Cohomology of group number 16 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 4.
  • Its center has rank 2.
  • It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 4.

Structure of the cohomology ring

General information

  • The cohomology ring is of dimension 4 and depth 2.
  • The depth coincides with the Duflot bound.
  • The Poincaré series is
    (2) · (t2  −  1/2·t  +  1/2)
    (t  +  1) · (t  −  1)4 · (t2  +  1)2
  • The a-invariants are -∞,-∞,-4,-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 22 minimal generators of maximal degree 6:

  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_2, a nilpotent element of degree 2
  5. b_2_1, an element of degree 2
  6. b_2_3, an element of degree 2
  7. a_3_2, a nilpotent element of degree 3
  8. a_3_3, 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. b_3_4, an element of degree 3
  12. b_3_7, an element of degree 3
  13. a_4_6, a nilpotent element of degree 4
  14. b_4_7, an element of degree 4
  15. b_4_10, an element of degree 4
  16. b_4_11, an element of degree 4
  17. c_4_12, a Duflot regular element of degree 4
  18. c_4_13, a Duflot regular element of degree 4
  19. a_5_15, a nilpotent element of degree 5
  20. a_5_12, a nilpotent element of degree 5
  21. b_5_18, an element of degree 5
  22. b_6_26, an element of degree 6

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

Ring relations

There are 169 minimal relations of maximal degree 12:

  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_2·a_1_1
  7. a_2_2·a_1_0
  8. b_2_1·a_1_1
  9. b_2_3·a_1_0
  10. a_2_02
  11. a_2_0·b_2_1 + a_2_22
  12. a_1_1·a_3_2
  13. a_1_0·a_3_2
  14. a_1_1·a_3_3 + a_2_0·a_2_2
  15. a_1_0·a_3_3
  16. a_1_1·a_3_5
  17. a_1_0·a_3_5 + a_2_0·a_2_2
  18. a_2_0·b_2_3 + a_1_1·a_3_6
  19. a_1_0·a_3_6
  20. a_1_1·b_3_4 + a_2_0·a_2_2
  21. a_1_0·b_3_4 + a_2_22
  22. a_1_1·b_3_7 + a_2_0·b_2_3
  23. a_1_0·b_3_7
  24. a_2_0·a_3_2
  25. b_2_1·a_3_3 + a_2_2·a_3_2
  26. a_2_2·a_3_3
  27. a_2_0·a_3_3
  28. a_2_2·a_3_5
  29. a_2_0·a_3_5
  30. b_2_3·a_3_2 + b_2_1·a_3_6
  31. a_2_2·a_3_6
  32. a_2_0·a_3_6
  33. b_2_1·a_3_5 + a_2_2·b_3_4
  34. a_2_0·b_3_4
  35. b_2_3·a_3_2 + a_2_2·b_3_7
  36. a_2_0·b_3_7
  37. a_4_6·a_1_1
  38. a_4_6·a_1_0
  39. b_4_7·a_1_1
  40. b_4_7·a_1_0
  41. b_4_10·a_1_1
  42. b_4_10·a_1_0 + a_2_2·a_3_2
  43. b_4_11·a_1_1 + b_2_3·a_3_3
  44. b_4_11·a_1_0
  45. a_3_22 + a_2_22·b_2_1
  46. a_3_2·a_3_3
  47. a_3_32
  48. a_3_52
  49. a_3_3·a_3_5
  50. a_3_2·a_3_6
  51. a_3_62
  52. a_3_5·a_3_6
  53. a_3_5·b_3_4 + a_2_2·b_2_1·b_2_3
  54. a_3_3·b_3_4 + a_3_2·a_3_5
  55. b_3_42 + b_2_12·b_2_3
  56. a_3_2·b_3_7 + a_2_2·b_2_1·b_2_3
  57. a_3_6·b_3_7 + a_2_2·b_2_32
  58. a_3_6·b_3_4 + a_3_5·b_3_7
  59. a_3_3·b_3_7 + a_3_3·a_3_6
  60. b_3_72 + b_2_1·b_2_32
  61. a_3_2·b_3_4 + b_2_1·a_4_6 + a_2_2·b_2_1·b_2_3
