1·a_1_1
  9. a_2_02
  10. a_2_22 + a_2_1·a_2_2 + a_2_12 + a_2_0·a_2_1
  11. a_1_1·a_3_3 + a_2_0·a_2_1
  12. a_1_0·a_3_3
  13. a_1_1·a_3_4 + a_2_0·a_2_2 + a_2_0·a_2_1
  14. a_1_0·a_3_4 + a_2_0·a_2_1
  15. a_1_1·a_3_5
  16. a_1_0·a_3_5 + a_2_0·a_2_2
  17. a_1_1·a_3_6 + a_2_22 + a_2_0·a_2_2
  18. a_1_0·a_3_6 + a_2_12 + a_2_0·a_2_1
  19. a_1_0·a_3_7 + a_2_22 + a_2_0·a_2_2
  20. a_2_2·a_3_3
  21. a_2_0·a_3_3
  22. a_2_1·a_3_3 + a_2_1·c_2_3·a_1_1
  23. a_2_0·a_3_4
  24. a_2_2·a_3_5 + a_2_2·a_3_4 + a_2_1·a_3_4
  25. a_2_0·a_3_5 + a_2_1·c_2_3·a_1_1
  26. a_2_2·a_3_4 + a_2_1·a_3_5 + a_2_1·c_2_3·a_1_1
  27. a_2_1·a_3_4 + a_2_0·a_3_6
  28. a_2_1·a_3_6 + a_2_1·a_3_4
  29. a_2_2·a_3_7 + a_2_1·c_2_3·a_1_1
  30. a_2_2·a_3_4 + a_2_1·a_3_4 + a_2_0·a_3_7 + a_2_1·c_2_3·a_1_1
  31. a_2_2·a_3_6 + a_2_1·a_3_7 + a_2_1·a_3_4 + a_2_1·c_2_3·a_1_1
  32. a_4_8·a_1_1 + a_2_2·a_3_4 + a_2_1·a_3_4 + a_2_1·c_2_3·a_1_1
  33. a_4_8·a_1_0 + a_2_1·a_3_4 + a_2_1·c_2_3·a_1_1
  34. b_4_9·a_1_0 + a_2_1·a_3_4
  35. a_3_32
  36. a_3_3·a_3_4 + a_2_0·a_2_2·c_2_3 + a_2_0·a_2_1·c_2_3
  37. a_3_42 + a_2_12·c_2_3
  38. a_3_3·a_3_5 + a_2_0·a_2_1·c_2_3
  39. a_3_52 + a_2_1·a_2_2·c_2_3 + a_2_12·c_2_3 + a_2_0·a_2_1·c_2_3
  40. a_3_4·a_3_5 + a_2_1·a_2_2·c_2_3 + a_2_0·a_2_2·c_2_3 + a_2_0·a_2_1·c_2_3
  41. a_3_3·a_3_6 + a_2_1·a_2_2·c_2_3 + a_2_12·c_2_3 + a_2_0·a_2_2·c_2_3 + a_2_0·a_2_1·c_2_3
  42. a_3_72 + a_3_6·a_3_7 + a_3_62 + a_2_1·a_2_2·c_2_3 + a_2_12·c_2_3 + a_2_0·a_2_2·c_2_3
       + a_2_0·a_2_1·c_2_3
  43. a_3_5·a_3_7 + a_3_4·a_3_6 + a_3_3·a_3_7 + a_2_12·c_2_3 + a_2_0·a_2_2·c_2_3
       + a_2_0·a_2_1·c_2_3
  44. a_3_5·a_3_6 + a_3_4·a_3_7 + a_3_4·a_3_6 + a_2_1·a_2_2·c_2_3 + a_2_12·c_2_3
       + a_2_0·a_2_1·c_2_3
  45. a_3_5·a_3_6 + a_2_2·a_4_8
  46. a_2_0·a_4_8 + a_2_12·c_2_3 + a_2_0·a_2_2·c_2_3 + a_2_0·a_2_1·c_2_3
  47. a_3_4·a_3_6 + a_2_1·a_4_8 + a_2_1·a_2_2·c_2_3 + a_2_0·a_2_1·c_2_3
  48. a_2_2·b_4_9 + a_3_72 + a_3_5·a_3_6
  49. a_2_0·b_4_9 + a_3_4·a_3_6 + a_3_3·a_3_7 + c_2_3·a_1_1·a_3_7
  50. a_2_1·b_4_9 + a_3_62 + a_3_4·a_3_6 + a_2_12·c_2_3 + a_2_0·a_2_1·c_2_3
  51. a_3_4·a_3_6 + a_3_3·a_3_7 + a_1_1·a_5_10 + c_2_3·a_1_1·a_3_7 + a_2_12·c_2_3
       + a_2_0·a_2_2·c_2_3
  52. a_1_0·a_5_10 + a_2_0·a_2_2·c_2_3 + a_2_0·a_2_1·c_2_3
