David J. Green

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Cohomology of group number 41 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 3 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

  ( − 2) · (t5  +  t3  +  1/2·t  +  1/2)

  (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 20 minimal generators of maximal degree 9:

1. a_1_0, a nilpotent element of degree 1
2. a_1_1, a nilpotent element of degree 1
3. a_1_2, a nilpotent element of degree 1
4. b_2_3, an element of degree 2
5. a_3_2, a nilpotent element of degree 3
6. a_3_3, a nilpotent element of degree 3
7. a_3_4, a nilpotent element of degree 3
8. a_3_5, a nilpotent element of degree 3
9. b_4_7, an element of degree 4
10. a_5_5, a nilpotent element of degree 5
11. a_5_6, a nilpotent element of degree 5
12. a_5_7, a nilpotent element of degree 5
13. a_5_8, a nilpotent element of degree 5
14. a_5_9, a nilpotent element of degree 5
15. b_6_11, an element of degree 6
16. c_6_12, a Duflot regular element of degree 6
17. c_6_13, a Duflot regular element of degree 6
18. a_7_16, a nilpotent element of degree 7
19. a_7_17, a nilpotent element of degree 7
20. a_9_23, a nilpotent element of degree 9

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 243

Ring relations

There are 15 "obvious" relations:
   a_1_0², a_1_1², a_1_2², a_3_2², a_3_3², a_3_4², a_3_5², a_5_5², a_5_6², a_5_7², a_5_8², a_5_9², a_7_16², a_7_17², a_9_23²

Apart from that, there are 125 minimal relations of maximal degree 16:

1. a_1_0·a_1_1·a_1_2
2. b_2_3·a_1_0
3. a_1_0·a_3_2
4. a_1_1·a_3_3 − a_1_1·a_3_2
5. a_1_1·a_3_2 + a_1_0·a_3_3
6. a_1_2·a_3_3 − a_1_2·a_3_2 + a_1_1·a_3_4
7. a_1_2·a_3_2 + a_1_1·a_3_2 + a_1_0·a_3_4
8. a_1_2·a_3_4 − a_1_2·a_3_3 + a_1_2·a_3_2
9. a_1_0·a_3_5
10. a_1_0·a_1_1·a_3_4
11. b_2_3·a_3_5
12.  − b_2_3·a_3_2 + a_1_1·a_1_2·a_3_5
13. b_4_7·a_1_1 − b_2_3·a_3_3 + b_2_3·a_3_2
14. b_4_7·a_1_0 + b_2_3·a_3_2
15. b_4_7·a_1_2 − b_2_3·a_3_4 + b_2_3·a_3_3
16. a_3_2·a_3_5
17.  − a_3_2·a_3_3 + a_1_1·a_5_5 − b_2_3·a_1_1·a_3_4
18. a_1_0·a_5_5
19.  − a_3_2·a_3_4 + a_3_2·a_3_3 + a_1_2·a_5_5 + b_2_3·a_1_1·a_3_4
20.  − a_3_2·a_3_3 + a_1_1·a_5_6 + b_2_3·a_1_1·a_3_4
21. a_3_2·a_3_3 + a_1_0·a_5_6
