Cohomology of group number 8 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
    (t2  +  1) · (t7  −  t3  −  1)
    (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 24 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. b_2_2, an element of degree 2
  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_3_4, a nilpotent element of degree 3
  10. a_4_2, a nilpotent element of degree 4
  11. a_4_3, a nilpotent element of degree 4
  12. a_4_4, a nilpotent element of degree 4
  13. a_5_3, a nilpotent element of degree 5
  14. a_5_4, a nilpotent element of degree 5
  15. a_5_5, a nilpotent element of degree 5
  16. a_5_6, a nilpotent element of degree 5
  17. a_6_3, a nilpotent element of degree 6
  18. a_6_4, a nilpotent element of degree 6
  19. a_6_5, a nilpotent element of degree 6
  20. a_6_6, a nilpotent element of degree 6
  21. c_6_8, a Duflot regular element of degree 6
  22. c_6_9, a Duflot regular element of degree 6
  23. a_7_10, a nilpotent element of degree 7
  24. 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 11 "obvious" relations:
   a_1_02, a_1_12, a_3_12, a_3_22, a_3_32, a_3_42, a_5_32, a_5_42, a_5_52, a_5_62, a_7_102

Apart from that, there are 219 minimal relations of maximal degree 16:

  1. a_1_0·a_1_1
  2. a_2_0·a_1_1
  3. a_2_1·a_1_1 − a_2_0·a_1_0
  4. a_2_1·a_1_0 + a_2_0·a_1_0
  5. b_2_2·a_1_1
  6. b_2_2·a_1_0
  7. a_2_02
  8. a_2_0·a_2_1
  9. a_2_12
  10.  − a_2_1·b_2_2 − a_2_0·b_2_2 + a_1_1·a_3_1
  11.  − a_2_0·b_2_2 + a_1_0·a_3_1
  12.  − a_2_0·b_2_2 + a_1_1·a_3_2
  13.  − a_2_1·b_2_2 − a_2_0·b_2_2 + a_1_0·a_3_2
  14. a_2_1·b_2_2 + a_2_0·b_2_2 + a_1_1·a_3_3
  15. a_2_0·b_2_2 + a_1_0·a_3_3
  16. a_2_1·b_2_2 + a_2_0·b_2_2 + a_1_1·a_3_4
  17.  − a_2_0·b_2_2 + a_1_0·a_3_4
  18. b_2_2·a_3_1
  19. a_2_1·a_3_1 + a_2_0·a_3_1
  20. a_2_0·a_3_2
  21. a_2_1·a_3_2 − a_2_0·a_3_1
  22. b_2_2·a_3_3 − b_2_2·a_3_2
  23. a_2_1·a_3_3 + a_2_0·a_3_3 + a_2_0·a_3_1
  24. b_2_2·a_3_4
  25. a_2_0·a_3_4
  26. a_2_1·a_3_4 − a_2_0·a_3_3 − a_2_0·a_3_1
  27. a_4_2·a_1_1 − a_2_0·a_3_1
  28. a_4_2·a_1_0
  29. a_4_3·a_1_1 − a_2_0·a_3_1
  30. a_4_3·a_1_0 − a_2_0·a_3_3 − a_2_0·a_3_1
  31. a_4_4·a_1_1 − a_2_0·a_3_3 + a_2_0·a_3_1
  32. a_4_4·a_1_0 + a_2_0·a_3_1
  33. a_3_1·a_3_2
  34. a_3_3·a_3_4 + a_3_2·a_3_4 + a_3_2·a_3_3
  35.  − a_3_2·a_3_3 + a_3_1·a_3_4
  36. b_2_2·a_4_2 + a_3_3·a_3_4 + a_3_2·a_3_3 + a_3_1·a_3_3
  37. a_2_0·a_4_2
  38. a_2_1·a_4_2
  39. b_2_2·a_4_3 − a_3_3·a_3_4 − a_3_2·a_3_3
  40. a_2_0·a_4_3
  41. a_2_1·a_4_3
  42. b_2_2·a_4_4 + a_3_3·a_3_4 − a_3_2·a_3_3 + a_3_1·a_3_3