  53. a_3_4·a_3_6 + a_3_3·a_3_7 + a_1_1·a_5_12 + c_2_3·a_1_1·a_3_7 + a_2_1·a_2_2·c_2_3
       + a_2_0·a_2_2·c_2_3
  54. a_1_0·a_5_12 + a_2_12·c_2_3 + a_2_0·a_2_1·c_2_3
  55. a_4_8·a_3_3 + a_2_0·c_2_3·a_3_7
  56. a_4_8·a_3_5 + a_2_1·c_2_3·a_3_7 + a_2_0·c_2_3·a_3_6 + a_2_1·c_2_32·a_1_1
  57. a_4_8·a_3_4 + a_2_0·c_2_3·a_3_7 + a_2_1·c_2_32·a_1_1
  58. b_4_9·a_3_5 + b_4_9·a_3_3 + a_4_8·a_3_7 + a_4_8·a_3_6 + a_2_0·a_2_1·a_3_7
       + a_2_0·c_2_3·a_3_7 + a_2_0·c_2_3·a_3_6 + a_2_1·c_2_32·a_1_1
  59. b_4_9·a_3_4 + a_4_8·a_3_6 + a_2_0·a_2_1·a_3_7 + a_2_1·c_2_3·a_3_7 + a_2_1·c_2_32·a_1_1
  60. a_2_0·a_5_10 + a_2_1·c_2_32·a_1_1
  61. a_2_2·a_5_10 + a_2_1·a_5_10 + a_2_0·c_2_3·a_3_7 + a_2_0·c_2_3·a_3_6
       + a_2_1·c_2_32·a_1_1
  62. a_2_2·a_5_12 + a_2_0·a_2_1·a_3_7 + a_2_1·c_2_3·a_3_7 + a_2_0·c_2_3·a_3_6
  63. a_2_0·a_5_12 + a_2_0·c_2_3·a_3_6
  64. a_2_2·a_5_10 + a_2_1·a_5_12 + a_2_0·a_2_1·a_3_7
  65. a_6_13·a_1_1
  66. a_6_13·a_1_0 + a_2_0·a_2_1·a_3_7
  67. b_6_16·a_1_1 + b_4_9·a_3_3 + a_2_2·a_5_10 + a_2_0·c_2_3·a_3_7 + a_2_0·c_2_3·a_3_6
       + a_2_1·c_2_32·a_1_1
  68. b_6_16·a_1_0 + a_2_0·a_2_1·a_3_7 + a_2_1·c_2_32·a_1_1
  69. a_4_82 + c_2_3·a_3_62 + a_2_1·a_2_2·c_2_32 + a_2_12·c_2_32
       + a_2_0·a_2_1·c_2_32
  70. a_3_5·a_5_10 + a_3_4·a_5_10 + b_4_9·a_1_1·a_3_7 + c_2_3·a_1_1·a_5_10
       + a_2_1·a_2_2·c_2_32 + a_2_0·a_2_1·c_2_32
  71. a_3_3·a_5_12 + a_3_3·a_5_10 + a_2_1·a_2_2·c_2_32 + a_2_12·c_2_32
  72. a_3_7·a_5_12 + a_3_7·a_5_10 + a_3_6·a_5_12 + a_3_5·a_5_10 + a_3_4·a_5_10
       + c_2_3·a_3_6·a_3_7 + c_2_3·a_3_62 + a_2_2·c_2_3·a_4_8 + a_2_1·c_2_3·a_4_8
       + a_2_12·c_2_32 + a_2_0·a_2_2·c_2_32
  73. a_3_5·a_5_12 + a_3_5·a_5_10 + a_3_4·a_5_10 + a_2_2·c_2_3·a_4_8 + a_2_1·a_2_2·c_2_32
  74. a_3_5·a_5_10 + a_3_4·a_5_12 + a_3_3·a_5_10 + a_2_1·c_2_3·a_4_8 + a_2_1·a_2_2·c_2_32
       + a_2_12·c_2_32 + a_2_0·a_2_1·c_2_32
  75. a_3_4·a_5_10 + a_2_2·a_6_13 + a_2_1·a_2_2·c_2_32 + a_2_12·c_2_32
       + a_2_0·a_2_2·c_2_32
  76. a_2_0·a_6_13
  77. a_3_5·a_5_10 + a_3_4·a_5_10 + a_3_3·a_5_10 + a_2_1·a_6_13 + a_2_1·a_2_2·c_2_32
  78. a_2_2·b_6_16 + a_3_6·a_5_10 + a_3_4·a_5_10 + c_2_3·a_3_6·a_3_7 + c_2_3·a_3_62
       + a_2_2·c_2_3·a_4_8 + a_2_1·c_2_3·a_4_8 + a_2_1·a_2_2·c_2_32 + a_2_12·c_2_32
