Cohomology of group number 5 of order 243

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 243

General information on the group

  • The group has 2 minimal generators and exponent 9.
  • 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
    t8  −  t7  +  2·t6  −  2·t5  +  t4  −  t3  −  t2  +  t  −  1
    (t  −  1)3 · (t2  −  t  +  1)2 · (t2  +  t  +  1)2
  • 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 243

Ring generators

The cohomology ring has 28 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_3_0, a nilpotent element of degree 3
  6. a_3_1, a nilpotent element of degree 3
  7. a_3_2, a nilpotent element of degree 3
  8. a_3_3, a nilpotent element of degree 3
  9. a_4_2, a nilpotent element of degree 4
  10. a_4_3, a nilpotent element of degree 4
  11. a_4_4, a nilpotent element of degree 4
  12. b_4_5, an element of degree 4
  13. a_5_2, a nilpotent element of degree 5
  14. a_5_3, a nilpotent element of degree 5
  15. a_5_4, a nilpotent element of degree 5
  16. a_5_5, a nilpotent element of degree 5
  17. a_5_6, a nilpotent element of degree 5
  18. a_6_3, a nilpotent element of degree 6
  19. a_6_4, a nilpotent element of degree 6
  20. a_6_5, a nilpotent element of degree 6
  21. a_6_6, a nilpotent element of degree 6
  22. b_6_7, an element of degree 6
  23. c_6_8, a Duflot regular element of degree 6
  24. c_6_9, a Duflot regular element of degree 6
  25. a_7_8, a nilpotent element of degree 7
  26. a_7_9, a nilpotent element of degree 7
  27. a_7_10, a nilpotent element of degree 7
  28. a_8_8, a nilpotent element of degree 8

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 243

Ring relations

There are 14 "obvious" relations:
   a_1_02, a_1_12, a_3_02, a_3_12, a_3_22, a_3_32, a_5_22, a_5_32, a_5_42, a_5_52, a_5_62, a_7_82, a_7_92, a_7_102

Apart from that, there are 308 minimal relations of maximal degree 16:

