Cohomology of group number 13 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 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
    t2  −  t  +  1
    (t  −  1)4 · (t2  +  t  +  1)
  • 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 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_2, a nilpotent element of degree 2
  5. b_2_1, an 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_7, a nilpotent element of degree 4
  13. a_4_8, a nilpotent element of degree 4
  14. a_4_10, a nilpotent element of degree 4
  15. b_4_11, an element of degree 4
  16. a_5_14, a nilpotent element of degree 5
  17. a_5_15, a nilpotent element of degree 5
  18. a_5_18, a nilpotent element of degree 5
  19. a_5_19, a nilpotent element of degree 5
  20. a_6_21, a nilpotent element of degree 6
  21. a_6_22, a nilpotent element of degree 6
  22. a_6_25, a nilpotent element of degree 6
  23. b_6_26, an element of degree 6
  24. c_6_29, a Duflot regular element of degree 6
  25. a_7_33, a nilpotent element of degree 7
  26. a_7_38, a nilpotent element of degree 7
  27. a_7_39, a nilpotent element of degree 7
  28. a_8_50, 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_32, a_3_42, a_3_52, a_3_62, a_3_72, a_5_142, a_5_152, a_5_182, a_5_192, a_7_332, a_7_382, a_7_392

Apart from that, there are 280 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_2·a_1_1
  5. a_2_2·a_1_0
  6. b_2_1·a_1_0
  7. a_2_02
  8. a_2_22
  9.  − a_2_0·b_2_1 + a_1_1·a_3_3
  10. a_1_0·a_3_3
  11. a_1_1·a_3_4 − a_2_0·a_2_2
  12. a_1_0·a_3_4
  13.  − a_2_2·b_2_1 + a_1_1·a_3_5
  14. a_1_0·a_3_5 − a_2_0·a_2_2
  15. a_1_1·a_3_6
  16. a_1_0·a_3_6 + a_2_0·a_2_2
  17. a_1_0·a_3_7
  18. a_2_0·a_3_3
  19. b_2_1·a_3_4 + a_2_2·a_3_3
  20. a_2_2·a_3_4
  21. a_2_0·a_3_4
  22. a_2_2·a_3_5
  23. a_2_2·a_3_3 + a_2_0·a_3_5
  24. a_2_2·a_3_6
  25. a_2_0·a_3_6
  26.  − b_2_1·a_3_6 + a_2_0·a_3_7
  27. a_4_7·a_1_1 + a_2_2·a_3_3
  28. a_4_7·a_1_0
  29.  − b_2_1·a_3_6 + a_4_8·a_1_1
  30. a_4_8·a_1_0
  31. a_4_10·a_1_1 + a_2_2·a_3_7
  32. a_4_10·a_1_0
  33. b_4_11·a_1_0
  34. a_3_3·a_3_4
  35. a_3_4·a_3_5
  36. a_3_4·a_3_6
  37. a_3_3·a_3_6
  38. a_3_6·a_3_7
  39.  − a_3_5·a_3_6 + a_3_4·a_3_7
  40. b_2_1·a_4_7 − a_3_3·a_3_5
  41. a_2_2·a_4_7
  42. a_2_0·a_4_7
  43. a_3_5·a_3_6 + a_2_2·a_4_8
  44. a_2_0·a_4_8
  45. b_2_1·a_4_10 + a_3_5·a_3_7
  46. a_2_2·a_4_10
  47. a_3_5·a_3_6 + a_2_0·a_4_10
  48.  − b_2_1·a_4_8 + a_2_0·b_4_11 + a_3_3·a_3_7
  49. a_3_5·a_3_6 + a_1_1·a_5_14
  50. a_1_0·a_5_14
  51.  − b_2_1·a_4_8 + a_3_3·a_3_7 + a_1_1·a_5_15
  52. a_1_0·a_5_15
  53.  − a_2_2·b_4_11 + a_1_1·a_5_18
  54. a_1_0·a_5_18
  55. a_1_0·a_5_19
  56. a_4_7·a_3_6
  57. a_4_7·a_3_5
  58. a_4_7·a_3_4
  59. a_4_7·a_3_3
  60. a_4_8·a_3_6
  61. a_4_8·a_3_4