       + a_2_0·a_2_1·c_2_32
  79. a_2_0·b_6_16 + a_3_3·a_5_10
  80. a_2_1·b_6_16 + a_3_7·a_5_12 + a_3_7·a_5_10 + a_3_6·a_5_10 + a_3_3·a_5_10
       + c_2_3·a_3_6·a_3_7 + c_2_3·a_3_62 + a_2_12·c_2_32
  81. a_3_3·a_5_10 + a_1_1·a_7_20 + c_2_32·a_1_1·a_3_7 + a_2_0·a_2_2·c_2_32
       + a_2_0·a_2_1·c_2_32
  82. a_1_0·a_7_20 + a_2_1·a_2_2·c_2_32 + a_2_12·c_2_32 + a_2_0·a_2_1·c_2_32
  83. a_6_13·a_3_3 + a_2_1·a_4_8·a_3_7 + a_2_0·a_2_1·c_2_3·a_3_7
  84. a_6_13·a_3_7 + a_4_8·a_5_12 + a_3_62·a_3_7 + c_2_3·a_4_8·a_3_6
       + a_2_0·a_2_1·c_2_3·a_3_7 + a_2_0·c_2_32·a_3_7 + a_2_0·c_2_32·a_3_6
       + a_2_1·c_2_33·a_1_1
  85. a_6_13·a_3_6 + a_4_8·a_5_12 + a_4_8·a_5_10 + c_2_3·a_4_8·a_3_6 + a_2_1·c_2_3·a_5_10
       + a_2_0·a_2_1·c_2_3·a_3_7 + a_2_1·c_2_32·a_3_7 + a_2_0·c_2_32·a_3_7
       + a_2_0·c_2_32·a_3_6
  86. a_6_13·a_3_5 + a_2_1·c_2_3·a_5_10 + a_2_0·c_2_32·a_3_7
  87. a_6_13·a_3_4 + a_2_0·a_2_1·c_2_3·a_3_7
  88. b_6_16·a_3_3 + b_4_92·a_1_1 + a_3_62·a_3_7 + a_2_1·a_4_8·a_3_7 + a_2_1·c_2_3·a_5_10
       + c_2_32·b_4_9·a_1_1
  89. b_6_16·a_3_6 + b_4_9·a_5_12 + b_4_9·a_5_10 + b_4_92·a_1_1 + a_3_62·a_3_7
       + a_2_1·a_4_8·a_3_7 + c_2_3·b_4_9·a_3_3 + c_2_3·a_4_8·a_3_7 + c_2_3·a_4_8·a_3_6
       + c_2_32·b_4_9·a_1_1 + a_2_1·c_2_32·a_3_7 + a_2_0·c_2_32·a_3_7 + a_2_0·c_2_32·a_3_6
  90. b_6_16·a_3_5 + b_4_92·a_1_1 + a_4_8·a_5_10 + a_3_62·a_3_7 + c_2_3·a_4_8·a_3_7
       + c_2_3·a_4_8·a_3_6 + a_2_1·c_2_3·a_5_10 + a_2_0·a_2_1·c_2_3·a_3_7
       + c_2_32·b_4_9·a_1_1 + a_2_0·c_2_32·a_3_7 + a_2_1·c_2_33·a_1_1
  91. b_6_16·a_3_4 + a_4_8·a_5_12 + a_4_8·a_5_10 + a_2_1·c_2_3·a_5_10
       + a_2_0·a_2_1·c_2_3·a_3_7 + a_2_0·c_2_32·a_3_6
  92. a_2_2·a_7_20 + a_3_62·a_3_7 + a_2_1·c_2_3·a_5_10
  93. a_2_0·a_7_20 + a_2_0·c_2_32·a_3_7 + a_2_1·c_2_33·a_1_1
  94. a_2_1·a_7_20 + a_3_62·a_3_7 + a_2_1·c_2_3·a_5_10 + a_2_1·c_2_32·a_3_7
       + a_2_0·c_2_32·a_3_6
  95. a_5_102 + b_4_9·a_3_6·a_3_7 + a_4_8·a_3_6·a_3_7 + a_2_1·a_2_2·c_2_33
       + a_2_0·a_2_1·c_2_33
  96. a_5_122 + b_4_9·a_3_6·a_3_7 + b_4_9·a_3_62 + a_4_8·a_3_6·a_3_7 + a_4_8·a_3_62
       + c_2_32·a_3_62
  97. a_4_8·a_6_13 + a_4_8·a_3_62 + c_2_3·a_3_6·a_5_12 + c_2_3·a_3_6·a_5_10