  43. a_2_0·a_4_4
  44. a_2_1·a_4_4
  45. a_3_2·a_3_3 + a_1_1·a_5_3
  46. a_3_1·a_3_3 + a_1_0·a_5_3
  47. a_3_3·a_3_4 − a_3_2·a_3_3 − a_3_1·a_3_3 + a_1_1·a_5_4
  48. a_3_2·a_3_3 + a_3_1·a_3_3 + a_1_0·a_5_4
  49.  − a_3_3·a_3_4 − a_3_2·a_3_3 − a_3_1·a_3_3 + a_1_1·a_5_5
  50. a_1_0·a_5_5
  51. a_3_3·a_3_4 − a_3_2·a_3_3 − a_3_1·a_3_3 + a_1_1·a_5_6
  52.  − a_3_3·a_3_4 − a_3_1·a_3_3 + a_1_0·a_5_6
  53. a_4_2·a_3_3 + a_4_2·a_3_2
  54. a_4_2·a_3_1
  55. a_4_3·a_3_2 + a_4_2·a_3_3
  56. a_4_3·a_3_1 − a_4_2·a_3_4
  57. a_4_3·a_3_4 − a_4_2·a_3_4
  58. a_4_4·a_3_3 − a_4_2·a_3_4
  59. a_4_4·a_3_2 − a_4_2·a_3_4 + a_4_2·a_3_3
  60. a_4_4·a_3_1 − a_4_2·a_3_3
  61. a_4_4·a_3_4 − a_4_3·a_3_3 + a_4_2·a_3_4 + a_4_2·a_3_3
  62. b_2_2·a_5_3 − b_2_22·a_3_2
  63.  − a_4_2·a_3_4 + a_2_0·a_5_3
  64. a_4_2·a_3_4 − a_4_2·a_3_3 + a_2_1·a_5_3
  65.  − a_4_2·a_3_4 − a_4_2·a_3_3 + a_2_0·a_5_4
  66. a_2_1·a_5_4
  67. b_2_2·a_5_5 − b_2_2·a_5_4
  68. a_4_3·a_3_3 + a_4_2·a_3_4 + a_4_2·a_3_3 + a_2_0·a_5_5
  69.  − a_4_3·a_3_3 − a_4_2·a_3_3 + a_2_1·a_5_5
  70. a_4_2·a_3_4 − a_4_2·a_3_3 + a_2_0·a_5_6
  71. a_4_3·a_3_3 − a_4_2·a_3_4 + a_2_1·a_5_6
  72. a_6_3·a_1_1 + a_4_2·a_3_3
  73. a_6_3·a_1_0
  74. a_6_4·a_1_1 + a_4_2·a_3_4 + a_4_2·a_3_3
  75. a_6_4·a_1_0 + a_4_3·a_3_3 + a_4_2·a_3_4 + a_4_2·a_3_3
  76. a_6_5·a_1_1 + a_4_3·a_3_3 − a_4_2·a_3_4 − a_4_2·a_3_3
  77. a_6_5·a_1_0 + a_4_3·a_3_3 − a_4_2·a_3_4 + a_4_2·a_3_3
  78. a_6_6·a_1_1 − a_4_2·a_3_4
  79. a_6_6·a_1_0 + a_4_2·a_3_3
  80. a_4_22
  81. a_4_32
  82. a_4_2·a_4_3
  83. a_4_3·a_4_4
  84. a_4_42
  85. a_4_2·a_4_4
  86. a_3_2·a_5_3
  87. a_3_1·a_5_3
  88. a_3_4·a_5_3 − a_3_3·a_5_4 + a_3_3·a_5_3 + a_3_2·a_5_4
  89. a_3_1·a_5_4
  90. a_3_4·a_5_4 − a_3_4·a_5_3 − a_3_3·a_5_3
  91. a_3_4·a_5_3 + a_3_3·a_5_5 − a_3_3·a_5_4 − a_3_3·a_5_3
  92.  − a_3_3·a_5_4 + a_3_3·a_5_3 + a_3_2·a_5_5
  93.  − a_3_3·a_5_3 + a_3_1·a_5_5
  94. a_3_4·a_5_5 − a_3_3·a_5_3
  95.  − a_3_3·a_5_6 − a_3_3·a_5_3 + a_3_2·a_5_6
  96.  − a_3_4·a_5_3 + a_3_1·a_5_6
  97. a_3_4·a_5_6 − a_3_3·a_5_3
  98. b_2_2·a_6_3 + a_3_4·a_5_3 − a_3_3·a_5_4 + a_3_3·a_5_3
  99. a_2_0·a_6_3
  100. a_2_1·a_6_3
  101. b_2_2·a_6_4 + a_3_4·a_5_3 − a_3_3·a_5_4 − a_3_3·a_5_3
  102. a_2_0·a_6_4
  103. a_2_1·a_6_4
  104. b_2_2·a_6_5 + a_3_3·a_5_4
  105. a_2_0·a_6_5
  106. a_2_1·a_6_5
  107. b_2_2·a_6_6 − a_3_4·a_5_3 − a_3_3·a_5_6 + a_3_3·a_5_4 − a_3_3·a_5_3
  108. a_2_0·a_6_6
  109. a_2_1·a_6_6
  110. a_3_4·a_5_3 + a_1_1·a_7_10
  111. a_3_3·a_5_3 + a_1_0·a_7_10
  112. a_4_3·a_5_4 − a_4_3·a_5_3
  113. a_4_4·a_5_4 + a_4_4·a_5_3 − a_4_3·a_5_3