  1. a_1_0·a_1_1
  2. a_2_0·a_1_1
  3. a_2_0·a_1_0
  4. a_2_1·a_1_1
  5. a_2_1·a_1_0
  6. a_2_02
  7. a_2_0·a_2_1
  8. a_2_12
  9. a_1_1·a_3_0
  10. a_1_0·a_3_0
  11. a_1_0·a_3_1
  12.  − a_1_1·a_3_1 + a_1_0·a_3_2
  13. a_1_1·a_3_3 − a_1_1·a_3_2
  14.  − a_1_1·a_3_2 + a_1_0·a_3_3
  15.  − a_2_1·a_3_0 + a_2_0·a_3_1
  16. a_2_1·a_3_1
  17.  − a_2_1·a_3_0 + a_2_0·a_3_2 + a_2_0·a_3_0
  18. a_2_1·a_3_2 − a_2_0·a_3_0
  19. a_2_1·a_3_0 + a_2_0·a_3_3
  20. a_2_1·a_3_3 − a_2_0·a_3_0
  21. a_4_2·a_1_1 − a_2_1·a_3_0
  22. a_4_2·a_1_0 − a_2_0·a_3_0
  23. a_4_3·a_1_1 − a_2_1·a_3_0
  24. a_4_3·a_1_0 − a_2_1·a_3_0 − a_2_0·a_3_0
  25. a_4_4·a_1_1 − a_2_1·a_3_0 + a_2_0·a_3_0
  26. a_4_4·a_1_0 + a_2_1·a_3_0 + a_2_0·a_3_0
  27. b_4_5·a_1_1 + a_2_1·a_3_0 − a_2_0·a_3_0
  28. b_4_5·a_1_0 + a_2_0·a_3_0
  29. a_3_2·a_3_3 + a_3_1·a_3_2 + a_3_0·a_3_1
  30. a_3_1·a_3_3 − a_3_1·a_3_2 + a_3_0·a_3_1
  31.  − a_3_1·a_3_2 + a_3_0·a_3_3 − a_3_0·a_3_2
  32. a_2_0·a_4_2
  33. a_2_1·a_4_2
  34. a_2_0·a_4_3
  35. a_2_1·a_4_3
  36. a_2_0·a_4_4
  37. a_2_1·a_4_4
  38. a_2_0·b_4_5
  39. a_2_1·b_4_5
  40. a_1_1·a_5_2
  41. a_1_0·a_5_2
  42.  − a_3_1·a_3_2 + a_1_1·a_5_3
  43. a_3_1·a_3_2 − a_3_0·a_3_2 + a_3_0·a_3_1 + a_1_0·a_5_3
  44.  − a_3_1·a_3_2 − a_3_0·a_3_1 + a_1_1·a_5_4
  45.  − a_3_0·a_3_2 + a_3_0·a_3_1 + a_1_0·a_5_4
  46. a_3_0·a_3_1 + a_1_1·a_5_5
  47. a_1_0·a_5_5
  48. a_3_1·a_3_2 + a_3_0·a_3_2 + a_3_0·a_3_1 + a_1_1·a_5_6
  49. a_3_0·a_3_2 + a_1_0·a_5_6
  50. a_4_2·a_3_3 − a_4_2·a_3_2 + a_4_2·a_3_1 + a_4_2·a_3_0
  51. a_4_3·a_3_3 + a_4_2·a_3_3 + a_4_2·a_3_2 + a_4_2·a_3_1
  52. a_4_3·a_3_2 − a_4_2·a_3_3 + a_4_2·a_3_1
  53. a_4_3·a_3_1 − a_4_2·a_3_1
  54. a_4_3·a_3_0 + a_4_2·a_3_3 − a_4_2·a_3_2
  55. a_4_4·a_3_3 − a_4_2·a_3_3
  56. a_4_4·a_3_2 − a_4_2·a_3_3 − a_4_2·a_3_2 − a_4_2·a_3_1
  57. a_4_4·a_3_1 + a_4_2·a_3_3 − a_4_2·a_3_2 + a_4_2·a_3_1
  58. a_4_4·a_3_0 + a_4_2·a_3_3 + a_4_2·a_3_1
  59. b_4_5·a_3_3 − a_4_2·a_3_3
  60. b_4_5·a_3_1 − a_4_2·a_3_3 + a_4_2·a_3_2
  61.  − b_4_5·a_3_2 + b_4_5·a_3_0 − a_4_2·a_3_2
  62. a_2_0·a_5_2
  63. a_2_1·a_5_2
  64.  − a_4_2·a_3_3 − a_4_2·a_3_1 + a_2_0·a_5_3
  65.  − a_4_2·a_3_3 + a_4_2·a_3_2 + a_2_1·a_5_3
  66. a_4_2·a_3_3 + a_4_2·a_3_2 − a_4_2·a_3_1 + a_2_0·a_5_4
  67.  − a_4_2·a_3_3 + a_4_2·a_3_2 + a_2_1·a_5_4
  68. a_4_2·a_3_3 − a_4_2·a_3_2 + a_4_2·a_3_1 + a_2_0·a_5_5
  69.  − a_4_2·a_3_1 + a_2_1·a_5_5
  70. a_4_2·a_3_2 + a_4_2·a_3_1 + a_2_0·a_5_6
  71.  − a_4_2·a_3_3 + a_4_2·a_3_1 + a_2_1·a_5_6