       + a_2_2·c_2_3·a_6_13 + c_2_32·a_3_62 + a_2_2·c_2_32·a_4_8 + a_2_1·c_2_32·a_4_8
       + a_2_12·c_2_33 + a_2_0·a_2_2·c_2_33
  98. a_5_10·a_5_12 + b_4_9·a_3_62 + b_4_9·a_1_1·a_5_10 + a_4_8·a_6_13 + c_2_3·a_3_6·a_5_12
       + c_2_3·b_4_9·a_1_1·a_3_7 + a_2_1·c_2_3·a_6_13 + c_2_32·a_3_62 + a_2_2·c_2_32·a_4_8
       + a_2_1·c_2_32·a_4_8 + a_2_12·c_2_33 + a_2_0·a_2_2·c_2_33
  99. a_3_3·a_7_20 + b_4_9·a_1_1·a_5_10 + a_2_1·c_2_32·a_4_8 + c_2_33·a_1_1·a_3_7
       + a_2_1·a_2_2·c_2_33 + a_2_12·c_2_33
  100. b_4_9·a_6_13 + a_4_8·b_6_16 + a_5_10·a_5_12 + a_3_7·a_7_20 + b_4_9·a_3_62
       + b_4_9·a_1_1·a_5_10 + a_4_8·a_3_62 + c_2_3·a_4_8·b_4_9 + c_2_3·a_3_7·a_5_10
       + c_2_32·a_3_6·a_3_7 + c_2_32·a_1_1·a_5_10 + a_2_1·c_2_32·a_4_8
       + a_2_1·a_2_2·c_2_33 + a_2_0·a_2_2·c_2_33
  101. a_3_6·a_7_20 + b_4_9·a_3_6·a_3_7 + b_4_9·a_3_62 + a_4_8·a_3_6·a_3_7 + a_4_8·a_3_62
       + c_2_3·a_3_6·a_5_10 + c_2_32·a_3_6·a_3_7 + a_2_1·c_2_32·a_4_8 + a_2_1·a_2_2·c_2_33
       + a_2_12·c_2_33
  102. a_5_10·a_5_12 + a_3_5·a_7_20 + b_4_9·a_3_62 + a_4_8·a_3_6·a_3_7 + a_4_8·a_3_62
       + c_2_3·a_3_6·a_5_10 + c_2_3·b_4_9·a_1_1·a_3_7 + c_2_33·a_1_1·a_3_7
       + a_2_1·a_2_2·c_2_33 + a_2_12·c_2_33 + a_2_0·a_2_2·c_2_33
  103. a_3_4·a_7_20 + a_4_8·a_6_13 + a_4_8·a_3_6·a_3_7 + c_2_3·a_3_6·a_5_12
       + c_2_3·a_3_6·a_5_10 + c_2_32·a_3_62 + a_2_1·a_2_2·c_2_33 + a_2_0·a_2_1·c_2_33
  104. a_6_13·a_5_12 + a_4_8·b_4_9·a_3_7 + a_3_62·a_5_10 + c_2_3·a_4_8·a_5_12
       + c_2_3·a_4_8·a_5_10 + c_2_3·a_3_62·a_3_7 + c_2_32·a_4_8·a_3_6 + a_2_1·c_2_32·a_5_10
       + a_2_1·c_2_33·a_3_7 + a_2_0·c_2_33·a_3_7 + a_2_0·c_2_33·a_3_6
  105. a_6_13·a_5_10 + a_4_8·b_4_9·a_3_7 + a_4_8·b_4_9·a_3_6 + a_3_62·a_5_10
       + a_2_1·a_4_8·a_5_10 + a_2_1·c_2_3·a_4_8·a_3_7 + a_2_1·c_2_32·a_5_10
       + a_2_0·c_2_33·a_3_7
  106. b_6_16·a_5_12 + b_6_16·a_5_10 + b_4_92·a_3_6 + b_4_92·a_3_3 + a_4_8·b_4_9·a_3_6
       + c_2_3·b_4_92·a_1_1 + c_2_3·a_4_8·a_5_10 + c_2_3·a_3_62·a_3_7 + c_2_32·b_4_9·a_3_6
       + c_2_32·b_4_9·a_3_3 + c_2_32·a_4_8·a_3_7 + c_2_32·a_4_8·a_3_6
       + a_2_1·c_2_32·a_5_10 + c_2_33·b_4_9·a_1_1 + a_2_1·c_2_33·a_3_7
       + a_2_0·c_2_33·a_3_7 + a_2_1·c_2_34·a_1_1
  107. a_4_8·a_7_20 + a_4_8·b_4_9·a_3_7 + a_4_8·b_4_9·a_3_6 + a_2_1·a_4_8·a_5_10