  62. a_4_8·a_3_3
  63. a_4_10·a_3_7
  64. a_4_10·a_3_6
  65. a_4_10·a_3_5
  66. a_4_10·a_3_4
  67. a_4_10·a_3_3 + a_4_7·a_3_7
  68. b_4_11·a_3_6 − a_4_8·a_3_7
  69. b_4_11·a_3_4 − a_4_8·a_3_5 − a_4_7·a_3_7
  70. b_2_1·a_5_14 − a_4_8·a_3_5 − a_1_1·a_3_3·a_3_5
  71. a_2_2·a_5_14
  72. a_2_0·a_5_14
  73. a_4_8·a_3_5 + a_4_7·a_3_7 + a_2_2·a_5_15
  74. a_2_0·a_5_15
  75.  − b_4_11·a_3_5 + b_2_1·a_5_18 − a_4_8·a_3_7
  76. a_2_2·a_5_18
  77.  − a_4_8·a_3_5 − a_4_7·a_3_7 + a_2_0·a_5_18
  78.  − a_4_8·a_3_7 + a_2_0·a_5_19
  79. a_6_21·a_1_1 − a_4_8·a_3_5 − a_4_7·a_3_7 − a_1_1·a_3_3·a_3_5
  80. a_6_21·a_1_0
  81. a_6_22·a_1_1 − a_4_8·a_3_7
  82. a_6_22·a_1_0
  83. a_6_25·a_1_1 + a_2_2·a_5_19 + a_1_1·a_3_3·a_3_5
  84. a_6_25·a_1_0
  85. b_6_26·a_1_1 + b_4_11·a_3_3 − b_2_1·a_5_15
  86. b_6_26·a_1_0
  87. a_4_72
  88. a_4_82
  89. a_4_7·a_4_8
  90. a_4_7·a_4_10
  91. a_4_102
  92. a_3_7·a_5_14 − a_4_8·a_4_10 + a_2_0·a_3_5·a_3_7
  93. a_3_6·a_5_14
  94. a_3_5·a_5_14
  95. a_3_4·a_5_14
  96. a_3_3·a_5_14
  97. a_3_6·a_5_15
  98. a_3_4·a_5_15
  99.  − a_4_10·b_4_11 + a_3_7·a_5_18
  100. a_3_6·a_5_18 − a_4_8·a_4_10
  101. a_3_5·a_5_18 + a_4_8·a_4_10
  102. a_3_4·a_5_18
  103.  − a_4_7·b_4_11 + a_3_3·a_5_18
  104. a_3_3·a_5_15 − b_4_11·a_1_1·a_3_7 + b_2_1·a_1_1·a_5_19 + b_2_1·a_1_1·a_5_15
  105. a_3_6·a_5_19
  106. a_3_4·a_5_19 + a_4_8·a_4_10
  107.  − a_4_8·b_4_11 + a_3_3·a_5_19
  108. b_2_1·a_6_21 + a_3_5·a_5_15 − b_2_1·a_3_3·a_3_5 + b_2_1·a_1_1·a_5_18
  109. a_2_2·a_6_21
  110. a_2_0·a_6_21
  111. b_2_1·a_6_22 + a_3_7·a_5_15 + a_3_3·a_5_15
  112.  − a_4_8·a_4_10 + a_2_2·a_6_22
  113. a_2_0·a_6_22
  114. b_2_1·a_6_25 + a_3_5·a_5_19 + b_2_1·a_3_3·a_3_5 − b_2_1·a_1_1·a_5_18
  115. a_2_2·a_6_25
  116.  − a_4_8·a_4_10 + a_2_0·a_6_25
  117. a_4_7·b_4_11 + a_2_2·b_6_26 + a_3_5·a_5_15
  118. a_2_0·b_6_26 + a_3_3·a_5_15
  119. a_1_1·a_7_33 − a_4_8·a_4_10
  120. a_1_0·a_7_33
  121. a_4_7·b_4_11 + a_3_5·a_5_15 + a_1_1·a_7_38 − b_2_1·a_1_1·a_5_18
  122. a_1_0·a_7_38
  123. a_4_8·b_4_11 + a_3_7·a_5_15 + a_1_1·a_7_39
  124. a_1_0·a_7_39
  125. a_4_8·a_5_14
  126. a_4_7·a_5_14
  127. a_4_10·a_5_14
  128. a_4_7·a_5_18
  129. a_4_10·a_5_18
  130.  − b_4_11·a_5_14 + a_4_8·a_5_18 + a_1_1·a_3_3·a_5_18
  131. a_4_8·a_5_19
  132. a_4_8·a_5_18 + a_4_7·a_5_19
  133.  − b_4_11·a_5_14 + a_4_8·a_5_18 + a_4_7·a_5_15 + a_1_1·a_3_7·a_5_18 + a_1_1·a_3_5·a_5_19
  134.  − a_4_8·a_5_15 − a_1_1·a_3_7·a_5_19 + a_1_1·a_3_3·a_5_19
  135. a_6_21·a_3_7 + a_4_10·a_5_15 − a_3_3·a_3_5·a_3_7 − a_1_1·a_3_7·a_5_18
  136. a_6_21·a_3_6
  137. a_6_21·a_3_5
  138. a_6_21·a_3_4
  139.  − b_4_11·a_5_14 + a_6_21·a_3_3 + a_4_8·a_5_18 + a_4_7·a_5_15
  140. a_6_22·a_3_7 − a_4_8·a_5_15
  141. a_6_22·a_3_6
  142. a_6_22·a_3_5 − a_4_10·a_5_15 − a_4_7·a_5_15