  72. a_6_3·a_1_1 + a_4_2·a_3_3 + a_4_2·a_3_2 − a_4_2·a_3_1
  73. a_6_3·a_1_0
  74. a_6_4·a_1_1 − a_4_2·a_3_1
  75. a_6_4·a_1_0 + a_4_2·a_3_3 − a_4_2·a_3_2 + a_4_2·a_3_1
  76. a_6_5·a_1_1 + a_4_2·a_3_1
  77. a_6_5·a_1_0 − a_4_2·a_3_3 + a_4_2·a_3_2 + a_4_2·a_3_1
  78. a_6_6·a_1_1 − a_4_2·a_3_3 + a_4_2·a_3_1
  79. a_6_6·a_1_0 − a_4_2·a_3_3 + a_4_2·a_3_1
  80. b_6_7·a_1_1 + a_4_2·a_3_1
  81. b_6_7·a_1_0 − a_4_2·a_3_3 + a_4_2·a_3_2 + a_4_2·a_3_1
  82. a_4_22
  83. a_4_32
  84. a_4_2·a_4_3
  85. a_4_3·a_4_4
  86. a_4_42
  87. a_4_2·a_4_4
  88. a_4_3·b_4_5
  89. a_4_4·b_4_5
  90. a_4_2·b_4_5
  91. a_3_3·a_5_2
  92. a_3_2·a_5_2
  93. a_3_1·a_5_2
  94. a_3_0·a_5_2
  95.  − a_3_3·a_5_3 + a_3_2·a_5_3
  96. a_3_3·a_5_3 − a_3_1·a_5_3 + a_3_0·a_5_3
  97. a_3_3·a_5_4 − a_3_3·a_5_3 + a_3_1·a_5_3
  98. a_3_1·a_5_4 − a_3_1·a_5_3
  99.  − a_3_3·a_5_3 − a_3_2·a_5_4 + a_3_0·a_5_4
  100. a_3_3·a_5_5 − a_3_3·a_5_3
  101. a_3_2·a_5_5 − a_3_2·a_5_4 + a_3_1·a_5_3
  102. a_3_1·a_5_5
  103. a_3_3·a_5_3 − a_3_2·a_5_4 − a_3_1·a_5_3 + a_3_0·a_5_5
  104. a_3_3·a_5_6 + a_3_3·a_5_3 + a_3_1·a_5_3
  105. a_3_3·a_5_3 + a_3_1·a_5_6 − a_3_1·a_5_3
  106. a_3_3·a_5_3 − a_3_2·a_5_6 − a_3_1·a_5_3 + a_3_0·a_5_6
  107. a_2_0·a_6_3
  108. a_2_1·a_6_3
  109. a_2_0·a_6_4
  110. a_2_1·a_6_4
  111. a_2_0·a_6_5
  112. a_2_1·a_6_5
  113. a_2_0·a_6_6
  114. a_2_1·a_6_6
  115. a_2_0·b_6_7
  116. a_2_1·b_6_7
  117. a_1_1·a_7_8
  118. a_1_0·a_7_8
  119. a_1_1·a_7_9
  120. a_1_0·a_7_9
  121.  − a_3_1·a_5_3 + a_1_1·a_7_10
  122. a_3_3·a_5_3 − a_3_1·a_5_3 + a_1_0·a_7_10
  123. a_4_3·a_5_2
  124. a_4_4·a_5_2
  125. a_4_2·a_5_2
  126. a_4_4·a_5_3 + a_4_3·a_5_3
  127. b_4_5·a_5_3 − b_4_5·a_5_2 + a_4_3·a_5_3 + a_4_2·a_5_3
  128.  − b_4_5·a_5_3 + b_4_5·a_5_2 + a_4_3·a_5_4
  129. b_4_5·a_5_3 − b_4_5·a_5_2 + a_4_4·a_5_4 − a_4_3·a_5_3
  130.  − a_4_3·a_5_3 + a_4_2·a_5_4
  131. a_4_3·a_5_5
  132. a_4_4·a_5_5 + a_4_3·a_5_3
  133. b_4_5·a_5_5 − b_4_5·a_5_4 − b_4_5·a_5_2 − a_4_3·a_5_3
  134. a_4_2·a_5_5
  135. b_4_5·a_5_3 − b_4_5·a_5_2 + a_4_3·a_5_6 + a_4_3·a_5_3
  136. a_4_4·a_5_6
  137. a_4_3·a_5_3 + a_4_2·a_5_6
  138. a_6_3·a_3_3 + a_4_3·a_5_3
  139. a_6_3·a_3_2 + a_4_3·a_5_3
  140. a_6_3·a_3_1 + a_4_3·a_5_3
  141. a_6_3·a_3_0
  142.  − b_4_5·a_5_3 + b_4_5·a_5_2 + a_6_4·a_3_3
  143. a_6_4·a_3_2 − a_4_3·a_5_3
  144. a_6_4·a_3_1
  145. a_6_4·a_3_0
  146. a_6_5·a_3_3 + a_4_3·a_5_3
  147.  − b_4_5·a_5_3 + b_4_5·a_5_2 + a_6_5·a_3_2 − a_4_3·a_5_3
  148. a_6_5·a_3_1
  149. a_6_5·a_3_0