22.  − a_3_3·a_3_4 + a_3_2·a_3_3 + a_1_2·a_5_6 + b_2_3·a_1_1·a_3_4
23. a_3_3·a_3_4 − a_3_2·a_3_4 + a_1_1·a_5_7
24. a_3_2·a_3_4 − a_3_2·a_3_3 + a_1_0·a_5_7
25. a_1_2·a_5_7 + b_2_3·a_1_1·a_3_4
26.  − a_3_3·a_3_5 + a_3_3·a_3_4 − a_3_2·a_3_4 + a_1_1·a_5_8 − b_2_3·a_1_1·a_3_4
27. a_3_2·a_3_4 + a_3_2·a_3_3 + a_1_0·a_5_8
28.  − a_3_4·a_3_5 + a_3_3·a_3_5 + a_3_3·a_3_4 − a_3_2·a_3_4 + a_1_2·a_5_8 + b_2_3·a_1_1·a_3_4
29. a_1_0·a_5_9
30. b_4_7·a_3_2
31. b_4_7·a_3_5
32. b_4_7·a_3_3 + b_2_3·a_5_6 − b_2_3·a_5_5 − b_2_3²·a_3_4 + b_2_3²·a_3_3
33. b_4_7·a_3_4 − b_4_7·a_3_3 + b_2_3·a_5_7 + b_2_3·a_5_5 − b_2_3²·a_3_4 − b_2_3²·a_3_3
34.  − b_2_3·a_5_5 + b_2_3²·a_3_4 + a_1_1·a_1_2·a_5_8
35.  − b_4_7·a_3_4 − b_4_7·a_3_3 − b_2_3·a_5_8 − b_2_3·a_5_5 − b_2_3²·a_3_4 + a_1_1·a_1_2·a_5_9
36. b_6_11·a_1_1 + b_4_7·a_3_3 + b_2_3·a_5_5 − b_2_3²·a_3_4 + b_2_3²·a_3_3
       − a_1_0·a_1_1·a_5_7
37. b_6_11·a_1_0 + b_2_3·a_5_5 − b_2_3²·a_3_4
38. b_6_11·a_1_2 − b_4_7·a_3_4 + b_2_3·a_5_8 − b_2_3·a_5_5 + b_2_3²·a_3_4 − b_2_3²·a_3_3
39. a_3_2·a_5_5
40. a_3_5·a_5_5
41. a_3_3·a_5_6 + a_3_3·a_5_5 + a_3_2·a_5_6
42. a_3_4·a_5_6 + a_3_3·a_5_7 − a_3_3·a_5_5
43. a_3_4·a_5_5 − a_3_3·a_5_6 − a_3_3·a_5_5 + a_3_2·a_5_7
44.  − a_3_3·a_5_6 + b_2_3·a_1_1·a_5_7
45. a_3_4·a_5_7 + a_3_4·a_5_6 − a_3_4·a_5_5
46.  − a_3_5·a_5_6 + a_3_4·a_5_6 + a_3_3·a_5_8 + a_3_3·a_5_5
47. a_3_4·a_5_5 + a_3_3·a_5_6 + a_3_3·a_5_5 + a_3_2·a_5_8
48. a_3_5·a_5_8 − a_3_5·a_5_7 + a_3_5·a_5_6
49.  − a_3_5·a_5_7 − a_3_5·a_5_6 + a_3_4·a_5_8 − a_3_4·a_5_6 + a_3_4·a_5_5 − a_3_3·a_5_6
50. a_3_2·a_5_9
51. a_3_5·a_5_9
52. a_3_3·a_5_6 + a_3_3·a_5_5 + c_6_12·a_1_0·a_1_1
53. a_3_4·a_5_6 − a_3_4·a_5_5 + a_3_3·a_5_6 + b_2_3²·a_1_1·a_3_4 + c_6_12·a_1_1·a_1_2
54. a_3_4·a_5_5 − a_3_3·a_5_6 − a_3_3·a_5_5 + c_6_12·a_1_0·a_1_2
55. b_4_7² + b_2_3·b_6_11 + b_2_3²·b_4_7 − b_2_3·a_1_1·a_5_9
56. a_3_5·a_5_6 − a_3_4·a_5_6 + a_3_4·a_5_5 + a_3_3·a_5_6 + a_1_1·a_7_16
57.  − a_3_4·a_5_5 − a_3_3·a_5_6 − a_3_3·a_5_5 + a_1_0·a_7_16 − c_6_13·a_1_0·a_1_1
58. a_3_5·a_5_7 − a_3_4·a_5_6 + a_3_4·a_5_5 + a_3_3·a_5_6 + a_1_2·a_7_16 + c_6_13·a_1_1·a_1_2
59.  − a_3_5·a_5_6 − a_3_4·a_5_6 + a_3_4·a_5_5 − a_3_3·a_5_9 + a_3_3·a_5_6 + a_1_1·a_7_17
       + b_2_3²·a_1_1·a_3_4