  114.  − a_4_4·a_5_3 + a_4_2·a_5_4
  115. a_4_4·a_5_5 + a_4_4·a_5_3
  116. a_4_4·a_5_3 + a_4_2·a_5_5 − a_4_2·a_5_3
  117. a_4_3·a_5_6 − a_4_3·a_5_3 + a_4_2·a_5_3
  118. a_4_4·a_5_6 − a_4_4·a_5_3 − a_4_3·a_5_5 − a_4_2·a_5_3
  119.  − a_4_4·a_5_3 + a_4_3·a_5_3 + a_4_2·a_5_6
  120. a_4_4·a_5_3 − a_4_3·a_5_5 − a_4_2·a_5_3 + a_2_0·c_6_8·a_1_0
  121. a_4_2·a_5_3 + a_2_0·c_6_9·a_1_0
  122. a_6_3·a_3_3 − a_4_2·a_5_3
  123. a_6_3·a_3_2 + a_4_2·a_5_3
  124. a_6_3·a_3_1
  125. a_6_3·a_3_4 − a_4_4·a_5_3 + a_4_2·a_5_3
  126. a_6_4·a_3_3 + a_4_4·a_5_3 − a_4_3·a_5_5 − a_4_3·a_5_3
  127. a_6_4·a_3_2 + a_4_4·a_5_3
  128. a_6_4·a_3_1 + a_4_3·a_5_3 + a_4_2·a_5_3
  129. a_6_4·a_3_4 − a_4_4·a_5_3 + a_4_3·a_5_3
  130. a_6_5·a_3_3 − a_4_3·a_5_5 − a_4_3·a_5_3 − a_4_2·a_5_3
  131. a_6_5·a_3_2 + a_4_4·a_5_3 + a_4_3·a_5_3 + a_4_2·a_5_3
  132. a_6_5·a_3_1 + a_4_4·a_5_3 + a_4_3·a_5_3
  133. a_6_5·a_3_4 + a_4_4·a_5_3 − a_4_3·a_5_5 + a_4_3·a_5_3 + a_4_2·a_5_3
  134. a_6_6·a_3_3 − a_4_2·a_5_3
  135. a_6_6·a_3_2 − a_4_4·a_5_3 + a_4_2·a_5_3
  136. a_6_6·a_3_1 + a_4_2·a_5_3
  137. a_6_6·a_3_4 − a_4_3·a_5_3 + a_4_2·a_5_3
  138. b_2_2·a_7_10 + b_2_23·a_3_2
  139.  − a_4_4·a_5_3 + a_4_3·a_5_5 + a_4_3·a_5_3 + a_4_2·a_5_3 + a_2_0·a_7_10
  140. a_4_4·a_5_3 − a_4_3·a_5_3 + a_4_2·a_5_3 + a_2_1·a_7_10
  141. a_8_8·a_1_1 + a_4_4·a_5_3 − a_4_3·a_5_5 − a_4_3·a_5_3 − a_4_2·a_5_3
  142. a_8_8·a_1_0 + a_4_4·a_5_3 + a_4_2·a_5_3
  143.  − a_5_5·a_5_6 + a_5_4·a_5_6 − a_5_4·a_5_5 − a_5_3·a_5_6 + b_2_2·a_3_2·a_5_6
  144. a_5_5·a_5_6 − a_5_4·a_5_6 + a_5_3·a_5_5 + a_5_3·a_5_4 + b_2_2·a_3_2·a_5_4
       + c_6_8·a_1_0·a_3_2
  145. a_5_3·a_5_5 − b_2_2·a_3_2·a_5_4 + c_6_8·a_1_0·a_3_1
  146. a_5_3·a_5_4 − b_2_2·a_3_2·a_5_4 + c_6_9·a_1_0·a_3_2
  147.  − a_5_5·a_5_6 + a_5_4·a_5_6 + a_5_4·a_5_5 + a_5_3·a_5_5 − a_5_3·a_5_4 + c_6_9·a_1_0·a_3_1
  148. a_4_3·a_6_3
  149. a_4_4·a_6_3
  150. a_4_2·a_6_3
  151. a_4_3·a_6_4
  152. a_4_4·a_6_4
  153. a_4_2·a_6_4
  154. a_4_3·a_6_5
  155. a_4_4·a_6_5
  156. a_4_2·a_6_5
  157. a_4_3·a_6_6
  158. a_4_4·a_6_6
  159. a_4_2·a_6_6
  160.  − a_5_4·a_5_5 + a_5_3·a_5_5 + a_5_3·a_5_4 + a_3_3·a_7_10 + b_2_2·a_3_2·a_5_4
  161. a_5_4·a_5_5 + a_3_2·a_7_10
  162. a_5_4·a_5_5 + a_5_3·a_5_4 + a_3_1·a_7_10 − b_2_2·a_3_2·a_5_4
  163. a_5_5·a_5_6 − a_5_4·a_5_6 − a_5_4·a_5_5 + a_3_4·a_7_10
  164. b_2_2·a_8_8 + a_5_4·a_5_6 − a_5_3·a_5_5 − a_5_3·a_5_4
  165. a_2_0·a_8_8
  166. a_2_1·a_8_8
  167. a_6_3·a_5_3 + a_2_0·c_6_9·a_3_1
  168. a_6_3·a_5_5 + a_2_0·c_6_9·a_3_3 + a_2_0·c_6_9·a_3_1
  169. a_6_3·a_5_4 − a_2_0·c_6_9·a_3_3