  150. b_4_5·a_5_3 − b_4_5·a_5_2 + a_6_6·a_3_3
  151.  − b_4_5·a_5_3 + b_4_5·a_5_2 + a_6_6·a_3_2
  152. b_4_5·a_5_3 − b_4_5·a_5_2 + a_6_6·a_3_1
  153. a_6_6·a_3_0
  154. b_6_7·a_3_3 + a_4_3·a_5_3
  155. b_6_7·a_3_2 − b_4_5·a_5_3 − a_4_3·a_5_3
  156. b_6_7·a_3_1
  157. b_6_7·a_3_0 − b_4_5·a_5_2
  158. a_2_0·a_7_8
  159. a_2_1·a_7_8
  160. a_2_0·a_7_9
  161. a_2_1·a_7_9
  162. b_4_5·a_5_3 − b_4_5·a_5_2 + a_4_3·a_5_3 + a_2_0·a_7_10
  163.  − b_4_5·a_5_3 + b_4_5·a_5_2 + a_4_3·a_5_3 + a_2_1·a_7_10
  164.  − b_4_5·a_5_3 + b_4_5·a_5_2 + a_8_8·a_1_1 + a_4_3·a_5_3
  165. a_8_8·a_1_0 + a_4_3·a_5_3
  166. a_5_2·a_5_3
  167. a_5_2·a_5_5 − a_5_2·a_5_4
  168. a_5_5·a_5_6 − a_5_4·a_5_6 + a_5_3·a_5_6 − a_5_3·a_5_5 − a_5_3·a_5_4 + a_5_2·a_5_6
       − a_5_2·a_5_4
  169. a_5_4·a_5_5 + a_5_3·a_5_5 − a_5_3·a_5_4 + a_5_2·a_5_4 + c_6_8·a_1_0·a_3_3
  170.  − a_5_4·a_5_5 − a_5_3·a_5_4 + c_6_8·a_1_0·a_3_2
  171. a_5_5·a_5_6 − a_5_4·a_5_6 + a_5_4·a_5_5 − a_5_3·a_5_6 + a_5_3·a_5_5 − a_5_3·a_5_4
       + a_5_2·a_5_4 + c_6_9·a_1_0·a_3_3
  172. a_5_4·a_5_5 − a_5_3·a_5_4 − a_5_2·a_5_4 + c_6_9·a_1_0·a_3_2
  173. a_4_3·a_6_3
  174. a_4_4·a_6_3
  175. b_4_5·a_6_3 − a_5_2·a_5_4
  176. a_4_2·a_6_3
  177. a_4_3·a_6_4
  178. a_4_4·a_6_4
  179. b_4_5·a_6_4
  180. a_4_2·a_6_4
  181. a_4_3·a_6_5
  182. a_4_4·a_6_5
  183. b_4_5·a_6_5 − a_5_2·a_5_4
  184. a_4_2·a_6_5
  185. a_4_3·a_6_6
  186. a_4_4·a_6_6
  187. b_4_5·a_6_6 + a_5_5·a_5_6 − a_5_4·a_5_6 + a_5_3·a_5_6 − a_5_3·a_5_5 − a_5_3·a_5_4
       − a_5_2·a_5_4
  188. a_4_2·a_6_6
  189. a_4_3·b_6_7
  190. a_4_4·b_6_7
  191. a_4_2·b_6_7
  192. a_5_5·a_5_6 − a_5_4·a_5_6 − a_5_3·a_5_6 + a_3_3·a_7_8
  193.  − a_5_3·a_5_5 + a_3_2·a_7_8
  194. a_5_4·a_5_5 + a_5_3·a_5_4 + a_3_1·a_7_8
  195.  − a_5_2·a_5_4 + a_3_0·a_7_8
  196. a_5_5·a_5_6 − a_5_4·a_5_6 + a_5_4·a_5_5 − a_5_3·a_5_6 + a_5_3·a_5_5 − a_5_3·a_5_4
       + a_5_2·a_5_4 + a_3_3·a_7_9
  197.  − a_5_5·a_5_6 + a_5_4·a_5_6 + a_5_4·a_5_5 + a_5_3·a_5_4 − a_5_2·a_5_4 + a_3_2·a_7_9
  198. a_5_4·a_5_5 − a_5_3·a_5_4 − a_5_2·a_5_4 + a_3_1·a_7_9
  199. a_5_5·a_5_6 − a_5_4·a_5_6 + a_5_3·a_5_6 − a_5_3·a_5_5 − a_5_3·a_5_4 + a_5_2·a_5_4
       + a_3_0·a_7_9
  200. a_5_5·a_5_6 − a_5_4·a_5_6 − a_5_3·a_5_6 + a_5_3·a_5_4 − a_5_2·a_5_4 + a_3_3·a_7_10
  201.  − a_5_5·a_5_6 + a_5_4·a_5_6 + a_5_4·a_5_5 + a_5_3·a_5_6 + a_3_2·a_7_10
  202.  − a_5_3·a_5_5 + a_5_3·a_5_4 + a_3_1·a_7_10
  203. a_5_3·a_5_5 − a_5_3·a_5_4 − a_5_2·a_5_4 + a_3_0·a_7_10
  204. a_2_0·a_8_8
  205. a_2_1·a_8_8