60.  − a_3_4·a_5_5 − a_3_3·a_5_6 − a_3_3·a_5_5 + a_1_0·a_7_17 + c_6_13·a_1_0·a_1_1
61.  − a_3_5·a_5_7 − a_3_4·a_5_9 − a_3_4·a_5_6 + a_3_4·a_5_5 + a_3_3·a_5_9 + a_1_2·a_7_17
       + b_2_3²·a_1_1·a_3_4 − c_6_13·a_1_1·a_1_2
62.  − b_4_7·a_5_6 + b_4_7·a_5_5 − b_2_3²·a_5_7 − b_2_3³·a_3_3 + b_2_3·c_6_12·a_1_1
63.  − b_4_7·a_5_7 − b_4_7·a_5_5 − b_2_3²·a_5_7 + b_2_3²·a_5_6 − b_2_3³·a_3_4 + b_2_3³·a_3_3
       + b_2_3·c_6_12·a_1_2
64. b_6_11·a_3_3 − b_4_7·a_5_6 + b_4_7·a_5_5 − b_2_3²·a_5_7 − b_2_3²·a_5_6 − b_2_3³·a_3_4
65. b_6_11·a_3_2 + b_4_7·a_5_5 + b_2_3²·a_5_7 + b_2_3²·a_5_6 + b_2_3³·a_3_4
66. b_6_11·a_3_5
67. b_6_11·a_3_4 + b_4_7·a_5_8 + b_4_7·a_5_7 + b_4_7·a_5_5 + b_2_3²·a_5_7 − b_2_3²·a_5_6
       − b_2_3³·a_3_4 + b_2_3³·a_3_3
68.  − b_4_7·a_5_8 − b_4_7·a_5_7 + b_4_7·a_5_6 + b_2_3·a_7_16 + b_2_3·a_1_1·a_1_2·a_5_9
       − b_2_3·c_6_13·a_1_1
69.  − b_4_7·a_5_5 − b_2_3²·a_5_7 − b_2_3²·a_5_6 − b_2_3³·a_3_4 + a_1_1·a_1_2·a_7_16
70.  − b_4_7·a_5_9 + b_4_7·a_5_8 + b_4_7·a_5_5 + b_2_3·a_7_17 + b_2_3²·a_5_7 + b_2_3²·a_5_6
       − b_2_3³·a_3_4 + b_2_3³·a_3_3 − b_2_3·a_1_1·a_1_2·a_5_9 + b_2_3·c_6_13·a_1_1
71.  − b_4_7·a_5_8 + b_4_7·a_5_7 − b_4_7·a_5_6 − b_4_7·a_5_5 − b_2_3²·a_5_7 + b_2_3²·a_5_6
       + b_2_3³·a_3_4 − b_2_3³·a_3_3 + a_1_1·a_1_2·a_7_17
72. a_5_5·a_5_8 − a_5_5·a_5_7 + a_5_5·a_5_6 − b_2_3²·a_1_1·a_5_7
73. a_5_5·a_5_6 + b_2_3²·a_1_1·a_5_7 + b_2_3³·a_1_1·a_3_4 + c_6_12·a_1_0·a_3_3
       + b_2_3·c_6_12·a_1_1·a_1_2
74.  − a_5_6·a_5_8 + a_5_6·a_5_7 − b_2_3²·a_1_1·a_5_7 + c_6_12·a_1_1·a_3_5
       + c_6_12·a_1_0·a_3_3
75.  − a_5_7·a_5_8 + a_5_6·a_5_7 − a_5_5·a_5_7 + b_2_3²·a_1_1·a_5_7 + c_6_12·a_1_2·a_3_5
76. a_5_6·a_5_7 − a_5_5·a_5_7 + b_2_3²·a_1_1·a_5_7 + c_6_12·a_1_1·a_3_4
77. a_5_5·a_5_7 + a_5_5·a_5_6 + c_6_12·a_1_0·a_3_4
78. b_4_7·b_6_11 − b_2_3²·b_6_11 + b_2_3³·b_4_7 − a_5_8·a_5_9 + a_5_7·a_5_9 − a_5_6·a_5_9
       + b_2_3²·a_1_1·a_5_9 − b_2_3²·c_6_12
79.  − a_5_6·a_5_8 + a_3_3·a_7_16 + b_2_3²·a_1_1·a_5_7 − c_6_13·a_1_0·a_3_3
80.  − a_5_5·a_5_7 − a_5_5·a_5_6 + a_3_2·a_7_16 − c_6_13·a_1_0·a_3_3 + c_6_12·a_1_0·a_3_3
81.  − a_5_7·a_5_8 + a_5_6·a_5_8 − a_5_5·a_5_7 + a_3_5·a_7_16 − b_2_3²·a_1_1·a_5_7
       + c_6_13·a_1_1·a_3_5 − c_6_12·a_1_0·a_3_3
82.  − a_5_7·a_5_8 − a_5_6·a_5_8 + a_5_5·a_5_7 − a_5_5·a_5_6 + a_3_4·a_7_16