  143. a_6_22·a_3_4
  144. a_6_22·a_3_3 + a_4_8·a_5_15
  145. a_6_25·a_3_7 + a_4_10·a_5_19 + a_3_3·a_3_5·a_3_7 + a_1_1·a_3_7·a_5_18
  146. a_6_25·a_3_6
  147. a_6_25·a_3_5
  148. a_6_25·a_3_4
  149. b_4_11·a_5_14 + a_6_25·a_3_3 + a_4_8·a_5_18
  150. b_6_26·a_3_6 − a_4_8·a_5_15
  151. b_6_26·a_3_4 − a_4_7·a_5_15
  152. b_6_26·a_3_3 − b_4_112·a_1_1 + b_2_1·b_4_11·a_3_7 − b_2_1·b_4_11·a_3_3 − b_2_12·a_5_19
  153.  − b_4_11·a_5_14 + b_2_1·a_7_33 − a_4_10·a_5_15 + a_4_8·a_5_18 + a_1_1·a_3_7·a_5_18
       + b_2_1·a_1_1·a_3_3·a_3_5
  154. a_2_2·a_7_33
  155. a_2_0·a_7_33
  156.  − b_6_26·a_3_5 + b_2_1·a_7_38 − b_2_12·a_5_18 − a_4_8·a_5_15
  157. a_2_2·a_7_38
  158.  − b_4_11·a_5_14 + a_4_8·a_5_18 − a_4_7·a_5_15 + a_2_0·a_7_38
  159.  − b_6_26·a_3_7 + b_4_11·a_5_15 − b_4_112·a_1_1 + b_2_1·a_7_39
  160. a_4_10·a_5_15 − a_4_8·a_5_18 + a_2_2·a_7_39
  161.  − a_4_8·a_5_15 + a_2_0·a_7_39
  162.  − b_4_11·a_5_14 + a_8_50·a_1_1 − a_4_10·a_5_15 − a_4_8·a_5_18 + a_4_7·a_5_15
       − a_1_1·a_3_7·a_5_18 + b_2_1·a_1_1·a_3_3·a_3_5
  163. a_8_50·a_1_0
  164. a_5_14·a_5_18
  165.  − a_5_14·a_5_15 + a_2_2·a_3_7·a_5_19 + a_2_0·a_3_5·a_5_19
  166. a_5_14·a_5_19 − a_5_14·a_5_15 + a_2_2·a_3_7·a_5_19 + a_2_0·a_2_2·c_6_29
  167. a_4_8·a_6_21 + a_2_2·a_3_7·a_5_19
  168. a_4_7·a_6_21
  169. a_4_10·a_6_21
  170. b_4_11·a_6_21 − a_5_15·a_5_18 + b_4_11·a_1_1·a_5_18 − b_2_1·a_3_3·a_5_18
  171. a_4_8·a_6_22
  172.  − a_5_14·a_5_15 + a_4_7·a_6_22
  173. a_5_14·a_5_19 + a_4_10·a_6_22 + a_2_2·a_3_7·a_5_19
  174. b_4_11·a_6_22 − a_5_15·a_5_19 − c_6_29·a_1_1·a_3_3
  175. a_5_14·a_5_19 + a_4_8·a_6_25
  176. a_4_7·a_6_25
  177. a_4_10·a_6_25
  178. b_4_11·a_6_25 + a_5_18·a_5_19 − b_4_11·a_1_1·a_5_18 + b_2_1·a_3_3·a_5_18
  179. a_4_8·b_6_26 − b_4_11·a_1_1·a_5_19 + b_2_1·a_3_7·a_5_19 − b_2_1·a_3_3·a_5_19
  180. a_4_7·b_6_26 − b_4_11·a_1_1·a_5_18 + b_2_1·a_3_7·a_5_18 + b_2_1·a_3_5·a_5_19
       − b_2_1·a_3_3·a_5_18
  181. a_5_14·a_5_19 + a_3_7·a_7_33 − a_1_1·a_3_3·a_3_5·a_3_7
  182. a_3_6·a_7_33
  183. a_3_5·a_7_33
  184. a_3_4·a_7_33
  185. a_3_3·a_7_33 − a_2_2·a_3_7·a_5_19
  186.  − a_4_10·b_6_26 + a_3_7·a_7_38 − b_2_1·a_3_7·a_5_18
  187.  − a_5_14·a_5_19 + a_3_6·a_7_38 + a_2_2·a_3_7·a_5_19
  188. a_5_14·a_5_19 + a_3_5·a_7_38 − a_2_2·a_3_7·a_5_19
  189. a_3_4·a_7_38
  190. a_3_3·a_7_38 − b_4_11·a_1_1·a_5_18 + b_2_1·a_3_7·a_5_18 + b_2_1·a_3_5·a_5_19
       + b_2_1·a_3_3·a_5_18
  191.  − a_5_15·a_5_19 + a_3_7·a_7_39 + b_4_11·a_1_1·a_5_19 + b_4_11·a_1_1·a_5_15
       − c_6_29·a_1_1·a_3_3
  192. a_3_6·a_7_39
  193. a_4_10·b_6_26 − a_5_15·a_5_18 + a_3_5·a_7_39 + b_4_11·a_1_1·a_5_18
  194.  − a_5_14·a_5_15 + a_3_4·a_7_39
  195. a_3_3·a_7_39 − b_4_11·a_1_1·a_5_19 + b_4_11·a_1_1·a_5_15 + b_2_1·a_3_7·a_5_19
       − b_2_1·a_3_3·a_5_19