       + c_2_3·a_4_8·a_5_10 + c_2_3·a_3_62·a_3_7 + c_2_32·a_4_8·a_3_7 + a_2_0·c_2_33·a_3_7
       + a_2_0·c_2_33·a_3_6
  108. b_6_16·a_5_12 + b_4_9·a_7_20 + b_4_92·a_3_6 + b_4_92·a_3_3 + a_4_8·b_4_9·a_3_6
       + a_3_62·a_5_10 + c_2_3·b_4_92·a_1_1 + c_2_3·a_4_8·a_5_12 + a_2_1·c_8_26·a_1_1
       + c_2_32·b_4_9·a_3_7 + c_2_32·b_4_9·a_3_6 + c_2_32·b_4_9·a_3_3 + c_2_32·a_4_8·a_3_7
       + c_2_32·a_4_8·a_3_6 + a_2_1·c_2_32·a_5_10 + c_2_33·b_4_9·a_1_1
       + a_2_1·c_2_34·a_1_1
  109. a_6_132 + c_2_3·b_4_9·a_3_62 + c_2_3·a_4_8·a_3_62
  110. a_6_13·b_6_16 + a_4_8·b_4_92 + a_5_12·a_7_20 + b_4_9·a_3_7·a_5_10 + b_4_9·a_3_6·a_5_12
       + b_4_9·a_3_6·a_5_10 + b_4_92·a_1_1·a_3_7 + c_2_3·a_4_8·b_6_16 + c_2_3·a_3_7·a_7_20
       + c_2_3·b_4_9·a_3_62 + c_2_3·b_4_9·a_1_1·a_5_10 + c_2_32·a_4_8·b_4_9
       + c_2_32·a_3_6·a_5_10 + a_2_1·c_2_32·a_6_13 + a_2_1·a_2_2·c_2_34 + a_2_12·c_2_34
  111. a_5_12·a_7_20 + a_5_10·a_7_20 + b_4_9·a_3_6·a_5_10 + b_4_92·a_1_1·a_3_7
       + a_4_8·a_3_6·a_5_12 + c_2_3·a_4_8·a_3_6·a_3_7 + c_2_32·a_3_6·a_5_12
       + c_2_32·a_3_6·a_5_10 + a_2_2·c_2_32·a_6_13 + c_2_33·a_3_6·a_3_7 + c_2_33·a_3_62
       + c_2_33·a_1_1·a_5_10 + a_2_2·c_2_33·a_4_8 + a_2_12·c_2_34 + a_2_0·a_2_1·c_2_34
  112. a_5_12·a_7_20 + b_4_9·a_3_6·a_5_12 + b_4_9·a_3_6·a_5_10 + b_4_92·a_1_1·a_3_7
       + c_2_3·b_4_9·a_3_6·a_3_7 + c_2_3·b_4_9·a_3_62 + a_2_0·a_2_1·c_8_26
       + c_2_32·a_3_7·a_5_10 + c_2_32·a_3_6·a_5_12 + c_2_32·a_3_6·a_5_10
       + c_2_33·a_3_6·a_3_7 + c_2_33·a_3_62 + c_2_33·a_1_1·a_5_10 + a_2_2·c_2_33·a_4_8
  113. b_6_162 + b_4_93 + a_4_8·b_4_92 + b_4_9·a_3_6·a_5_12 + b_4_9·a_3_6·a_5_10
       + c_2_3·b_4_9·a_3_6·a_3_7 + c_2_3·b_4_9·a_3_62 + a_2_12·c_8_26 + c_2_32·b_4_92
       + c_2_33·a_3_62 + a_2_1·a_2_2·c_2_34 + a_2_12·c_2_34 + a_2_0·a_2_1·c_2_34
  114. a_6_13·a_7_20 + a_4_8·b_4_9·a_5_10 + b_4_9·a_3_62·a_3_7 + c_2_3·a_4_8·b_4_9·a_3_7
       + c_2_3·a_4_8·b_4_9·a_3_6 + c_2_3·a_3_62·a_5_10 + c_2_32·a_4_8·a_5_12
       + c_2_33·a_4_8·a_3_6 + a_2_1·c_2_33·a_5_10 + a_2_0·a_2_1·c_2_33·a_3_7
       + a_2_0·c_2_34·a_3_6 + a_2_1·c_2_35·a_1_1
  115. b_6_16·a_7_20 + b_4_92·a_5_10 + a_4_8·b_4_9·a_5_10 + c_2_3·a_4_8·b_4_9·a_3_6