       + b_2_3²·a_1_1·a_5_7 + c_6_13·a_1_1·a_3_4
83. a_5_8·a_5_9 − a_5_7·a_5_9 − a_5_6·a_5_9 + a_5_6·a_5_8 + a_5_6·a_5_7 + a_5_5·a_5_9
       + a_3_3·a_7_17 − b_2_3²·a_1_1·a_5_7 + c_6_13·a_1_0·a_3_3 + c_6_12·a_1_0·a_3_3
84.  − a_5_5·a_5_7 − a_5_5·a_5_6 + a_3_2·a_7_17 + c_6_13·a_1_0·a_3_3 + c_6_12·a_1_0·a_3_3
85.  − a_5_8·a_5_9 + a_5_7·a_5_9 − a_5_6·a_5_9 − a_5_5·a_5_6 + b_2_3·a_1_1·a_7_17
       + b_2_3²·a_1_1·a_5_7 + b_2_3³·a_1_1·a_3_4 − c_6_12·a_1_0·a_3_3
86. a_5_8·a_5_9 − a_5_7·a_5_9 + a_5_6·a_5_9 − a_5_5·a_5_9 − a_5_5·a_5_6 + b_2_3·a_1_2·a_7_17
       + b_2_3³·a_1_1·a_3_4 − c_6_12·a_1_0·a_3_3 − b_2_3·c_6_13·a_1_1·a_1_2
87.  − a_5_7·a_5_8 + a_5_6·a_5_8 − a_5_5·a_5_7 + a_3_5·a_7_17 − b_2_3²·a_1_1·a_5_7
       − c_6_13·a_1_1·a_3_5 − c_6_12·a_1_0·a_3_3
88. a_5_7·a_5_9 + a_5_7·a_5_8 + a_5_6·a_5_9 + a_5_6·a_5_8 − a_5_6·a_5_7 + a_5_5·a_5_9
       + a_3_4·a_7_17 − c_6_13·a_1_1·a_3_4 − c_6_12·a_1_0·a_3_3
89.  − a_5_6·a_5_9 + a_5_6·a_5_8 − a_5_5·a_5_9 − a_5_5·a_5_7 + a_1_1·a_9_23 − c_6_13·a_1_1·a_3_4
       + c_6_13·a_1_0·a_3_3 + c_6_12·a_1_0·a_3_3
90. a_5_5·a_5_7 + a_5_5·a_5_6 + a_1_0·a_9_23 − c_6_13·a_1_0·a_3_4 − c_6_13·a_1_0·a_3_3
91.  − a_5_7·a_5_9 + a_5_7·a_5_8 + a_5_6·a_5_7 + a_5_5·a_5_9 + a_1_2·a_9_23 − c_6_13·a_1_1·a_3_4
       + c_6_13·a_1_0·a_3_4 − c_6_13·a_1_0·a_3_3
92. b_6_11·a_5_6 + b_6_11·a_5_5 − b_2_3⁴·a_3_3 + b_2_3·c_6_12·a_3_3 − b_2_3²·c_6_12·a_1_1
93. b_6_11·a_5_8 + b_6_11·a_5_5 − b_2_3³·a_5_6 + b_2_3⁴·a_3_4 + b_2_3·c_6_12·a_3_4
       + b_2_3·c_6_12·a_3_3 − b_2_3²·c_6_12·a_1_2 + c_6_12·a_1_1·a_1_2·a_3_5
94.  − b_6_11·a_5_7 − b_6_11·a_5_5 + b_4_7·a_7_16 + b_2_3³·a_5_7 − b_2_3⁴·a_3_4
       − b_2_3·c_6_13·a_3_3 − b_2_3·c_6_12·a_3_3 − b_2_3²·c_6_12·a_1_2
       + c_6_13·a_1_1·a_1_2·a_3_5
95. b_6_11·a_5_5 − b_2_3³·a_5_7 − b_2_3³·a_5_6 + b_2_3⁴·a_3_4 + b_2_3·a_1_1·a_1_2·a_7_17
       − b_2_3²·c_6_12·a_1_2 − b_2_3²·c_6_12·a_1_1
96. b_6_11·a_5_9 + b_6_11·a_5_7 + b_4_7·a_7_17 + b_2_3²·a_7_17 − b_2_3³·a_5_7
       − b_2_3⁴·a_3_4 + b_2_3⁴·a_3_3 − b_2_3²·a_1_1·a_1_2·a_5_9 + b_2_3·c_6_13·a_3_3
       − b_2_3·c_6_12·a_3_4 + b_2_3²·c_6_13·a_1_1 + b_2_3²·c_6_12·a_1_2
       − c_6_13·a_1_1·a_1_2·a_3_5 − c_6_12·a_1_1·a_1_2·a_3_5