  196.  − a_4_10·b_6_26 + b_2_1·a_8_50 + a_5_15·a_5_18 − b_4_11·a_1_1·a_5_18 + b_2_1·a_3_7·a_5_18
       − b_2_1·a_3_5·a_5_19 + b_2_12·a_3_3·a_3_5 + b_2_12·a_1_1·a_5_18 − c_6_29·a_1_1·a_3_5
  197. a_2_2·a_8_50
  198. a_5_14·a_5_19 + a_5_14·a_5_15 + a_2_0·a_8_50
  199. a_6_21·a_5_18
  200. a_6_21·a_5_14
  201. a_6_22·a_5_18 + a_6_21·a_5_19 − a_3_3·a_3_5·a_5_19 + a_1_1·a_5_18·a_5_19
       − a_2_0·c_6_29·a_3_5
  202. a_6_22·a_5_19 − a_2_0·c_6_29·a_3_7
  203. a_6_22·a_5_15
  204. a_6_22·a_5_14
  205. a_6_25·a_5_18
  206. a_6_25·a_5_19 + a_3_3·a_3_5·a_5_19 − a_1_1·a_5_18·a_5_19
  207. a_6_25·a_5_15 + a_6_21·a_5_19 + a_6_21·a_5_15 − a_3_3·a_3_5·a_5_19 + a_1_1·a_5_18·a_5_19
  208. a_6_25·a_5_14
  209. b_6_26·a_5_15 + b_4_112·a_3_7 + b_4_112·a_3_3 − b_2_1·b_4_11·a_5_19
       + b_2_1·b_4_11·a_5_15 + b_2_12·c_6_29·a_1_1
  210. b_6_26·a_5_14 + a_3_5·a_3_7·a_5_19 + a_3_3·a_3_5·a_5_19 + a_1_1·a_5_18·a_5_19
       + b_2_1·a_1_1·a_3_7·a_5_18 + b_2_1·a_1_1·a_3_5·a_5_19 − b_2_1·a_1_1·a_3_3·a_5_18
  211. a_4_8·a_7_33
  212. a_4_7·a_7_33
  213. a_4_10·a_7_33
  214. b_4_11·a_7_33 + a_6_21·a_5_19 − a_3_3·a_3_5·a_5_19 + b_2_1·a_1_1·a_3_3·a_5_18
       − a_2_0·c_6_29·a_3_5
  215. a_4_8·a_7_38 + a_3_5·a_3_7·a_5_19 − a_3_3·a_3_5·a_5_19 + a_1_1·a_5_18·a_5_19
  216. a_4_7·a_7_38
  217. a_4_10·a_7_38
  218.  − b_6_26·a_5_18 + b_4_11·a_7_38 − b_2_1·b_4_11·a_5_18
  219.  − a_6_21·a_5_15 + a_4_7·a_7_39 − a_3_5·a_3_7·a_5_19 − a_3_3·a_3_5·a_5_19
       − a_1_1·a_5_18·a_5_19 − b_2_1·a_1_1·a_3_7·a_5_18 − b_2_1·a_1_1·a_3_5·a_5_19
       + b_2_1·a_1_1·a_3_3·a_5_18
  220.  − a_6_21·a_5_19 − a_6_21·a_5_15 + a_4_10·a_7_39 + a_3_3·a_3_5·a_5_19
       − b_2_1·a_1_1·a_3_7·a_5_18 − b_2_1·a_1_1·a_3_5·a_5_19 + b_2_1·a_1_1·a_3_3·a_5_18
       + a_2_0·c_6_29·a_3_5
  221.  − b_6_26·a_5_19 + b_4_11·a_7_39 − b_4_112·a_3_7 + b_4_112·a_3_3 + b_2_1·b_4_11·a_5_19
       − b_2_1·c_6_29·a_3_3
  222.  − a_4_8·a_7_39 + a_1_1·a_3_7·a_7_39
  223.  − a_6_21·a_5_15 + a_1_1·a_3_7·a_7_38 + a_1_1·a_3_5·a_7_39 + b_2_1·a_1_1·a_3_7·a_5_18
       − b_2_1·a_1_1·a_3_5·a_5_19 + b_2_1·a_1_1·a_3_3·a_5_18
  224. a_8_50·a_3_7 + a_6_21·a_5_19 + a_6_21·a_5_15 + a_3_5·a_3_7·a_5_19 − a_3_3·a_3_5·a_5_19
       + b_2_1·a_3_3·a_3_5·a_3_7 + b_2_1·a_1_1·a_3_5·a_5_19 − b_2_1·a_1_1·a_3_3·a_5_18
       − a_2_2·c_6_29·a_3_7 − a_2_0·c_6_29·a_3_5
  225. a_8_50·a_3_6
  226. a_8_50·a_3_5
  227. a_8_50·a_3_4
  228. a_8_50·a_3_3 + a_3_5·a_3_7·a_5_19 − a_3_3·a_3_5·a_5_19 + a_1_1·a_5_18·a_5_19
       − b_2_1·a_1_1·a_3_3·a_5_18 + a_2_0·c_6_29·a_3_5
  229. a_6_212
  230. a_6_21·a_6_22 − a_2_0·a_5_18·a_5_19 − a_1_1·a_3_5·a_3_7·a_5_19
       − a_1_1·a_3_3·a_3_5·a_5_19
  231. a_6_222
  232. a_6_252
  233. a_6_21·a_6_25
  234. a_6_22·a_6_25 + a_2_0·a_5_18·a_5_19 + a_1_1·a_3_5·a_3_7·a_5_19