  62. a_3_6·b_3_4 + b_2_3·a_4_6 + a_2_2·b_2_32 + a_3_3·a_3_6
  63. a_3_2·a_3_5 + a_2_2·a_4_6
  64. a_2_0·a_4_6
  65. b_3_4·b_3_7 + b_2_1·b_4_7 + a_3_6·b_3_4 + a_3_2·b_3_4
  66. a_3_6·b_3_4 + a_2_2·b_4_7 + a_3_2·a_3_5
  67. a_2_0·b_4_7
  68. a_3_2·b_3_4 + a_2_2·b_4_10 + a_2_2·b_2_1·b_2_3 + a_3_2·a_3_5 + a_2_22·b_2_1
  69. a_2_0·b_4_10 + a_3_2·a_3_5
  70. b_3_4·b_3_7 + b_2_3·b_4_10 + b_2_1·b_2_32 + a_2_2·b_4_11 + a_2_2·b_2_1·b_2_3
       + a_3_3·a_3_6 + b_2_3·a_1_1·a_3_6
  71. a_2_0·b_4_11 + a_3_3·a_3_6
  72. a_1_1·a_5_15
  73. a_3_2·a_3_5 + a_1_0·a_5_15
  74. a_3_3·a_3_6 + a_1_1·a_5_12
  75. a_1_0·a_5_12
  76. a_1_1·b_5_18
  77. a_1_0·b_5_18 + a_3_2·a_3_5 + a_2_22·b_2_1
  78. a_4_6·a_3_2
  79. a_4_6·a_3_6
  80. a_4_6·a_3_5
  81. a_4_6·a_3_3
  82. a_4_6·b_3_4 + a_2_2·b_2_3·b_3_4 + a_2_2·b_2_1·b_3_7
  83. a_4_6·b_3_7 + a_2_2·b_2_3·b_3_7 + a_2_2·b_2_3·b_3_4
  84. b_4_7·a_3_2 + a_2_2·b_2_3·b_3_4
  85. b_4_7·a_3_6 + b_2_32·a_3_5
  86. b_4_7·a_3_5 + a_2_2·b_2_3·b_3_7
  87. b_4_7·a_3_3
  88. b_4_7·b_3_4 + b_2_1·b_2_3·b_3_7 + a_2_2·b_2_3·b_3_7 + a_2_2·b_2_1·b_3_7
  89. b_4_7·b_3_7 + b_2_32·b_3_4 + b_2_32·a_3_5 + a_2_2·b_2_3·b_3_4
  90. b_4_10·a_3_2 + a_2_2·b_2_1·b_3_7 + a_2_2·b_2_1·b_3_4 + a_2_2·b_2_1·a_3_2
  91. b_4_10·a_3_6 + a_2_2·b_2_3·b_3_7 + a_2_2·b_2_3·b_3_4
  92. b_4_10·a_3_5 + a_2_2·b_2_3·b_3_4 + a_2_2·b_2_1·b_3_7
  93. b_4_10·a_3_3
  94. b_4_10·b_3_7 + b_2_1·b_2_3·b_3_7 + b_2_1·b_2_3·b_3_4 + b_4_11·a_3_2 + a_2_2·b_2_1·b_3_7
  95. b_4_11·a_3_3 + b_2_32·a_3_3 + b_2_3·c_4_12·a_1_1
  96. b_4_10·b_3_4 + b_2_1·b_2_3·b_3_4 + b_2_12·b_3_7 + b_2_1·a_5_15 + b_2_12·a_3_2
       + a_2_2·b_2_3·b_3_4 + a_2_2·b_2_1·b_3_7 + a_2_2·b_2_1·a_3_2 + b_2_1·c_4_12·a_1_0
  97. b_4_11·a_3_5 + b_2_3·a_5_15 + b_2_32·a_3_5 + b_2_32·a_3_3 + a_2_2·b_2_3·b_3_7
       + a_2_2·b_2_3·b_3_4 + a_2_2·b_2_1·b_3_7 + b_2_3·c_4_12·a_1_1
  98. a_2_2·a_5_15 + a_2_2·b_2_1·a_3_2
  99. a_2_0·a_5_15
  100. b_4_10·b_3_7 + b_2_1·b_2_3·b_3_7 + b_2_1·b_2_3·b_3_4 + b_2_1·a_5_12 + b_2_12·a_3_2
       + a_2_2·b_2_3·b_3_4 + b_2_1·c_4_13·a_1_0 + b_2_1·c_4_12·a_1_0
  101. b_4_11·a_3_6 + b_2_3·a_5_12 + b_2_32·a_3_5 + a_2_2·b_2_3·b_3_7 + a_2_2·b_2_1·b_3_7
       + b_2_3·c_4_13·a_1_1
  102. a_2_2·a_5_12 + a_2_2·b_2_1·a_3_2
  103. a_2_0·a_5_12
  104. b_4_11·b_3_7 + b_2_3·b_5_18 + b_2_32·b_3_7 + b_2_32·b_3_4 + b_2_1·b_2_3·b_3_4
       + b_4_11·a_3_6 + b_4_11·a_3_5 + b_2_32·a_3_6 + b_2_32·a_3_5 + a_2_2·b_2_3·b_3_7
       + a_2_2·b_2_3·b_3_4 + b_2_3·c_4_13·a_1_1 + b_2_3·c_4_12·a_1_1
  105. b_4_10·b_3_7 + b_2_1·b_2_3·b_3_7 + b_2_1·b_2_3·b_3_4 + a_2_2·b_5_18 + a_2_2·b_2_3·b_3_7
       + a_2_2·b_2_3·b_3_4 + a_2_2·b_2_1·b_3_7 + a_2_2·b_2_1·b_3_4
  106. a_2_0·b_5_18
  107. b_6_26·a_1_1
  108. b_6_26·a_1_0 + a_2_2·b_2_1·a_3_2 + b_2_1·c_4_13·a_1_0
  109. a_4_62
  110. a_4_6·b_4_7 + a_2_2·b_2_3·b_4_7 + a_2_2·b_2_1·b_2_32
  111. b_4_72 + b_2_1·b_2_33