97.  − b_6_11·a_5_9 + b_6_11·a_5_5 + b_2_3·a_9_23 − b_2_3³·a_5_6 + b_2_3⁴·a_3_4
       + b_2_3⁴·a_3_3 − b_2_3²·a_1_1·a_1_2·a_5_9 − b_2_3·c_6_13·a_3_4 − b_2_3·c_6_13·a_3_3
       − b_2_3·c_6_12·a_3_4 − b_2_3²·c_6_12·a_1_2 + b_2_3²·c_6_12·a_1_1
       + c_6_12·a_1_1·a_1_2·a_3_5
98. b_6_11·a_5_7 − b_6_11·a_5_5 − b_2_3⁴·a_3_4 + b_2_3⁴·a_3_3 + a_1_1·a_1_2·a_9_23
       + b_2_3·c_6_12·a_3_4 − b_2_3·c_6_12·a_3_3 − b_2_3²·c_6_12·a_1_2
       − c_6_12·a_1_1·a_1_2·a_3_5
99. a_5_5·a_7_16 + b_2_3³·a_1_1·a_5_7 + b_2_3⁴·a_1_1·a_3_4 − c_6_13·a_1_0·a_5_6
       + c_6_12·a_1_0·a_5_7 − c_6_12·a_1_0·a_5_6 + b_2_3·c_6_13·a_1_1·a_3_4
       − b_2_3·c_6_12·a_1_1·a_3_4 + b_2_3²·c_6_12·a_1_1·a_1_2
100. a_5_7·a_7_16 + b_2_3³·a_1_1·a_5_7 + c_6_13·a_1_1·a_5_7 + c_6_12·a_1_2·a_5_8
101. a_5_6·a_7_16 − b_2_3³·a_1_1·a_5_7 + b_2_3⁴·a_1_1·a_3_4 − c_6_13·a_1_0·a_5_6
        + c_6_12·a_1_1·a_5_8 + c_6_12·a_1_0·a_5_7 − c_6_12·a_1_0·a_5_6
        − b_2_3·c_6_13·a_1_1·a_3_4 + b_2_3²·c_6_12·a_1_1·a_1_2
102. a_5_8·a_7_16 + b_2_3³·a_1_1·a_5_7 + b_2_3⁴·a_1_1·a_3_4 + c_6_13·a_1_1·a_5_8
        − b_2_3·c_6_12·a_1_1·a_3_4 + b_2_3²·c_6_12·a_1_1·a_1_2
103. b_6_11² − b_2_3⁴·b_4_7 − a_5_9·a_7_16 + a_5_7·a_7_17 + a_5_6·a_7_17 − a_5_5·a_7_17
        − b_2_3³·a_1_1·a_5_7 + b_2_3⁴·a_1_1·a_3_4 + b_2_3·b_4_7·c_6_12 − b_2_3³·c_6_12
        − c_6_13·a_1_1·a_5_9 − c_6_13·a_1_1·a_5_7 − c_6_12·a_1_2·a_5_8 + c_6_12·a_1_1·a_5_9
        − c_6_12·a_1_1·a_5_8 + c_6_12·a_1_1·a_5_7 − b_2_3·c_6_13·a_1_1·a_3_4
        − b_2_3·c_6_12·a_1_1·a_3_4 + b_2_3²·c_6_12·a_1_1·a_1_2
104. a_5_9·a_7_16 − a_5_7·a_7_17 − a_5_5·a_7_17 + b_2_3²·a_1_1·a_7_17 + b_2_3³·a_1_1·a_5_7
        + b_2_3⁴·a_1_1·a_3_4 + c_6_13·a_1_1·a_5_9 + c_6_13·a_1_1·a_5_7 − c_6_13·a_1_0·a_5_6
        + c_6_12·a_1_2·a_5_8 + c_6_12·a_1_1·a_5_9 + c_6_12·a_1_1·a_5_7 − c_6_12·a_1_0·a_5_7
        + c_6_12·a_1_0·a_5_6 + b_2_3·c_6_13·a_1_1·a_3_4 − b_2_3·c_6_12·a_1_1·a_3_4
105. b_6_11² − b_2_3⁴·b_4_7 + a_5_9·a_7_16 + a_5_7·a_7_17 − a_5_5·a_7_17
        + b_2_3²·a_1_2·a_7_17 + b_2_3³·a_1_1·a_5_7 + b_2_3·b_4_7·c_6_12 − b_2_3³·c_6_12
        + c_6_13·a_1_1·a_5_9 − c_6_13·a_1_1·a_5_7 − c_6_13·a_1_0·a_5_6 + c_6_12·a_1_2·a_5_9
        − c_6_12·a_1_2·a_5_8 + c_6_12·a_1_1·a_5_9 − c_6_12·a_1_1·a_5_7 − c_6_12·a_1_0·a_5_7