       + a_1_1·a_3_3·a_3_5·a_5_19 − a_2_0·a_4_10·c_6_29
  235. b_6_262 + b_4_113 + b_2_1·b_4_11·b_6_26 + b_2_13·c_6_29
  236. a_5_18·a_7_33
  237. a_5_19·a_7_33 − a_1_1·a_3_3·a_3_5·a_5_19 − a_2_0·a_4_10·c_6_29
  238. a_5_15·a_7_33 + a_2_0·a_5_18·a_5_19
  239. a_5_14·a_7_33
  240. a_5_18·a_7_38
  241.  − a_6_25·b_6_26 + a_5_19·a_7_38 + b_4_11·a_1_1·a_7_38 + b_2_1·a_5_18·a_5_19
       + b_2_1·b_4_11·a_1_1·a_5_18 + b_2_12·a_3_7·a_5_18 + b_2_12·a_3_5·a_5_19
       − b_2_12·a_3_3·a_5_18
  242. a_6_21·b_6_26 + a_5_15·a_7_38 − b_4_11·a_3_7·a_5_18 − b_4_11·a_1_1·a_7_38
       − b_2_1·a_5_18·a_5_19 + b_2_12·a_3_7·a_5_18 + b_2_12·a_3_5·a_5_19
       − b_2_12·a_3_3·a_5_18 − b_2_1·c_6_29·a_1_1·a_3_5
  243. a_5_14·a_7_38
  244. a_6_25·b_6_26 − a_6_21·b_6_26 + a_5_18·a_7_39 − b_2_1·b_4_11·a_1_1·a_5_18
       + b_2_12·a_3_7·a_5_18 + b_2_12·a_3_5·a_5_19 − b_2_12·a_3_3·a_5_18
       + c_6_29·a_3_3·a_3_5 − b_2_1·c_6_29·a_1_1·a_3_5
  245.  − a_6_22·b_6_26 − b_4_11·a_3_7·a_5_19 + b_4_11·a_1_1·a_7_39 − b_4_112·a_1_1·a_3_7
       + b_2_1·a_3_7·a_7_39 − b_2_1·b_4_11·a_1_1·a_5_19 − b_2_1·b_4_11·a_1_1·a_5_15
       − b_2_1·c_6_29·a_1_1·a_3_7 − b_2_1·c_6_29·a_1_1·a_3_3
  246.  − a_6_21·b_6_26 − b_4_11·a_3_7·a_5_18 − b_2_1·a_5_18·a_5_19 + b_2_1·a_3_7·a_7_38
       + b_2_1·a_3_5·a_7_39 − b_2_1·b_4_11·a_1_1·a_5_18 + b_2_12·a_3_7·a_5_18
       − b_2_12·a_3_5·a_5_19 + b_2_12·a_3_3·a_5_18 − b_2_1·c_6_29·a_1_1·a_3_5
  247.  − a_6_22·b_6_26 + a_5_19·a_7_39 − b_4_11·a_1_1·a_7_39 + b_4_112·a_1_1·a_3_7
       − b_2_1·b_4_11·a_1_1·a_5_19 − b_2_1·b_4_11·a_1_1·a_5_15 + c_6_29·a_3_3·a_3_7
       + c_6_29·a_1_1·a_5_15 − b_2_1·c_6_29·a_1_1·a_3_7 − b_2_1·c_6_29·a_1_1·a_3_3
  248.  − a_6_22·b_6_26 + a_5_15·a_7_39 + b_4_112·a_1_1·a_3_7 − b_2_1·b_4_11·a_1_1·a_5_19
       − b_2_1·b_4_11·a_1_1·a_5_15 − b_2_1·c_6_29·a_1_1·a_3_3
  249. a_5_14·a_7_39 − a_2_0·a_5_18·a_5_19 − a_1_1·a_3_5·a_3_7·a_5_19
       − a_1_1·a_3_3·a_3_5·a_5_19
  250. a_4_8·a_8_50 − a_1_1·a_3_3·a_3_5·a_5_19 − a_2_0·a_4_10·c_6_29
  251. a_4_7·a_8_50
  252. a_4_10·a_8_50
  253.  − a_6_25·b_6_26 + a_6_21·b_6_26 + b_4_11·a_8_50 + b_4_11·a_3_7·a_5_18
       − b_2_1·a_5_18·a_5_19 − b_2_1·b_4_11·a_1_1·a_5_18 − b_2_12·a_3_7·a_5_18
       − b_2_12·a_3_5·a_5_19 − b_2_12·a_3_3·a_5_18 − c_6_29·a_3_3·a_3_5
       − c_6_29·a_1_1·a_5_18 + b_2_1·c_6_29·a_1_1·a_3_5
  254. a_6_25·a_7_33
  255. a_6_21·a_7_33
  256. a_6_22·a_7_33
  257. b_6_26·a_7_38 + b_4_112·a_5_18 + b_2_12·c_6_29·a_3_5 + c_6_29·a_1_1·a_3_3·a_3_7
  258. a_6_25·a_7_38
  259. a_6_21·a_7_38
  260. a_6_22·a_7_38 − a_3_7·a_5_18·a_5_19 − a_3_3·a_5_18·a_5_19 − c_6_29·a_1_1·a_3_5·a_3_7
       + c_6_29·a_1_1·a_3_3·a_3_5
  261. b_6_26·a_7_39 + b_4_112·a_5_19 − b_4_112·a_5_15 − b_4_113·a_1_1 + b_2_1·b_4_11·a_7_39
       − b_2_1·b_4_11·c_6_29·a_1_1 + b_2_12·c_6_29·a_3_7