  112. b_4_102 + b_2_12·b_2_32 + b_2_13·b_2_3 + a_2_22·b_2_12 + a_2_22·c_4_12
  113. a_4_6·b_4_10 + a_2_2·b_2_1·b_2_32 + a_2_2·b_2_12·b_2_3 + a_2_22·b_4_10
       + a_2_0·a_2_2·c_4_12
  114. b_4_112 + b_4_7·b_4_10 + b_2_32·b_4_11 + b_2_32·b_4_7 + b_2_12·b_2_32
       + a_4_6·b_4_11 + a_2_2·b_2_3·b_4_11 + a_2_2·b_2_3·b_4_7 + a_2_2·b_2_33
       + a_2_2·b_2_12·b_2_3 + b_2_32·a_1_1·a_3_6 + a_2_22·b_4_10 + b_2_32·c_4_12
       + c_4_12·a_1_1·a_3_6 + a_2_22·c_4_13 + a_2_0·a_2_2·c_4_12
  115. a_3_2·a_5_15 + a_2_22·b_4_10 + a_2_22·b_2_12 + a_2_0·a_2_2·c_4_12
  116. b_4_112 + b_2_32·b_4_11 + b_2_32·b_4_7 + b_2_1·b_2_3·b_4_7 + a_2_2·b_2_3·b_4_7
       + a_2_2·b_2_33 + a_2_2·b_2_1·b_4_7 + a_2_2·b_2_1·b_2_32 + a_3_6·a_5_15
       + b_2_32·a_1_1·a_3_6 + a_2_22·b_4_10 + b_2_32·c_4_12 + c_4_12·a_1_1·a_3_6
       + a_2_22·c_4_13
  117. a_3_5·a_5_15 + a_2_22·b_4_10 + a_2_0·a_2_2·c_4_12
  118. a_3_3·a_5_15 + a_2_0·a_2_2·c_4_12
  119. b_3_4·a_5_15 + a_2_2·b_2_1·b_4_11 + a_2_2·b_2_1·b_4_10 + a_2_2·b_2_1·b_4_7
       + a_2_2·b_2_1·b_2_32 + a_2_22·b_4_10 + a_2_22·b_2_12 + a_2_22·c_4_12
       + a_2_0·a_2_2·c_4_12
  120. b_4_112 + b_4_7·b_4_10 + b_2_32·b_4_11 + b_2_32·b_4_7 + b_2_12·b_2_32
       + b_3_7·a_5_15 + a_2_2·b_2_33 + a_2_2·b_2_1·b_4_7 + a_2_2·b_2_1·b_2_32
       + b_2_32·a_1_1·a_3_6 + b_2_32·c_4_12 + c_4_12·a_1_1·a_3_6 + a_2_22·c_4_13
       + a_2_0·a_2_2·c_4_12
  121. a_3_2·a_5_12 + a_2_22·b_2_12
  122. b_4_112 + b_2_32·b_4_11 + b_2_32·b_4_7 + b_2_1·b_2_3·b_4_7 + a_2_2·b_2_3·b_4_7
       + a_2_2·b_2_33 + a_2_2·b_2_1·b_4_7 + a_2_2·b_2_1·b_2_32 + b_2_3·a_1_1·a_5_12
       + b_2_32·a_1_1·a_3_6 + a_2_22·b_4_10 + b_2_32·c_4_12 + a_2_22·c_4_13
  123. a_3_6·a_5_12 + c_4_13·a_1_1·a_3_6 + a_2_0·a_2_2·c_4_13
  124. a_3_5·a_5_12 + a_2_22·b_4_10 + a_2_0·a_2_2·c_4_13 + a_2_0·a_2_2·c_4_12
  125. b_4_112 + b_2_32·b_4_11 + b_2_32·b_4_7 + b_2_1·b_2_3·b_4_7 + a_2_2·b_2_3·b_4_7
       + a_2_2·b_2_33 + a_2_2·b_2_1·b_4_7 + a_2_2·b_2_1·b_2_32 + a_3_3·a_5_12
       + b_2_32·a_1_1·a_3_6 + a_2_22·b_4_10 + b_2_32·c_4_12 + c_4_12·a_1_1·a_3_6
       + a_2_22·c_4_13 + a_2_0·a_2_2·c_4_13
  126. b_4_7·b_4_10 + b_2_1·b_2_3·b_4_7 + b_2_12·b_2_32 + b_3_4·a_5_12 + a_2_2·b_2_1·b_4_10
       + a_2_22·b_2_12 + a_2_22·c_4_13 + a_2_22·c_4_12 + a_2_0·a_2_2·c_4_13
       + a_2_0·a_2_2·c_4_12
  127. b_3_7·a_5_12 + a_2_2·b_2_3·b_4_11 + a_2_2·b_2_3·b_4_7 + a_2_2·b_2_1·b_2_32
       + a_2_2·b_2_12·b_2_3 + c_4_13·a_1_1·a_3_6 + a_2_0·a_2_2·c_4_13
  128. a_3_2·b_5_18 + a_2_2·b_2_1·b_4_11 + a_2_2·b_2_1·b_4_10 + a_2_2·b_2_1·b_4_7
       + a_2_2·b_2_1·b_2_32 + a_2_2·b_2_12·b_2_3 + a_2_22·b_4_10 + a_2_22·b_2_12
       + a_2_0·a_2_2·c_4_12
  129. a_3_6·b_5_18 + a_2_2·b_2_3·b_4_11 + a_2_2·b_2_3·b_4_7 + a_2_2·b_2_33
       + a_2_2·b_2_1·b_4_7 + a_2_22·b_4_10 + c_4_13·a_1_1·a_3_6 + c_4_12·a_1_1·a_3_6
  130. b_4_7·b_4_10 + b_2_1·b_2_3·b_4_7 + b_2_12·b_2_32 + a_3_5·b_5_18 + a_2_2·b_2_3·b_4_7