       + a_2_1·c_2_3·a_4_8·a_5_10 + c_2_32·b_6_16·a_3_7 + c_2_32·b_4_9·a_5_10
       + a_2_1·c_2_3·c_8_26·a_1_1 + c_2_32·a_3_62·a_3_7 + c_2_33·a_4_8·a_3_6
       + a_2_1·c_2_33·a_5_10 + a_2_0·c_2_34·a_3_7
  116. a_7_202 + b_4_92·a_3_6·a_3_7 + c_2_32·b_4_9·a_3_6·a_3_7
       + c_2_32·a_4_8·a_3_6·a_3_7 + c_2_34·a_3_6·a_3_7 + c_2_34·a_3_62
       + a_2_0·a_2_2·c_2_35

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 14.
  • 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_2_3, a Duflot regular element of degree 2
    2. c_8_26, a Duflot regular element of degree 8
    3. b_4_9, an element of degree 4
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 7, 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_2_0 → 0, an element of degree 2
  4. a_2_1 → 0, an element of degree 2
  5. a_2_2 → 0, an element of degree 2
  6. c_2_3 → c_1_12, an element of degree 2
  7. a_3_3 → 0, an element of degree 3
  8. a_3_4 → 0, an element of degree 3
  9. a_3_5 → 0, an element of degree 3
  10. a_3_6 → 0, an element of degree 3
  11. a_3_7 → 0, an element of degree 3
  12. a_4_8 → 0, an element of degree 4
  13. b_4_9 → 0, an element of degree 4
  14. a_5_10 → 0, an element of degree 5
  15. a_5_12 → 0, an element of degree 5
  16. a_6_13 → 0, an element of degree 6
  17. b_6_16 → 0, an element of degree 6
  18. a_7_20 → 0, an element of degree 7
  19. c_8_26 → c_1_08, an element of degree 8

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_2_0 → 0, an element of degree 2
  4. a_2_1 → 0, an element of degree 2
  5. a_2_2 → 0, an element of degree 2
  6. c_2_3 → c_1_22 + c_1_12, an element of degree 2
  7. a_3_3 → 0, an element of degree 3
  8. a_3_4 → 0, an element of degree 3
  9. a_3_5 → 0, an element of degree 3
  10. a_3_6 → 0, an element of degree 3
  11. a_3_7 → 0, an element of degree 3
  12. a_4_8 → 0, an element of degree 4
  13. b_4_9 → c_1_24, an element of degree 4
  14. a_5_10 → 0, an element of degree 5
  15. a_5_12 → 0, an element of degree 5
  16. a_6_13 → 0, an element of degree 6
  17. b_6_16 → c_1_12·c_1_24, an element of degree 6
  18. a_7_20 → 0, an element of degree 7
  19. c_8_26 → c_1_14·c_1_24 + c_1_04·c_1_24 + c_1_08, 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