  262. b_6_26·a_7_33 + a_6_25·a_7_39 + b_4_11·a_1_1·a_3_7·a_5_18 + b_2_1·a_3_5·a_3_7·a_5_19
       + b_2_1·a_3_3·a_3_5·a_5_19 − b_2_1·a_1_1·a_5_18·a_5_19 − b_2_12·a_1_1·a_3_7·a_5_18
       − b_2_12·a_1_1·a_3_5·a_5_19 + b_2_12·a_1_1·a_3_3·a_5_18 + a_4_7·c_6_29·a_3_7
       − a_2_0·c_6_29·a_5_18 − c_6_29·a_1_1·a_3_5·a_3_7
  263. a_6_22·a_7_39 − b_4_11·a_1_1·a_3_7·a_5_19 + b_2_1·a_1_1·a_3_7·a_7_39
       + c_6_29·a_1_1·a_3_3·a_3_7
  264. b_6_26·a_7_33 + a_6_21·a_7_39 + b_4_11·a_1_1·a_3_7·a_5_18 − b_2_1·a_3_5·a_3_7·a_5_19
       − b_2_1·a_3_3·a_3_5·a_5_19 + b_2_1·a_1_1·a_3_7·a_7_38 + b_2_1·a_1_1·a_3_5·a_7_39
       + b_2_12·a_1_1·a_3_7·a_5_18 − b_2_12·a_1_1·a_3_5·a_5_19
       + b_2_12·a_1_1·a_3_3·a_5_18
  265.  − b_6_26·a_7_33 + a_6_21·a_7_39 − a_3_7·a_5_18·a_5_19 + a_3_3·a_5_18·a_5_19
       + a_1_1·a_5_18·a_7_39 − b_4_11·a_1_1·a_3_7·a_5_18 − b_2_1·a_3_5·a_3_7·a_5_19
       − b_2_1·a_3_3·a_3_5·a_5_19 + b_2_1·a_1_1·a_5_18·a_5_19 + b_2_12·a_1_1·a_3_7·a_5_18
       + b_2_12·a_1_1·a_3_5·a_5_19 − b_2_12·a_1_1·a_3_3·a_5_18 − c_6_29·a_1_1·a_3_5·a_3_7
  266. a_8_50·a_5_18
  267. a_8_50·a_5_19 + a_3_3·a_5_18·a_5_19 + b_2_1·a_3_3·a_3_5·a_5_19
       + b_2_1·a_1_1·a_5_18·a_5_19 − a_4_7·c_6_29·a_3_7 − a_2_2·c_6_29·a_5_19
       + a_2_0·c_6_29·a_5_18
  268. a_8_50·a_5_15 − a_3_7·a_5_18·a_5_19 − a_3_3·a_5_18·a_5_19 − b_4_11·a_1_1·a_3_7·a_5_18
       − b_2_1·a_1_1·a_5_18·a_5_19 − b_2_12·a_1_1·a_3_7·a_5_18 − b_2_12·a_1_1·a_3_5·a_5_19
       + b_2_12·a_1_1·a_3_3·a_5_18 + a_2_0·c_6_29·a_5_18 − c_6_29·a_1_1·a_3_5·a_3_7
       − c_6_29·a_1_1·a_3_3·a_3_5
  269. a_8_50·a_5_14
  270. a_7_33·a_7_38
  271. a_7_33·a_7_39 + a_1_1·a_3_7·a_5_18·a_5_19 − a_1_1·a_3_3·a_5_18·a_5_19
       + b_2_1·a_1_1·a_3_5·a_3_7·a_5_19 + b_2_1·a_1_1·a_3_3·a_3_5·a_5_19
  272. a_7_38·a_7_39 + b_4_11·a_5_18·a_5_19 + b_4_11·a_3_7·a_7_38 − b_4_112·a_1_1·a_5_18
       + b_2_1·a_5_18·a_7_39 − b_2_1·b_4_11·a_3_7·a_5_18 + b_2_1·c_6_29·a_3_5·a_3_7
       + b_2_1·c_6_29·a_1_1·a_5_18
  273. b_6_26·a_8_50 − b_4_11·a_5_18·a_5_19 + b_4_112·a_1_1·a_5_18
       − b_2_1·b_4_11·a_3_7·a_5_18 − b_2_12·a_5_18·a_5_19 + b_2_12·a_3_7·a_7_38
       + b_2_12·a_3_5·a_7_39 − b_2_12·b_4_11·a_1_1·a_5_18 + b_2_13·a_3_7·a_5_18
       − b_2_13·a_3_5·a_5_19 + b_2_13·a_3_3·a_5_18 − c_6_29·a_1_1·a_7_38
       − b_2_1·c_6_29·a_3_5·a_3_7 − b_2_1·c_6_29·a_3_3·a_3_5
  274. a_6_25·a_8_50
  275. a_6_21·a_8_50
  276. a_6_22·a_8_50 − a_1_1·a_3_7·a_5_18·a_5_19 + b_2_1·a_1_1·a_3_5·a_3_7·a_5_19
       + b_2_1·a_1_1·a_3_3·a_3_5·a_5_19 − a_2_0·a_6_25·c_6_29 + a_2_0·c_6_29·a_3_5·a_3_7
  277. a_8_50·a_7_39 + a_3_7·a_5_18·a_7_39 − b_2_1·a_3_7·a_5_18·a_5_19
       + b_2_1·a_3_3·a_5_18·a_5_19 + b_2_1·a_1_1·a_5_18·a_7_39 + b_2_12·a_3_5·a_3_7·a_5_19