       + a_2_2·b_2_1·b_4_7 + a_2_0·a_2_2·c_4_13
  131. a_3_3·b_5_18 + a_2_22·b_4_10 + a_2_0·a_2_2·c_4_13 + a_2_0·a_2_2·c_4_12
  132. b_3_4·b_5_18 + b_4_10·b_4_11 + b_4_7·b_4_10 + b_2_1·b_2_3·b_4_11 + b_2_13·b_2_3
       + a_2_2·b_2_3·b_4_11 + a_2_2·b_2_3·b_4_7 + a_2_22·b_4_10 + a_2_2·b_2_3·c_4_12
       + c_4_12·a_1_1·a_3_6 + a_2_22·c_4_13 + a_2_22·c_4_12 + a_2_0·a_2_2·c_4_13
  133. b_3_7·b_5_18 + b_4_7·b_4_10 + b_2_1·b_2_3·b_4_11 + b_2_1·b_2_33 + b_2_12·b_4_7
       + b_2_12·b_2_32 + a_2_2·b_2_3·b_4_11 + a_2_2·b_2_33 + a_2_2·b_2_1·b_4_10
       + a_2_22·b_2_12 + c_4_13·a_1_1·a_3_6 + c_4_12·a_1_1·a_3_6 + a_2_0·a_2_2·c_4_12
  134. b_4_10·b_4_11 + b_4_7·b_4_10 + b_2_1·b_6_26 + b_2_1·b_2_3·b_4_11 + b_2_1·b_2_3·b_4_7
       + b_2_12·b_4_11 + b_2_12·b_4_10 + a_2_2·b_2_3·b_4_11 + a_2_2·b_2_3·b_4_7
       + a_2_2·b_2_1·b_4_11 + a_2_2·b_2_1·b_4_10 + a_2_2·b_2_1·b_4_7 + a_2_2·b_2_1·b_2_32
       + a_2_2·b_2_13 + a_2_22·b_4_10 + a_2_22·b_2_12 + b_2_12·c_4_13
       + a_2_2·b_2_3·c_4_12 + a_2_2·b_2_1·c_4_13 + c_4_12·a_1_1·a_3_6 + a_2_22·c_4_13
       + a_2_22·c_4_12 + a_2_0·a_2_2·c_4_12
  135. b_4_7·b_4_11 + b_4_7·b_4_10 + b_2_3·b_6_26 + b_2_1·b_2_3·b_4_11 + b_2_1·b_2_3·b_4_7
       + b_2_1·b_2_33 + b_2_12·b_4_7 + a_2_2·b_2_3·b_4_7 + a_2_2·b_2_1·b_4_11
       + a_2_2·b_2_1·b_4_10 + a_2_2·b_2_1·b_4_7 + a_2_2·b_2_1·b_2_32 + a_2_22·b_4_10
       + a_2_22·b_2_12 + b_2_1·b_2_3·c_4_13 + a_2_2·b_2_3·c_4_13 + c_4_13·a_1_1·a_3_6
       + a_2_0·a_2_2·c_4_13 + a_2_0·a_2_2·c_4_12
  136. b_4_7·b_4_10 + b_2_1·b_2_3·b_4_7 + b_2_12·b_2_32 + a_2_2·b_6_26 + a_2_2·b_2_1·b_4_11
       + a_2_2·b_2_1·b_4_10 + a_2_2·b_2_1·b_4_7 + a_2_2·b_2_12·b_2_3 + a_2_22·b_4_10
       + a_2_22·b_2_12 + a_2_2·b_2_1·c_4_13 + a_2_22·c_4_13 + a_2_0·a_2_2·c_4_13
  137. a_2_0·b_6_26 + a_2_22·b_4_10 + a_2_22·c_4_13 + a_2_0·a_2_2·c_4_13
  138. a_4_6·a_5_15
  139. b_4_11·a_5_15 + b_4_10·a_5_15 + b_4_7·a_5_15 + b_2_33·a_3_3 + a_2_2·b_2_1·b_2_3·b_3_4
       + a_2_2·b_2_12·b_3_4 + a_2_2·b_2_12·a_3_2 + b_2_3·c_4_12·a_3_5 + b_2_3·c_4_12·a_3_3
       + b_2_32·c_4_12·a_1_1 + a_2_2·c_4_12·a_3_2
  140. b_4_11·a_5_12 + b_4_11·a_5_15 + b_2_32·a_5_12 + b_2_32·a_5_15 + b_2_33·a_3_5
       + a_2_2·b_4_11·b_3_4 + a_2_2·b_2_32·b_3_7 + a_2_2·b_2_1·b_2_3·b_3_7
       + b_2_3·c_4_13·a_3_3 + b_2_3·c_4_12·a_3_6 + b_2_3·c_4_12·a_3_5 + b_2_3·c_4_12·a_3_3
       + b_2_32·c_4_13·a_1_1
  141. b_4_11·a_5_15 + b_4_10·a_5_12 + b_4_10·a_5_15 + b_2_33·a_3_3 + a_2_2·b_4_11·b_3_4
       + a_2_2·b_2_32·b_3_7 + a_2_2·b_2_1·b_2_3·b_3_7 + a_2_2·b_2_1·b_2_3·b_3_4
       + a_2_2·b_2_12·b_3_7 + b_2_3·c_4_12·a_3_5 + b_2_3·c_4_12·a_3_3 + b_2_32·c_4_12·a_1_1
       + a_2_2·c_4_13·a_3_2
  142. a_4_6·a_5_12
  143. b_4_7·a_5_12 + b_2_32·a_5_15 + b_2_33·a_3_5 + b_2_33·a_3_3 + a_2_2·b_2_1·b_2_3·b_3_7
       + a_2_2·b_2_1·b_2_3·b_3_4 + b_2_32·c_4_12·a_1_1