  170. a_6_4·a_5_3 + a_2_0·c_6_9·a_3_1 − a_2_0·c_6_8·a_3_1
  171. a_6_4·a_5_5 + a_2_0·c_6_9·a_3_3 + a_2_0·c_6_8·a_3_3 + a_2_0·c_6_8·a_3_1
  172. a_6_4·a_5_4 − a_2_0·c_6_9·a_3_3 − a_2_0·c_6_9·a_3_1 − a_2_0·c_6_8·a_3_1
  173. a_6_4·a_5_6 − a_6_3·a_5_6 − a_2_0·c_6_9·a_3_3 + a_2_0·c_6_9·a_3_1 − a_2_0·c_6_8·a_3_1
  174. a_6_5·a_5_3 − a_2_0·c_6_8·a_3_1
  175. a_6_5·a_5_5 − a_2_0·c_6_9·a_3_3 + a_2_0·c_6_9·a_3_1 + a_2_0·c_6_8·a_3_3
  176. a_6_5·a_5_4 − a_2_0·c_6_9·a_3_3 + a_2_0·c_6_9·a_3_1 + a_2_0·c_6_8·a_3_1
  177. a_6_5·a_5_6 + a_6_3·a_5_6 − a_2_0·c_6_9·a_3_1 + a_2_0·c_6_8·a_3_3 − a_2_0·c_6_8·a_3_1
  178. a_6_6·a_5_3
  179. a_6_6·a_5_5 + a_6_3·a_5_6 − a_2_0·c_6_9·a_3_3
  180. a_6_6·a_5_4 + a_6_3·a_5_6 − a_2_0·c_6_9·a_3_3
  181. a_6_6·a_5_6 + a_6_3·a_5_6 + a_2_0·c_6_9·a_3_1 − a_2_0·c_6_8·a_3_1
  182. a_4_3·a_7_10 + a_2_0·c_6_9·a_3_1 + a_2_0·c_6_8·a_3_3 + a_2_0·c_6_8·a_3_1
  183. a_4_4·a_7_10 − a_2_0·c_6_9·a_3_3 + a_2_0·c_6_9·a_3_1 + a_2_0·c_6_8·a_3_3
       + a_2_0·c_6_8·a_3_1
  184. a_4_2·a_7_10 + a_2_0·c_6_9·a_3_3 − a_2_0·c_6_9·a_3_1 + a_2_0·c_6_8·a_3_1
  185. a_8_8·a_3_3 + a_6_3·a_5_6 − a_2_0·c_6_9·a_3_3 + a_2_0·c_6_8·a_3_1
  186. a_8_8·a_3_2 + a_6_3·a_5_6 − a_2_0·c_6_9·a_3_3 − a_2_0·c_6_8·a_3_1
  187. a_8_8·a_3_1 + a_2_0·c_6_9·a_3_3 − a_2_0·c_6_9·a_3_1
  188. a_8_8·a_3_4 + a_2_0·c_6_9·a_3_3 + a_2_0·c_6_9·a_3_1 − a_2_0·c_6_8·a_3_3
  189. a_6_32
  190. a_6_3·a_6_4
  191. a_6_42
  192. a_6_3·a_6_5
  193. a_6_4·a_6_5
  194. a_6_52
  195. a_6_3·a_6_6
  196. a_6_62
  197. a_6_4·a_6_6
  198. a_6_5·a_6_6
  199. a_5_3·a_7_10 + c_6_9·a_1_0·a_5_6 + 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_4
  200. a_5_5·a_7_10 − b_2_22·a_3_2·a_5_4 − c_6_9·a_1_0·a_5_6 − 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 + c_6_8·a_1_0·a_5_4
  201. a_5_4·a_7_10 − b_2_22·a_3_2·a_5_4 − c_6_9·a_1_0·a_5_6 − 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 + c_6_8·a_1_0·a_5_3
  202. a_5_6·a_7_10 − b_2_22·a_3_2·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_4
  203. a_4_3·a_8_8
  204. a_4_4·a_8_8
  205. a_4_2·a_8_8
  206. 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
  207. a_6_6·a_7_10 + 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_4
       − a_2_0·c_6_8·a_5_3
  208. a_6_4·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_4
       + a_2_0·c_6_8·a_5_3
  209. 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
  210. a_8_8·a_5_3 + a_3_2·a_5_4·a_5_6 + 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_4 + a_2_0·c_6_8·a_5_3
  211. a_8_8·a_5_5 + 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_3