  206. a_6_3·a_5_3 + a_2_0·c_6_9·a_3_1 + a_2_0·c_6_9·a_3_0 + a_2_0·c_6_8·a_3_1
       + a_2_0·c_6_8·a_3_0
  207. a_6_3·a_5_5 − a_2_0·c_6_9·a_3_1 − a_2_0·c_6_8·a_3_1
  208. a_6_3·a_5_4 − a_2_0·c_6_9·a_3_1 + a_2_0·c_6_9·a_3_0 − a_2_0·c_6_8·a_3_1
       + a_2_0·c_6_8·a_3_0
  209. a_6_3·a_5_2
  210. a_6_4·a_5_3 − a_2_0·c_6_8·a_3_1 − a_2_0·c_6_8·a_3_0
  211. a_6_4·a_5_5 + a_2_0·c_6_8·a_3_0
  212. a_6_4·a_5_4
  213. a_6_4·a_5_6 − a_2_0·c_6_9·a_3_1 + a_2_0·c_6_8·a_3_0
  214. a_6_4·a_5_2
  215. a_6_5·a_5_3 − a_2_0·c_6_8·a_3_1 − a_2_0·c_6_8·a_3_0
  216. a_6_5·a_5_5 + a_2_0·c_6_8·a_3_1 − a_2_0·c_6_8·a_3_0
  217. a_6_5·a_5_4 + a_2_0·c_6_8·a_3_1 + a_2_0·c_6_8·a_3_0
  218. a_6_5·a_5_6 − a_6_3·a_5_6 − a_2_0·c_6_8·a_3_1 + a_2_0·c_6_8·a_3_0
  219. a_6_5·a_5_2
  220. a_6_6·a_5_3 + a_2_0·c_6_9·a_3_1 + a_2_0·c_6_9·a_3_0 − a_2_0·c_6_8·a_3_1
       − a_2_0·c_6_8·a_3_0
  221. a_6_6·a_5_5 + a_6_3·a_5_6 − a_2_0·c_6_8·a_3_1 + a_2_0·c_6_8·a_3_0
  222. a_6_6·a_5_4 + a_6_3·a_5_6 + a_2_0·c_6_9·a_3_0 − a_2_0·c_6_8·a_3_1
  223. a_6_6·a_5_6 + a_2_0·c_6_9·a_3_0 − a_2_0·c_6_8·a_3_0
  224. a_6_6·a_5_2
  225. b_6_7·a_5_3 + b_4_52·a_3_0 − a_2_0·c_6_8·a_3_1 − a_2_0·c_6_8·a_3_0
  226.  − b_6_7·a_5_5 + b_6_7·a_5_4 − b_4_52·a_3_0 − a_2_0·c_6_8·a_3_0
  227. b_6_7·a_5_2 + b_4_52·a_3_0
  228. a_4_3·a_7_8 + a_2_0·c_6_9·a_3_1 + a_2_0·c_6_9·a_3_0 − a_2_0·c_6_8·a_3_1
  229. a_4_4·a_7_8 − a_2_0·c_6_9·a_3_1 − a_2_0·c_6_9·a_3_0 − a_2_0·c_6_8·a_3_1
       + a_2_0·c_6_8·a_3_0
  230.  − b_6_7·a_5_5 + b_4_5·a_7_8 + b_4_52·a_3_0 − a_2_0·c_6_9·a_3_0
  231. a_4_2·a_7_8 + a_2_0·c_6_9·a_3_0 − a_2_0·c_6_8·a_3_1
  232. a_4_3·a_7_9 + a_2_0·c_6_9·a_3_1
  233. a_4_4·a_7_9 + a_2_0·c_6_9·a_3_1 − a_2_0·c_6_9·a_3_0
  234.  − b_6_7·a_5_6 − b_6_7·a_5_5 + b_4_5·a_7_9 + a_6_3·a_5_6 − a_2_0·c_6_9·a_3_1
       + a_2_0·c_6_9·a_3_0
  235. a_4_2·a_7_9 + a_2_0·c_6_9·a_3_1
  236. a_4_3·a_7_10 + a_2_0·c_6_9·a_3_1 − a_2_0·c_6_9·a_3_0 + a_2_0·c_6_8·a_3_1
       + a_2_0·c_6_8·a_3_0
  237. a_4_4·a_7_10 + a_2_0·c_6_9·a_3_1 − a_2_0·c_6_9·a_3_0 − a_2_0·c_6_8·a_3_0
  238.  − b_6_7·a_5_5 + b_4_5·a_7_10 + b_4_52·a_3_0 − a_6_3·a_5_6 − a_2_0·c_6_9·a_3_1
       + a_2_0·c_6_8·a_3_1 + a_2_0·c_6_8·a_3_0
  239. a_4_2·a_7_10 − a_2_0·c_6_9·a_3_1 − a_2_0·c_6_9·a_3_0 − a_2_0·c_6_8·a_3_1
  240. a_8_8·a_3_3 − a_2_0·c_6_9·a_3_1 + a_2_0·c_6_8·a_3_1 + a_2_0·c_6_8·a_3_0
  241. a_8_8·a_3_2 + a_6_3·a_5_6 + a_2_0·c_6_9·a_3_1 + a_2_0·c_6_9·a_3_0 + a_2_0·c_6_8·a_3_1
  242. a_8_8·a_3_1 − a_2_0·c_6_9·a_3_1 − a_2_0·c_6_8·a_3_1 + a_2_0·c_6_8·a_3_0