  144. b_4_11·b_5_18 + b_2_3·b_4_11·b_3_4 + b_2_33·b_3_4 + b_2_1·b_4_11·b_3_4
       + b_2_1·b_2_32·b_3_4 + b_4_11·a_5_15 + b_4_10·a_5_15 + b_2_33·a_3_3
       + a_2_2·b_4_11·b_3_4 + a_2_2·b_2_32·b_3_7 + a_2_2·b_2_32·b_3_4
       + a_2_2·b_2_1·b_2_3·b_3_7 + a_2_2·b_2_12·b_3_4 + a_2_2·b_2_12·a_3_2
       + b_2_3·c_4_12·b_3_7 + b_2_3·c_4_13·a_3_3 + b_2_3·c_4_12·a_3_6 + b_2_32·c_4_12·a_1_1
       + a_2_2·c_4_13·a_3_2 + a_2_2·c_4_12·a_3_2
  145. b_4_10·a_5_15 + a_2_2·b_4_11·b_3_4 + a_2_2·b_2_32·b_3_4 + a_2_2·b_2_1·b_5_18
       + a_2_2·b_2_1·b_2_3·b_3_7 + a_2_2·b_2_1·b_2_3·b_3_4 + a_2_2·b_2_12·a_3_2
       + a_2_2·c_4_12·a_3_2
  146. b_4_11·a_5_15 + b_4_10·a_5_15 + b_2_33·a_3_3 + a_2_2·b_2_3·b_5_18
       + a_2_2·b_2_1·b_2_3·b_3_7 + a_2_2·b_2_1·b_2_3·b_3_4 + a_2_2·b_2_12·b_3_4
       + a_2_2·b_2_12·a_3_2 + b_2_3·c_4_12·a_3_5 + b_2_3·c_4_12·a_3_3 + b_2_32·c_4_12·a_1_1
       + a_2_2·c_4_12·a_3_2
  147. b_4_10·b_5_18 + b_2_1·b_4_11·b_3_4 + b_2_1·b_2_3·b_5_18 + b_2_1·b_2_32·b_3_4
       + b_2_12·b_2_3·b_3_7 + b_2_13·b_3_7 + b_2_12·a_5_15 + b_2_13·a_3_2
       + a_2_2·b_2_12·a_3_2 + b_2_12·c_4_12·a_1_0 + a_2_2·c_4_12·b_3_7 + a_2_2·c_4_13·a_3_2
       + a_2_2·c_4_12·a_3_2
  148. b_4_11·a_5_15 + b_4_10·a_5_15 + a_4_6·b_5_18 + b_2_33·a_3_3 + a_2_2·b_4_11·b_3_4
       + a_2_2·b_2_32·b_3_4 + a_2_2·b_2_1·b_2_3·b_3_4 + a_2_2·b_2_12·b_3_7
       + a_2_2·b_2_12·b_3_4 + a_2_2·b_2_12·a_3_2 + b_2_3·c_4_12·a_3_5 + b_2_3·c_4_12·a_3_3
       + b_2_32·c_4_12·a_1_1 + a_2_2·c_4_12·a_3_2
  149. b_4_7·b_5_18 + b_2_3·b_4_11·b_3_4 + b_2_33·b_3_4 + b_2_1·b_2_32·b_3_7
       + b_2_12·b_2_3·b_3_7 + b_4_11·a_5_15 + b_4_10·a_5_15 + b_2_33·a_3_3
       + a_2_2·b_4_11·b_3_4 + a_2_2·b_2_32·b_3_7 + a_2_2·b_2_32·b_3_4 + a_2_2·b_2_12·b_3_7
       + a_2_2·b_2_12·b_3_4 + a_2_2·b_2_12·a_3_2 + b_2_3·c_4_12·a_3_5 + b_2_3·c_4_12·a_3_3
       + b_2_32·c_4_12·a_1_1 + a_2_2·c_4_12·a_3_2
  150. b_6_26·a_3_2 + b_4_10·a_5_15 + a_2_2·b_2_32·b_3_4 + a_2_2·b_2_1·b_2_3·b_3_7
       + a_2_2·b_2_12·b_3_7 + a_2_2·b_2_12·a_3_2 + b_2_1·c_4_13·a_3_2 + a_2_2·c_4_13·a_3_2
       + a_2_2·c_4_12·a_3_2
  151. b_6_26·a_3_6 + b_4_11·a_5_15 + b_4_10·a_5_15 + b_2_32·a_5_15 + b_2_33·a_3_5
       + a_2_2·b_2_32·b_3_7 + a_2_2·b_2_1·b_2_3·b_3_7 + a_2_2·b_2_1·b_2_3·b_3_4
       + a_2_2·b_2_12·b_3_4 + a_2_2·b_2_12·a_3_2 + b_2_3·c_4_12·a_3_5 + b_2_3·c_4_12·a_3_3
       + a_2_2·c_4_13·b_3_7 + a_2_2·c_4_12·a_3_2
  152. b_6_26·a_3_5 + b_4_11·a_5_15 + b_4_10·a_5_15 + b_2_33·a_3_3 + a_2_2·b_4_11·b_3_4
       + a_2_2·b_2_32·b_3_7 + a_2_2·b_2_1·b_2_3·b_3_7 + a_2_2·b_2_1·b_2_3·b_3_4
       + a_2_2·b_2_12·b_3_7 + a_2_2·b_2_12·b_3_4 + a_2_2·b_2_12·a_3_2 + b_2_3·c_4_12·a_3_5
       + b_2_3·c_4_12·a_3_3 + b_2_32·c_4_12·a_1_1 + a_2_2·c_4_13·b_3_4 + a_2_2·c_4_12·a_3_2
  153. b_6_26·a_3_3 + a_2_2·c_4_13·a_3_2
  154. b_6_26·b_3_4 + b_2_1·b_4_11·b_3_4 + b_2_1·b_2_3·b_5_18 + b_2_1·b_2_32·b_3_7