       + b_2_12·a_3_3·a_3_5·a_5_19 + b_2_12·a_1_1·a_5_18·a_5_19
       + b_2_12·a_1_1·a_3_7·a_7_38 + b_2_12·a_1_1·a_3_5·a_7_39
       − b_2_13·a_1_1·a_3_7·a_5_18 − a_2_2·c_6_29·a_7_39 − c_6_29·a_3_3·a_3_5·a_3_7
       + c_6_29·a_1_1·a_3_3·a_5_18
  278. a_8_50·a_7_38
  279. a_8_50·a_7_33
  280. a_8_502

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_2_3, a Duflot regular element of degree 2
    2. c_6_29, a Duflot regular element of degree 6
    3. b_4_113 − b_2_12·b_4_112 − b_2_16 + b_2_13·c_6_29, an element of degree 12
    4. b_2_1, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 16, 18].
  • The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
  • We found that there exists some filter regular HSOP formed by the first 2 terms of the above HSOP, together with 2 elements of degree 4.

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_2 → 0, an element of degree 2
  5. b_2_1 → 0, an element of degree 2
  6. c_2_3 → c_2_1, 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_7 → 0, an element of degree 4
  13. a_4_8 → 0, an element of degree 4
  14. a_4_10 → 0, an element of degree 4
  15. b_4_11 → 0, an element of degree 4
  16. a_5_14 → 0, an element of degree 5
  17. a_5_15 → 0, an element of degree 5
  18. a_5_18 → 0, an element of degree 5
  19. a_5_19 → 0, an element of degree 5
  20. a_6_21 → 0, an element of degree 6
  21. a_6_22 → 0, an element of degree 6
  22. a_6_25 → 0, an element of degree 6
  23. b_6_26 → 0, an element of degree 6
  24. c_6_29 → c_2_23, an element of degree 6
  25. a_7_33 → 0, an element of degree 7
  26. a_7_38 → 0, an element of degree 7
  27. a_7_39 → 0, an element of degree 7
  28. a_8_50 → 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_1 → a_1_2, an element of degree 1
  3. a_2_0 →  − a_1_2·a_1_3, an element of degree 2
  4. a_2_2 →  − a_1_0·a_1_2, an element of degree 2
  5. b_2_1 → c_2_8, an element of degree 2
  6. c_2_3 → c_2_6, an element of degree 2
  7. a_3_3 → c_2_9·a_1_2 − c_2_8·a_1_3, an element of degree 3
  8. a_3_4 →  − a_1_0·a_1_2·a_1_3, an element of degree 3
  9. a_3_5 → a_1_1·a_1_2·a_1_3 + c_2_8·a_1_0, an element of degree 3
  10. a_3_6 →  − a_1_1·a_1_2·a_1_3, an element of degree 3
  11. a_3_7 →  − c_2_9·a_1_3 − c_2_9·a_1_2 − c_2_8·a_1_3 + c_2_8·a_1_1 + c_2_7·a_1_2, an element of degree 3
  12. a_4_7 →  − c_2_9·a_1_0·a_1_2 + c_2_8·a_1_0·a_1_3, an element of degree 4
  13. a_4_8 →  − c_2_9·a_1_1·a_1_2 + c_2_8·a_1_1·a_1_3 − c_2_7·a_1_2·a_1_3, an element of degree 4
  14. a_4_10 → c_2_9·a_1_0·a_1_3 + c_2_9·a_1_0·a_1_2 + c_2_8·a_1_0·a_1_3 − c_2_8·a_1_0·a_1_1
       − c_2_7·a_1_0·a_1_2, an element of degree 4
  15. b_4_11 →  − c_2_92 + c_2_8·c_2_9 − c_2_7·c_2_8, an element of degree 4
  16. a_5_14 →  − c_2_9·a_1_0·a_1_1·a_1_2 − c_2_8·a_1_0·a_1_2·a_1_3 + c_2_8·a_1_0·a_1_1·a_1_3