        + c_6_12·a_1_0·a_5_6 + b_2_3·c_6_13·a_1_1·a_3_4 − b_2_3²·c_6_13·a_1_1·a_1_2
        − b_2_3²·c_6_12·a_1_1·a_1_2
106.  − b_6_11² + b_2_3⁴·b_4_7 + a_5_9·a_7_17 + a_5_7·a_7_17 − a_5_5·a_7_17
        + b_2_3⁴·a_1_1·a_3_4 − b_2_3·b_4_7·c_6_12 + b_2_3³·c_6_12 − c_6_13·a_1_1·a_5_9
        − c_6_13·a_1_1·a_5_7 − c_6_13·a_1_0·a_5_6 + c_6_12·a_1_2·a_5_9 − c_6_12·a_1_2·a_5_8
        + c_6_12·a_1_1·a_5_9 − c_6_12·a_1_1·a_5_7 − c_6_12·a_1_0·a_5_7 + c_6_12·a_1_0·a_5_6
        + b_2_3·c_6_13·a_1_1·a_3_4 + b_2_3²·c_6_12·a_1_1·a_1_2
107. b_6_11² − b_2_3⁴·b_4_7 + a_5_8·a_7_17 − a_5_7·a_7_17 − a_5_5·a_7_17
        − b_2_3⁴·a_1_1·a_3_4 + b_2_3·b_4_7·c_6_12 − b_2_3³·c_6_12 − c_6_13·a_1_1·a_5_8
        + c_6_13·a_1_1·a_5_7 − c_6_13·a_1_0·a_5_6 − c_6_12·a_1_2·a_5_8 + c_6_12·a_1_1·a_5_9
        − c_6_12·a_1_1·a_5_8 + c_6_12·a_1_1·a_5_7 − c_6_12·a_1_0·a_5_7 + c_6_12·a_1_0·a_5_6
        + b_2_3·c_6_13·a_1_1·a_3_4 + b_2_3·c_6_12·a_1_1·a_3_4 − b_2_3²·c_6_12·a_1_1·a_1_2
108. b_6_11² − b_2_3⁴·b_4_7 − a_5_9·a_7_16 + a_5_7·a_7_17 + a_5_5·a_7_17 + a_3_3·a_9_23
        + b_2_3³·a_1_1·a_5_7 − b_2_3⁴·a_1_1·a_3_4 + b_2_3·b_4_7·c_6_12 − b_2_3³·c_6_12
        − c_6_13·a_1_1·a_5_9 + c_6_13·a_1_0·a_5_7 − c_6_13·a_1_0·a_5_6 − c_6_12·a_1_2·a_5_8
        + c_6_12·a_1_1·a_5_9 + c_6_12·a_1_1·a_5_8 − c_6_12·a_1_1·a_5_7 − c_6_12·a_1_0·a_5_7
        + c_6_12·a_1_0·a_5_6 − b_2_3·c_6_13·a_1_1·a_3_4 − b_2_3²·c_6_12·a_1_1·a_1_2
109. a_3_2·a_9_23 + c_6_13·a_1_0·a_5_7 − c_6_13·a_1_0·a_5_6 + c_6_12·a_1_0·a_5_7
        + c_6_12·a_1_0·a_5_6
110. b_6_11² − b_2_3⁴·b_4_7 + b_2_3·a_1_1·a_9_23 + b_2_3³·a_1_1·a_5_7
        + b_2_3·b_4_7·c_6_12 − b_2_3³·c_6_12 − b_2_3·c_6_13·a_1_1·a_3_4
        − b_2_3·c_6_12·a_1_1·a_3_4
111. b_6_11² − b_2_3⁴·b_4_7 + a_5_9·a_7_16 + b_2_3·a_1_2·a_9_23 + b_2_3⁴·a_1_1·a_3_4
        + b_2_3·b_4_7·c_6_12 − b_2_3³·c_6_12 + c_6_13·a_1_1·a_5_9 − c_6_12·a_1_2·a_5_9
        + c_6_12·a_1_1·a_5_9 − b_2_3·c_6_13·a_1_1·a_3_4 + b_2_3·c_6_12·a_1_1·a_3_4
        + b_2_3²·c_6_12·a_1_1·a_1_2
112. a_3_5·a_9_23 + c_6_13·a_1_2·a_5_8 − c_6_13·a_1_1·a_5_8 + c_6_12·a_1_2·a_5_8
        + c_6_12·a_1_1·a_5_8 + c_6_12·a_1_1·a_5_7 − b_2_3·c_6_13·a_1_1·a_3_4