       + b_2_13·b_3_7 + b_2_12·a_5_15 + b_2_13·a_3_2 + a_2_2·b_4_11·b_3_4
       + a_2_2·b_2_32·b_3_7 + a_2_2·b_2_32·b_3_4 + a_2_2·b_2_12·b_3_4 + a_2_2·b_2_12·a_3_2
       + b_2_1·c_4_13·b_3_4 + b_2_12·c_4_12·a_1_0 + a_2_2·c_4_13·b_3_4
  155. b_6_26·b_3_7 + b_2_3·b_4_11·b_3_4 + b_2_1·b_2_3·b_5_18 + b_2_1·b_2_32·b_3_4
       + b_2_12·b_2_3·b_3_7 + b_4_11·a_5_15 + b_2_32·a_5_15 + b_2_33·a_3_5
       + a_2_2·b_2_32·b_3_7 + a_2_2·b_2_32·b_3_4 + a_2_2·b_2_1·b_2_3·b_3_7
       + b_2_1·c_4_13·b_3_7 + b_2_3·c_4_12·a_3_5 + b_2_3·c_4_12·a_3_3 + a_2_2·c_4_13·b_3_7
  156. a_5_152 + a_2_22·b_2_13
  157. a_5_122 + a_2_22·b_2_13
  158. a_5_15·a_5_12 + b_2_32·a_1_1·a_5_12 + a_2_22·b_2_1·b_4_10 + a_2_22·b_2_13
       + c_4_12·a_1_1·a_5_12 + b_2_3·c_4_12·a_1_1·a_3_6 + a_2_2·a_4_6·c_4_13
       + a_2_2·a_4_6·c_4_12
  159. b_5_182 + b_2_1·b_2_32·b_4_11 + b_2_1·b_2_32·b_4_7 + b_2_1·b_2_34
       + b_2_12·b_2_3·b_4_7 + b_2_12·b_2_33 + b_2_14·b_2_3 + a_2_2·b_2_1·b_2_3·b_4_7
       + a_2_2·b_2_1·b_2_33 + a_2_2·b_2_12·b_4_7 + a_2_2·b_2_12·b_2_32
       + a_2_22·b_2_1·b_4_10 + b_2_1·b_2_32·c_4_12 + a_2_22·b_2_1·c_4_13
  160. a_5_12·b_5_18 + a_2_2·b_2_1·b_6_26 + a_2_2·b_2_1·b_2_3·b_4_11 + a_2_2·b_2_12·b_2_32
       + a_2_2·b_2_13·b_2_3 + a_2_22·b_2_1·b_4_10 + a_2_2·b_2_32·c_4_12
       + a_2_2·b_2_12·c_4_13 + c_4_13·a_1_1·a_5_12 + c_4_12·a_1_1·a_5_12 + a_2_2·a_4_6·c_4_13
       + a_2_2·a_4_6·c_4_12 + a_2_22·b_2_1·c_4_12
  161. b_4_11·b_6_26 + b_2_32·b_6_26 + b_2_1·b_2_32·b_4_11 + b_2_1·b_2_32·b_4_7
       + b_2_12·b_6_26 + b_2_12·b_2_3·b_4_11 + b_2_13·b_4_11 + b_2_13·b_4_10
       + b_2_13·b_2_32 + a_5_12·b_5_18 + a_5_15·b_5_18 + a_2_2·b_2_32·b_4_7
       + a_2_2·b_2_1·b_2_3·b_4_11 + a_2_2·b_2_1·b_2_3·b_4_7 + a_2_2·b_2_12·b_4_11
       + a_2_2·b_2_12·b_4_10 + a_2_2·b_2_12·b_2_32 + a_2_2·b_2_14 + a_2_22·b_2_13
       + b_2_3·b_4_7·c_4_12 + b_2_1·b_4_11·c_4_13 + b_2_1·b_2_32·c_4_13
       + b_2_1·b_2_32·c_4_12 + b_2_13·c_4_13 + a_2_2·b_4_11·c_4_13 + a_2_2·b_2_32·c_4_13
       + a_2_2·b_2_32·c_4_12 + a_2_2·b_2_1·b_2_3·c_4_12 + a_2_2·b_2_12·c_4_13
       + c_4_12·a_1_1·a_5_12 + b_2_3·c_4_13·a_1_1·a_3_6 + a_2_2·a_4_6·c_4_13
       + a_2_2·a_4_6·c_4_12 + a_2_22·b_2_1·c_4_13 + a_2_22·b_2_1·c_4_12
  162. a_5_12·b_5_18 + a_5_15·b_5_18 + a_2_2·b_2_3·b_6_26 + a_2_2·b_2_32·b_4_7
       + a_2_2·b_2_12·b_4_11 + a_2_2·b_2_12·b_4_7 + a_2_2·b_2_12·b_2_32
       + a_2_2·b_2_13·b_2_3 + a_2_2·b_4_7·c_4_12 + a_2_2·b_2_32·c_4_12
       + a_2_2·b_2_1·b_2_3·c_4_13 + c_4_13·a_1_1·a_5_12 + c_4_12·a_1_1·a_5_12
       + a_2_22·b_2_1·c_4_13
  163. b_4_10·b_6_26 + b_2_1·b_2_3·b_6_26 + b_2_12·b_6_26 + b_2_12·b_2_3·b_4_11
       + b_2_12·b_2_3·b_4_7 + b_2_13·b_4_11 + b_2_13·b_4_10 + b_2_13·b_4_7
       + b_2_13·b_2_32 + b_2_14·b_2_3 + a_5_12·b_5_18 + a_5_15·b_5_18 + a_2_2·b_2_32·b_4_7