       − c_2_7·a_1_0·a_1_2·a_1_3, an element of degree 5
  17. a_5_15 → c_2_92·a_1_3 − c_2_8·c_2_9·a_1_3 − c_2_7·c_2_9·a_1_2 + c_2_7·c_2_8·a_1_3
       − c_2_7·c_2_8·a_1_2, an element of degree 5
  18. a_5_18 →  − c_2_92·a_1_0 + c_2_8·c_2_9·a_1_0 − c_2_7·c_2_8·a_1_0, an element of degree 5
  19. a_5_19 →  − c_2_92·a_1_1 + c_2_8·c_2_9·a_1_1 + c_2_7·c_2_9·a_1_3 + c_2_7·c_2_9·a_1_2
       + c_2_7·c_2_8·a_1_3 − c_2_7·c_2_8·a_1_1 + c_2_72·a_1_2, an element of degree 5
  20. a_6_21 →  − c_2_92·a_1_0·a_1_3 − c_2_92·a_1_0·a_1_2 + c_2_8·c_2_9·a_1_0·a_1_3
       + c_2_82·a_1_0·a_1_3 + c_2_7·c_2_9·a_1_0·a_1_2 − c_2_7·c_2_8·a_1_0·a_1_3, an element of degree 6
  21. a_6_22 →  − c_2_92·a_1_1·a_1_3 + c_2_8·c_2_9·a_1_1·a_1_3 + c_2_7·c_2_9·a_1_2·a_1_3
       + c_2_7·c_2_9·a_1_1·a_1_2 − c_2_7·c_2_8·a_1_2·a_1_3 − c_2_7·c_2_8·a_1_1·a_1_3
       + c_2_7·c_2_8·a_1_1·a_1_2 − c_2_72·a_1_2·a_1_3, an element of degree 6
  22. a_6_25 → c_2_92·a_1_0·a_1_2 + c_2_92·a_1_0·a_1_1 − c_2_8·c_2_9·a_1_0·a_1_1
       − c_2_82·a_1_0·a_1_3 − c_2_7·c_2_9·a_1_0·a_1_3 − c_2_7·c_2_9·a_1_0·a_1_2
       − c_2_7·c_2_8·a_1_0·a_1_3 + c_2_7·c_2_8·a_1_0·a_1_2 + c_2_7·c_2_8·a_1_0·a_1_1
       − c_2_72·a_1_0·a_1_2, an element of degree 6
  23. b_6_26 → c_2_93 − c_2_8·c_2_92 − c_2_7·c_2_82, an element of degree 6
  24. c_6_29 →  − c_2_7·c_2_92 + c_2_7·c_2_8·c_2_9 + c_2_72·c_2_8 + c_2_73, an element of degree 6
  25. a_7_33 →  − c_2_92·a_1_0·a_1_1·a_1_3 − c_2_92·a_1_0·a_1_1·a_1_2
       + c_2_8·c_2_9·a_1_0·a_1_1·a_1_3 + c_2_8·c_2_9·a_1_0·a_1_1·a_1_2
       + c_2_82·a_1_0·a_1_2·a_1_3 + c_2_7·c_2_9·a_1_0·a_1_1·a_1_2
       + c_2_7·c_2_8·a_1_0·a_1_2·a_1_3 − c_2_7·c_2_8·a_1_0·a_1_1·a_1_3
       − c_2_72·a_1_0·a_1_2·a_1_3, an element of degree 7
  26. a_7_38 → c_2_93·a_1_0 + c_2_8·c_2_92·a_1_0 + c_2_82·c_2_9·a_1_0 + c_2_7·c_2_82·a_1_0, an element of degree 7
  27. a_7_39 → c_2_93·a_1_3 − c_2_93·a_1_2 + c_2_93·a_1_1 − c_2_8·c_2_92·a_1_3
       + c_2_8·c_2_92·a_1_2 − c_2_8·c_2_92·a_1_1 − c_2_7·c_2_92·a_1_3 + c_2_7·c_2_92·a_1_2
       − c_2_7·c_2_8·c_2_9·a_1_3 + c_2_7·c_2_82·a_1_3 − c_2_7·c_2_82·a_1_1
       − c_2_72·c_2_9·a_1_2 + c_2_72·c_2_8·a_1_3 − c_2_72·c_2_8·a_1_2, an element of degree 7
  28. a_8_50 →  − c_2_93·a_1_0·a_1_2 − c_2_93·a_1_0·a_1_1 + c_2_8·c_2_92·a_1_0·a_1_3
       − c_2_8·c_2_92·a_1_0·a_1_1 − c_2_82·c_2_9·a_1_0·a_1_3 − c_2_82·c_2_9·a_1_0·a_1_2
       − c_2_82·c_2_9·a_1_0·a_1_1 − c_2_83·a_1_0·a_1_3 + c_2_7·c_2_92·a_1_0·a_1_3
       − c_2_7·c_2_92·a_1_0·a_1_2 − c_2_7·c_2_8·c_2_9·a_1_0·a_1_2
       + c_2_7·c_2_82·a_1_0·a_1_3 − c_2_7·c_2_82·a_1_0·a_1_2 − c_2_7·c_2_82·a_1_0·a_1_1
       + c_2_72·c_2_9·a_1_0·a_1_2 − c_2_72·c_2_8·a_1_0·a_1_3 − c_2_73·a_1_0·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