  212. a_8_8·a_5_4 + 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_4
       + a_2_0·c_6_8·a_5_3
  213. a_8_8·a_5_6 + a_3_2·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
  214. a_6_3·a_8_8
  215. a_6_6·a_8_8
  216. a_6_4·a_8_8
  217. a_6_5·a_8_8
  218. a_8_8·a_7_10 − b_2_2·a_3_2·a_5_4·a_5_6 − a_2_0·c_6_8·a_7_10
  219. 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_2_2, an element of degree 2
  • The Raw Filter Degree Type of that HSOP is [-1, -1, 9, 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 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. b_2_2 → 0, an element of degree 2
  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_3_4 → 0, an element of degree 3
  10. a_4_2 → 0, an element of degree 4
  11. a_4_3 → 0, an element of degree 4
  12. a_4_4 → 0, an element of degree 4
  13. a_5_3 → 0, an element of degree 5
  14. a_5_4 → 0, an element of degree 5
  15. a_5_5 → 0, an element of degree 5
  16. a_5_6 → 0, an element of degree 5
  17. a_6_3 → 0, an element of degree 6
  18. a_6_4 → 0, an element of degree 6
  19. a_6_5 → 0, an element of degree 6
  20. a_6_6 → 0, an element of degree 6
  21. c_6_8 → c_2_23, an element of degree 6
  22. c_6_9 →  − c_2_13, an element of degree 6
  23. a_7_10 → 0, an element of degree 7
  24. 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. b_2_2 →  − c_2_5, an element of degree 2
  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 → c_2_5·a_1_2, an element of degree 3
  9. a_3_4 → 0, an element of degree 3
  10. a_4_2 → 0, an element of degree 4
  11. a_4_3 → 0, an element of degree 4
  12. a_4_4 → 0, an element of degree 4
  13. a_5_3 →  − c_2_52·a_1_2, an element of degree 5
  14. a_5_4 → c_2_52·a_1_1 − c_2_4·c_2_5·a_1_2, an element of degree 5
  15. a_5_5 → c_2_52·a_1_1 − c_2_4·c_2_5·a_1_2, an element of degree 5
  16. a_5_6 → c_2_52·a_1_2 + c_2_52·a_1_1 + c_2_52·a_1_0 − c_2_4·c_2_5·a_1_2 − c_2_3·c_2_5·a_1_2, an element of degree 5
  17. a_6_3 → c_2_52·a_1_1·a_1_2, an element of degree 6
  18. a_6_4 → c_2_52·a_1_1·a_1_2, an element of degree 6
  19. a_6_5 →  − c_2_52·a_1_1·a_1_2, an element of degree 6
  20. a_6_6 → c_2_52·a_1_0·a_1_2, an element of degree 6
  21. c_6_8 → c_2_52·a_1_0·a_1_2 + c_2_53 − c_2_4·c_2_52 + c_2_43, an element of degree 6
  22. c_6_9 →  − c_2_52·a_1_0·a_1_2 + c_2_3·c_2_52 − c_2_33, an element of degree 6
  23. a_7_10 →  − c_2_53·a_1_2, an element of degree 7
  24. a_8_8 →  − c_2_53·a_1_1·a_1_2 − 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