       + a_2_2·b_2_12·b_4_11 + a_2_2·b_2_12·b_4_10 + a_2_2·b_2_12·b_2_32 + a_2_2·b_2_14
       + a_2_22·b_2_13 + b_2_1·b_4_10·c_4_13 + b_2_12·b_2_3·c_4_13 + b_2_13·c_4_13
       + a_2_2·b_4_10·c_4_13 + a_2_2·b_2_32·c_4_12 + a_2_2·b_2_1·b_2_3·c_4_13
       + a_2_2·b_2_1·b_2_3·c_4_12 + a_2_2·b_2_12·c_4_13 + c_4_13·a_1_1·a_5_12
       + c_4_12·a_1_1·a_5_12 + a_2_2·a_4_6·c_4_13
  164. a_5_15·b_5_18 + a_4_6·b_6_26 + a_2_2·b_2_32·b_4_7 + a_2_2·b_2_1·b_2_3·b_4_7
       + a_2_2·b_2_12·b_4_10 + a_2_2·b_2_12·b_2_32 + a_2_2·b_2_13·b_2_3
       + a_2_22·b_2_1·b_4_10 + a_2_22·b_2_13 + a_2_2·b_4_10·c_4_13 + a_2_2·b_4_7·c_4_12
       + a_2_2·a_4_6·c_4_13 + a_2_2·a_4_6·c_4_12 + a_2_22·b_2_1·c_4_13 + a_2_22·b_2_1·c_4_12
  165. b_4_7·b_6_26 + b_2_1·b_2_3·b_6_26 + b_2_1·b_2_32·b_4_11 + b_2_1·b_2_32·b_4_7
       + b_2_12·b_2_3·b_4_11 + b_2_12·b_2_3·b_4_7 + b_2_12·b_2_33 + b_2_13·b_4_7
       + a_2_2·b_2_1·b_2_3·b_4_11 + a_2_2·b_2_1·b_2_3·b_4_7 + a_2_2·b_2_1·b_2_33
       + a_2_2·b_2_12·b_4_11 + a_2_2·b_2_12·b_4_10 + a_2_22·b_2_1·b_4_10 + a_2_22·b_2_13
       + b_2_1·b_4_7·c_4_13 + b_2_12·b_2_3·c_4_13 + a_2_2·b_4_7·c_4_13
       + a_2_2·b_2_1·b_2_3·c_4_13
  166. b_6_26·b_5_18 + b_2_1·b_2_3·b_4_11·b_3_4 + b_2_13·b_2_3·b_3_7 + b_2_13·b_2_3·b_3_4
       + b_2_14·b_3_7 + b_2_13·a_5_15 + b_2_14·a_3_2 + a_2_2·b_2_3·b_4_11·b_3_4
       + a_2_2·b_2_1·b_4_11·b_3_4 + a_2_2·b_2_1·b_2_32·b_3_4 + a_2_2·b_2_12·b_5_18
       + a_2_2·b_2_12·b_2_3·b_3_7 + a_2_2·b_2_12·b_2_3·b_3_4 + a_2_2·b_2_13·b_3_7
       + a_2_2·b_2_13·a_3_2 + b_2_32·c_4_12·b_3_4 + b_2_1·c_4_13·b_5_18
       + b_2_1·b_2_3·c_4_12·b_3_7 + b_2_13·c_4_12·a_1_0 + a_2_2·c_4_13·b_5_18
       + a_2_2·b_2_3·c_4_12·b_3_4 + a_2_2·b_2_1·c_4_12·b_3_7 + a_2_2·b_2_1·c_4_12·a_3_2
  167. b_6_26·a_5_12 + b_2_33·a_5_15 + b_2_34·a_3_5 + b_2_34·a_3_3 + a_2_2·b_2_32·b_5_18
       + a_2_2·b_2_33·b_3_7 + a_2_2·b_2_1·b_2_32·b_3_7 + a_2_2·b_2_1·b_2_32·b_3_4
       + a_2_2·b_2_12·b_5_18 + a_2_2·b_2_13·b_3_7 + b_2_32·c_4_12·a_3_5
       + b_2_33·c_4_12·a_1_1 + b_2_12·c_4_13·a_3_2 + a_2_2·c_4_13·b_5_18
       + a_2_2·b_2_3·c_4_13·b_3_7 + a_2_2·b_2_3·c_4_12·b_3_7 + a_2_2·b_2_1·c_4_13·b_3_7
       + a_2_2·b_2_1·c_4_13·b_3_4 + a_2_2·b_2_1·c_4_12·a_3_2 + b_2_1·c_4_132·a_1_0
       + b_2_1·c_4_12·c_4_13·a_1_0
  168. b_6_26·a_5_15 + a_2_2·b_2_1·b_4_11·b_3_4 + a_2_2·b_2_1·b_2_32·b_3_4
       + a_2_2·b_2_13·b_3_4 + b_2_1·c_4_13·a_5_15 + a_2_2·b_2_3·c_4_12·b_3_7
       + a_2_2·b_2_3·c_4_12·b_3_4 + a_2_2·b_2_1·c_4_13·a_3_2 + a_2_2·b_2_1·c_4_12·a_3_2
  169. b_6_262 + b_2_1·b_2_33·b_4_11 + b_2_1·b_2_33·b_4_7 + b_2_12·b_2_32·b_4_11
       + b_2_12·b_2_34 + b_2_13·b_2_3·b_4_7 + b_2_14·b_2_32 + b_2_15·b_2_3
       + a_2_2·b_2_1·b_2_32·b_4_7 + a_2_2·b_2_1·b_2_34 + a_2_2·b_2_13·b_4_7
       + a_2_2·b_2_13·b_2_32 + a_2_22·b_2_12·b_4_10 + b_2_1·b_2_33·c_4_12
       + b_2_12·b_2_32·c_4_12 + a_2_22·b_2_12·c_4_13 + a_2_22·b_2_12·c_4_12