113. b_6_11² − b_2_3⁴·b_4_7 − a_5_9·a_7_16 − a_5_5·a_7_17 + a_3_4·a_9_23
        + b_2_3⁴·a_1_1·a_3_4 + b_2_3·b_4_7·c_6_12 − b_2_3³·c_6_12 − c_6_13·a_1_1·a_5_9
        − c_6_13·a_1_1·a_5_7 − c_6_13·a_1_0·a_5_7 + c_6_13·a_1_0·a_5_6 + c_6_12·a_1_2·a_5_8
        + c_6_12·a_1_1·a_5_9 + c_6_12·a_1_1·a_5_8 + c_6_12·a_1_1·a_5_7 − c_6_12·a_1_0·a_5_6
        + b_2_3·c_6_13·a_1_1·a_3_4 − b_2_3·c_6_12·a_1_1·a_3_4 + b_2_3²·c_6_12·a_1_1·a_1_2
114.  − b_6_11·a_7_17 + b_6_11·a_7_16 + b_2_3²·a_9_23 − b_2_3³·a_7_17 + b_2_3⁴·a_5_7
        + b_2_3⁴·a_5_6 + b_2_3⁵·a_3_4 + b_2_3²·a_1_1·a_1_2·a_7_17 + b_2_3·c_6_13·a_5_6
        + b_2_3·c_6_12·a_5_9 − b_2_3²·c_6_13·a_3_3 − b_2_3²·c_6_12·a_3_4
        + b_2_3²·c_6_12·a_3_3 − b_2_3³·c_6_13·a_1_1 + b_2_3³·c_6_12·a_1_2
        − b_2_3³·c_6_12·a_1_1 + c_6_13·a_1_1·a_1_2·a_5_8 + c_6_12·a_1_1·a_1_2·a_5_9
115. b_6_11·a_7_16 − b_2_3⁵·a_3_4 − b_2_3⁵·a_3_3 + b_2_3·a_1_1·a_1_2·a_9_23
        + b_2_3²·a_1_1·a_1_2·a_7_17 − b_2_3·c_6_13·a_5_6 + b_2_3·c_6_12·a_5_7
        − b_2_3·c_6_12·a_5_6 − b_2_3²·c_6_13·a_3_4 − b_2_3²·c_6_12·a_3_4
        + b_2_3²·c_6_12·a_3_3 − b_2_3³·c_6_12·a_1_2 + b_2_3³·c_6_12·a_1_1
        − c_6_13·a_1_1·a_1_2·a_5_8 − c_6_13·a_1_0·a_1_1·a_5_7 + c_6_12·a_1_1·a_1_2·a_5_9
        − c_6_12·a_1_1·a_1_2·a_5_8 − c_6_12·a_1_0·a_1_1·a_5_7
116.  − b_6_11·a_7_17 + b_4_7·a_9_23 − b_2_3⁴·a_5_7 + b_2_3⁴·a_5_6 − b_2_3⁵·a_3_4
        − b_2_3⁵·a_3_3 − b_2_3²·a_1_1·a_1_2·a_7_17 + b_2_3·c_6_13·a_5_7 + b_2_3·c_6_13·a_5_6
        − b_2_3·c_6_12·a_5_6 + b_2_3²·c_6_13·a_3_4 + b_2_3²·c_6_13·a_3_3
        − b_2_3²·c_6_12·a_3_4 − b_2_3²·c_6_12·a_3_3 + b_2_3³·c_6_12·a_1_2
        + b_2_3³·c_6_12·a_1_1 + c_6_13·a_1_1·a_1_2·a_5_8 + c_6_12·a_1_1·a_1_2·a_5_8
        + c_6_12·a_1_0·a_1_1·a_5_7
117. a_5_7·a_9_23 − a_5_5·a_9_23 − b_2_3³·a_1_2·a_7_17 − b_2_3⁵·a_1_1·a_3_4
        + c_6_12·a_1_2·a_7_17 − c_6_12·a_1_2·a_7_16 − b_2_3·c_6_13·a_1_1·a_5_7
        − b_2_3·c_6_12·a_1_2·a_5_9 + b_2_3·c_6_12·a_1_1·a_5_7 − b_2_3²·c_6_13·a_1_1·a_3_4
        + b_2_3²·c_6_12·a_1_1·a_3_4 + b_2_3³·c_6_13·a_1_1·a_1_2 − c_6_12·c_6_13·a_1_0·a_1_2
        − c_6_12·c_6_13·a_1_0·a_1_1 + c_6_12²·a_1_1·a_1_2 + c_6_12²·a_1_0·a_1_1
118. a_5_6·a_9_23 + a_5_5·a_9_23 − b_2_3³·a_1_1·a_7_17 − b_2_3⁴·a_1_1·a_5_7