  243. a_8_8·a_3_0 + a_6_3·a_5_6 + a_2_0·c_6_9·a_3_1 − a_2_0·c_6_9·a_3_0 + a_2_0·c_6_8·a_3_1
       + a_2_0·c_6_8·a_3_0
  244. a_6_32
  245. a_6_42
  246. a_6_3·a_6_4
  247. a_6_4·a_6_5
  248. a_6_52
  249. a_6_3·a_6_5
  250. a_6_62
  251. a_6_4·a_6_6
  252. a_6_5·a_6_6
  253. a_6_3·a_6_6
  254. b_6_72 + b_4_53 − b_4_5·a_3_0·a_5_4
  255. a_6_6·b_6_7 + b_4_5·a_3_0·a_5_6
  256. a_6_4·b_6_7
  257. a_6_5·b_6_7 + b_4_5·a_3_0·a_5_4
  258. a_6_3·b_6_7 + b_4_5·a_3_0·a_5_4
  259. a_5_3·a_7_8 + b_4_5·a_3_0·a_5_4 − c_6_9·a_1_0·a_5_3 + c_6_8·a_1_0·a_5_4
       − c_6_8·a_1_0·a_5_3
  260. a_5_5·a_7_8 − b_4_5·a_3_0·a_5_4 + c_6_8·a_1_0·a_5_6 + c_6_8·a_1_0·a_5_4
  261. a_5_4·a_7_8 + b_4_5·a_3_0·a_5_4 − c_6_9·a_1_0·a_5_4 − c_6_8·a_1_0·a_5_6
       − c_6_8·a_1_0·a_5_3
  262. a_5_2·a_7_8 + b_4_5·a_3_0·a_5_4
  263. a_5_3·a_7_9 + b_4_5·a_3_0·a_5_6 + b_4_5·a_3_0·a_5_4 − c_6_9·a_1_0·a_5_4
       + c_6_9·a_1_0·a_5_3
  264. a_5_6·a_7_8 + a_5_5·a_7_9 − b_4_5·a_3_0·a_5_6 + c_6_9·a_1_0·a_5_6 − c_6_9·a_1_0·a_5_4
       − c_6_8·a_1_0·a_5_6 + c_6_8·a_1_0·a_5_3
  265. a_5_6·a_7_8 + a_5_4·a_7_9 + b_4_5·a_3_0·a_5_6 − b_4_5·a_3_0·a_5_4 + c_6_9·a_1_0·a_5_3
       − c_6_8·a_1_0·a_5_6 + c_6_8·a_1_0·a_5_3
  266. a_5_6·a_7_9 − a_5_6·a_7_8 + b_4_5·a_3_0·a_5_6 − c_6_9·a_1_0·a_5_6 − c_6_9·a_1_0·a_5_3
       + c_6_8·a_1_0·a_5_6 − c_6_8·a_1_0·a_5_3
  267. a_5_2·a_7_9 + b_4_5·a_3_0·a_5_6 + b_4_5·a_3_0·a_5_4
  268. a_5_3·a_7_10 + b_4_5·a_3_0·a_5_4 + c_6_9·a_1_0·a_5_6 − c_6_9·a_1_0·a_5_4
       + c_6_8·a_1_0·a_5_6 + c_6_8·a_1_0·a_5_4 − c_6_8·a_1_0·a_5_3
  269. a_5_5·a_7_10 − b_4_5·a_3_0·a_5_4 + c_6_9·a_1_0·a_5_6 + c_6_9·a_1_0·a_5_4
       + c_6_8·a_1_0·a_5_3
  270. a_5_4·a_7_10 + b_4_5·a_3_0·a_5_4 − c_6_9·a_1_0·a_5_4 − c_6_9·a_1_0·a_5_3
       + c_6_8·a_1_0·a_5_6
  271. a_5_6·a_7_10 − a_5_6·a_7_8 + c_6_9·a_1_0·a_5_6 + c_6_9·a_1_0·a_5_4 − c_6_8·a_1_0·a_5_4
       + c_6_8·a_1_0·a_5_3
  272. a_5_2·a_7_10 + b_4_5·a_3_0·a_5_4
  273. a_4_3·a_8_8
  274. a_4_4·a_8_8
  275. b_4_5·a_8_8 − a_5_6·a_7_8 − b_4_5·a_3_0·a_5_6 + b_4_5·a_3_0·a_5_4 + c_6_9·a_1_0·a_5_6
       + c_6_8·a_1_0·a_5_6 − c_6_8·a_1_0·a_5_3
  276. a_4_2·a_8_8
  277. b_6_7·a_7_8 + b_4_52·a_5_4 − b_4_52·a_5_2 + a_2_0·c_6_9·a_5_5 − a_2_0·c_6_9·a_5_4
       + a_2_0·c_6_9·a_5_3 − a_2_0·c_6_8·a_5_5 − a_2_0·c_6_8·a_5_4 + a_2_0·c_6_8·a_5_3
  278. a_6_6·a_7_8 − a_3_0·a_5_4·a_5_6 − a_2_0·c_6_9·a_5_5 − a_2_0·c_6_9·a_5_4
       − a_2_0·c_6_9·a_5_3 + a_2_0·c_6_8·a_5_5 + a_2_0·c_6_8·a_5_4 + a_2_0·c_6_8·a_5_3