       + b_2_12·c_4_132 + a_2_22·c_4_132

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 12.
  • 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_12, a Duflot regular element of degree 4
    2. c_4_13, a Duflot regular element of degree 4
    3. b_2_32 + b_2_1·b_2_3 + b_2_12, an element of degree 4
    4. b_3_7 + b_3_4, an element of degree 3
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 8, 11].
  • 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 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_2 → 0, an element of degree 2
  5. b_2_1 → 0, an element of degree 2
  6. b_2_3 → 0, an element of degree 2
  7. a_3_2 → 0, an element of degree 3
  8. a_3_3 → 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. b_3_4 → 0, an element of degree 3
  12. b_3_7 → 0, an element of degree 3
  13. a_4_6 → 0, an element of degree 4
  14. b_4_7 → 0, an element of degree 4
  15. b_4_10 → 0, an element of degree 4
  16. b_4_11 → 0, an element of degree 4
  17. c_4_12 → c_1_14 + c_1_04, an element of degree 4
  18. c_4_13 → c_1_04, an element of degree 4
  19. a_5_15 → 0, an element of degree 5
  20. a_5_12 → 0, an element of degree 5
  21. b_5_18 → 0, an element of degree 5
  22. b_6_26 → 0, an element of degree 6

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

  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_2 → 0, an element of degree 2
  5. b_2_1 → c_1_22, an element of degree 2
  6. b_2_3 → c_1_32, an element of degree 2
  7. a_3_2 → 0, an element of degree 3
  8. a_3_3 → 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. b_3_4 → c_1_22·c_1_3, an element of degree 3
  12. b_3_7 → c_1_2·c_1_32, an element of degree 3
  13. a_4_6 → 0, an element of degree 4
  14. b_4_7 → c_1_2·c_1_33, an element of degree 4
  15. b_4_10 → c_1_22·c_1_32 + c_1_23·c_1_3, an element of degree 4
  16. b_4_11 → c_1_2·c_1_33 + c_1_1·c_1_2·c_1_32 + c_1_12·c_1_32 + c_1_0·c_1_2·c_1_32
       + c_1_02·c_1_32, an element of degree 4
  17. c_4_12 → c_1_22·c_1_32 + c_1_23·c_1_3 + c_1_1·c_1_2·c_1_32 + c_1_12·c_1_32
       + c_1_12·c_1_22 + c_1_14 + c_1_0·c_1_2·c_1_32 + c_1_02·c_1_32
       + c_1_02·c_1_22 + c_1_04, an element of degree 4
  18. c_4_13 → c_1_02·c_1_32 + c_1_04, an element of degree 4
  19. a_5_15 → 0, an element of degree 5
  20. a_5_12 → 0, an element of degree 5
  21. b_5_18 → c_1_2·c_1_34 + c_1_24·c_1_3 + c_1_1·c_1_22·c_1_32 + c_1_12·c_1_2·c_1_32
       + c_1_0·c_1_22·c_1_32 + c_1_02·c_1_2·c_1_32, an element of degree 5
  22. b_6_26 → c_1_23·c_1_33 + c_1_24·c_1_32 + c_1_25·c_1_3 + c_1_1·c_1_22·c_1_33
       + c_1_1·c_1_23·c_1_32 + c_1_12·c_1_2·c_1_33 + c_1_12·c_1_22·c_1_32
       + c_1_0·c_1_22·c_1_33 + c_1_0·c_1_23·c_1_32 + c_1_02·c_1_2·c_1_33
       + c_1_04·c_1_22, an element of degree 6

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