  279. a_6_4·a_7_8 + a_2_0·c_6_9·a_5_5 + a_2_0·c_6_8·a_5_5 + a_2_0·c_6_8·a_5_4
       − a_2_0·c_6_8·a_5_3
  280. a_6_5·a_7_8 + a_2_0·c_6_9·a_5_5 − a_2_0·c_6_9·a_5_4 + a_2_0·c_6_9·a_5_3
       − a_2_0·c_6_8·a_5_5 − a_2_0·c_6_8·a_5_4 + a_2_0·c_6_8·a_5_3
  281. a_6_3·a_7_8 − a_2_0·c_6_8·a_5_4
  282. b_6_7·a_7_9 + b_4_52·a_5_6 + b_4_52·a_5_4 + b_4_52·a_5_2 + a_3_0·a_5_4·a_5_6
       + a_2_0·c_6_9·a_5_5 + a_2_0·c_6_9·a_5_4 − a_2_0·c_6_9·a_5_3
  283. a_6_6·a_7_9 − a_3_0·a_5_4·a_5_6 − a_2_0·c_6_9·a_5_5 − a_2_0·c_6_9·a_5_4
       − a_2_0·c_6_9·a_5_3
  284. a_6_4·a_7_9 − a_2_0·c_6_9·a_5_5 − a_2_0·c_6_9·a_5_4 + a_2_0·c_6_9·a_5_3
  285. a_6_5·a_7_9 + a_3_0·a_5_4·a_5_6 + a_2_0·c_6_9·a_5_5 + a_2_0·c_6_9·a_5_4
       − a_2_0·c_6_9·a_5_3
  286. a_6_3·a_7_9 + a_3_0·a_5_4·a_5_6 + a_2_0·c_6_9·a_5_4
  287. b_6_7·a_7_10 + b_4_52·a_5_4 − b_4_52·a_5_2 + a_3_0·a_5_4·a_5_6 + a_2_0·c_6_9·a_5_5
       + a_2_0·c_6_8·a_5_5 − a_2_0·c_6_8·a_5_3
  288. a_6_6·a_7_10 − a_3_0·a_5_4·a_5_6 − a_2_0·c_6_9·a_5_4 − a_2_0·c_6_9·a_5_3
       + a_2_0·c_6_8·a_5_4 + a_2_0·c_6_8·a_5_3
  289. a_6_4·a_7_10 + a_2_0·c_6_9·a_5_4 − a_2_0·c_6_9·a_5_3 + a_2_0·c_6_8·a_5_5
       + a_2_0·c_6_8·a_5_3
  290. a_6_5·a_7_10 + a_2_0·c_6_9·a_5_5 + a_2_0·c_6_8·a_5_5 − a_2_0·c_6_8·a_5_3
  291. a_6_3·a_7_10 + a_2_0·c_6_9·a_5_4 + a_2_0·c_6_9·a_5_3 − a_2_0·c_6_8·a_5_4
       + a_2_0·c_6_8·a_5_3
  292. a_8_8·a_5_3 − a_3_0·a_5_4·a_5_6 + a_2_0·c_6_9·a_5_5 + a_2_0·c_6_9·a_5_4
       + a_2_0·c_6_9·a_5_3 + a_2_0·c_6_8·a_5_5 + a_2_0·c_6_8·a_5_4
  293. a_8_8·a_5_5 − a_3_0·a_5_4·a_5_6 − a_2_0·c_6_9·a_5_5 + a_2_0·c_6_8·a_5_5
  294. a_8_8·a_5_4 + a_2_0·c_6_9·a_5_5 − a_2_0·c_6_9·a_5_3 + a_2_0·c_6_8·a_5_5
       − a_2_0·c_6_8·a_5_3
  295. a_8_8·a_5_6 + a_3_0·a_5_4·a_5_6 + a_2_0·c_6_9·a_5_5 + a_2_0·c_6_9·a_5_4
       + a_2_0·c_6_8·a_5_5 + a_2_0·c_6_8·a_5_3
  296. a_8_8·a_5_2 − a_3_0·a_5_4·a_5_6
  297. a_7_8·a_7_9 + b_4_5·a_5_4·a_5_6 − b_4_5·a_3_0·a_7_9 − b_4_5·a_3_0·a_7_8
  298. a_7_9·a_7_10 − b_4_5·a_5_4·a_5_6 + b_4_5·a_3_0·a_7_9 + b_4_5·a_3_0·a_7_8
       + c_6_9·a_1_1·a_7_10
  299. a_7_8·a_7_10 + c_6_9·a_1_0·a_7_10 − c_6_8·a_1_1·a_7_10
  300. b_6_7·a_8_8 − b_4_5·a_5_4·a_5_6 + b_4_5·a_3_0·a_7_8
  301. a_6_6·a_8_8
  302. a_6_4·a_8_8
  303. a_6_5·a_8_8
  304. a_6_3·a_8_8
  305. a_8_8·a_7_10 + a_3_0·a_5_4·a_7_9 − a_2_1·c_6_8·a_7_10
  306. a_8_8·a_7_9 + a_2_1·c_6_9·a_7_10
  307. a_8_8·a_7_8 + a_3_0·a_5_4·a_7_9 − a_2_1·c_6_9·a_7_10 − a_2_1·c_6_8·a_7_10
       − a_2_0·c_6_9·a_7_10
  308. a_8_82

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 243

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_6_8, a Duflot regular element of degree 6
    2. c_6_9, a Duflot regular element of degree 6
    3. b_4_5, an element of degree 4
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 9, 13].
  • 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 243

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_3_0 → 0, an element of degree 3
  6. a_3_1 → 0, an element of degree 3
  7. a_3_2 → 0, an element of degree 3
  8. a_3_3 → 0, an element of degree 3
  9. a_4_2 → 0, an element of degree 4
  10. a_4_3 → 0, an element of degree 4
  11. a_4_4 → 0, an element of degree 4
  12. b_4_5 → 0, an element of degree 4
  13. a_5_2 → 0, an element of degree 5
  14. a_5_3 → 0, an element of degree 5
  15. a_5_4 → 0, an element of degree 5
  16. a_5_5 → 0, an element of degree 5
  17. a_5_6 → 0, an element of degree 5
  18. a_6_3 → 0, an element of degree 6
  19. a_6_4 → 0, an element of degree 6
  20. a_6_5 → 0, an element of degree 6
  21. a_6_6 → 0, an element of degree 6
  22. b_6_7 → 0, an element of degree 6
  23. c_6_8 →  − c_2_23, an element of degree 6
  24. c_6_9 → c_2_23 + c_2_13, an element of degree 6
  25. a_7_8 → 0, an element of degree 7
  26. a_7_9 → 0, an element of degree 7
  27. a_7_10 → 0, an element of degree 7
  28. a_8_8 → 0, 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_3_0 →  − c_2_5·a_1_2, an element of degree 3
  6. a_3_1 → 0, an element of degree 3
  7. a_3_2 →  − c_2_5·a_1_2, an element of degree 3
  8. a_3_3 → 0, an element of degree 3
  9. a_4_2 → 0, an element of degree 4
  10. a_4_3 → 0, an element of degree 4
  11. a_4_4 → 0, an element of degree 4
  12. b_4_5 →  − c_2_52, an element of degree 4
  13. a_5_2 → c_2_52·a_1_2, an element of degree 5
  14. a_5_3 → c_2_52·a_1_2, an element of degree 5
  15. a_5_4 →  − c_2_52·a_1_1 + c_2_4·c_2_5·a_1_2, an element of degree 5
  16. a_5_5 → c_2_52·a_1_2 − c_2_52·a_1_1 + c_2_4·c_2_5·a_1_2, an element of degree 5
  17. a_5_6 →  − c_2_52·a_1_2 − c_2_52·a_1_0 + c_2_3·c_2_5·a_1_2, an element of degree 5
  18. a_6_3 →  − c_2_52·a_1_1·a_1_2, an element of degree 6
  19. a_6_4 → 0, an element of degree 6
  20. a_6_5 →  − c_2_52·a_1_1·a_1_2, an element of degree 6
  21. a_6_6 →  − c_2_52·a_1_0·a_1_2, an element of degree 6
  22. b_6_7 →  − c_2_52·a_1_1·a_1_2 + c_2_53, an element of degree 6
  23. c_6_8 → c_2_4·c_2_52 − c_2_43, an element of degree 6
  24. c_6_9 →  − c_2_52·a_1_1·a_1_2 + c_2_52·a_1_0·a_1_2 − c_2_4·c_2_52 + c_2_43 − c_2_3·c_2_52
       + c_2_33, an element of degree 6
  25. a_7_8 → c_2_53·a_1_2 + c_2_53·a_1_1 − c_2_4·c_2_52·a_1_2, an element of degree 7
  26. a_7_9 → c_2_53·a_1_1 + c_2_53·a_1_0 − c_2_4·c_2_52·a_1_2 − c_2_3·c_2_52·a_1_2, an element of degree 7
  27. a_7_10 →  − c_2_52·a_1_0·a_1_1·a_1_2 + c_2_53·a_1_2 + c_2_53·a_1_1 − c_2_4·c_2_52·a_1_2, an element of degree 7
  28. a_8_8 → c_2_53·a_1_0·a_1_1 − c_2_4·c_2_52·a_1_0·a_1_2 + c_2_3·c_2_52·a_1_1·a_1_2, an element of degree 8

About the group Ring generators Ring relations Completion information Restriction maps Back to groups of order 243


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