        + c_6_12·a_1_1·a_7_17 − c_6_12·a_1_1·a_7_16 − b_2_3·c_6_13·a_1_1·a_5_7
        − b_2_3·c_6_12·a_1_1·a_5_9 − b_2_3·c_6_12·a_1_1·a_5_7 − b_2_3²·c_6_13·a_1_1·a_3_4
        − b_2_3²·c_6_12·a_1_1·a_3_4 + b_2_3³·c_6_12·a_1_1·a_1_2 − c_6_12·c_6_13·a_1_1·a_1_2
        + c_6_12·c_6_13·a_1_0·a_1_2 − c_6_12²·a_1_1·a_1_2 + c_6_12²·a_1_0·a_1_2
119.  − a_7_16·a_7_17 + a_5_5·a_9_23 + b_2_3²·a_1_1·a_9_23 − b_2_3³·a_1_1·a_7_17
        − b_2_3⁴·a_1_1·a_5_7 + b_2_3⁵·a_1_1·a_3_4 + c_6_13·a_1_1·a_7_17 + c_6_13·a_1_1·a_7_16
        − c_6_12·a_1_2·a_7_17 + c_6_12·a_1_2·a_7_16 + c_6_12·a_1_1·a_7_17 − c_6_12·a_1_1·a_7_16
        − b_2_3·c_6_13·a_1_1·a_5_7 + b_2_3·c_6_12·a_1_1·a_5_9 − b_2_3²·c_6_13·a_1_1·a_3_4
        − b_2_3²·c_6_12·a_1_1·a_3_4 − b_2_3³·c_6_12·a_1_1·a_1_2 − c_6_12·c_6_13·a_1_1·a_1_2
        − c_6_12·c_6_13·a_1_0·a_1_2 + c_6_12·c_6_13·a_1_0·a_1_1 − c_6_12²·a_1_0·a_1_2
        − c_6_12²·a_1_0·a_1_1
120. a_7_16·a_7_17 + a_5_5·a_9_23 + b_2_3²·a_1_2·a_9_23 − b_2_3³·a_1_2·a_7_17
        + b_2_3⁵·a_1_1·a_3_4 − c_6_13·a_1_1·a_7_17 − c_6_13·a_1_1·a_7_16 + c_6_12·a_1_2·a_7_17
        − c_6_12·a_1_2·a_7_16 − c_6_12·a_1_1·a_7_17 + c_6_12·a_1_1·a_7_16
        − b_2_3·c_6_13·a_1_1·a_5_7 + b_2_3·c_6_12·a_1_2·a_5_9 − b_2_3²·c_6_13·a_1_1·a_3_4
        + b_2_3³·c_6_13·a_1_1·a_1_2 − b_2_3³·c_6_12·a_1_1·a_1_2 + c_6_12·c_6_13·a_1_1·a_1_2
        − c_6_12·c_6_13·a_1_0·a_1_2 + c_6_12·c_6_13·a_1_0·a_1_1 − c_6_12²·a_1_0·a_1_2
        − c_6_12²·a_1_0·a_1_1
121. a_7_16·a_7_17 + a_5_9·a_9_23 + a_5_5·a_9_23 + b_2_3³·a_1_2·a_7_17
        + b_2_3³·a_1_1·a_7_17 + b_2_3⁴·a_1_1·a_5_7 + c_6_13·a_1_2·a_7_17
        + c_6_13·a_1_2·a_7_16 + c_6_13·a_1_1·a_7_17 + c_6_13·a_1_1·a_7_16 − c_6_12·a_1_2·a_7_17
        − c_6_12·a_1_1·a_7_16 + b_2_3·c_6_13·a_1_1·a_5_7 + b_2_3·c_6_12·a_1_2·a_5_9
        + b_2_3·c_6_12·a_1_1·a_5_9 + b_2_3·c_6_12·a_1_1·a_5_7 − b_2_3²·c_6_12·a_1_1·a_3_4
        − b_2_3³·c_6_13·a_1_1·a_1_2 + b_2_3³·c_6_12·a_1_1·a_1_2 + c_6_12·c_6_13·a_1_1·a_1_2
        − c_6_12·c_6_13·a_1_0·a_1_2 + c_6_12·c_6_13·a_1_0·a_1_1 + c_6_12²·a_1_1·a_1_2
        − c_6_12²·a_1_0·a_1_2 − c_6_12²·a_1_0·a_1_1
122.  − a_7_16·a_7_17 + a_5_8·a_9_23 − a_5_5·a_9_23 − b_2_3³·a_1_2·a_7_17
        + b_2_3³·a_1_1·a_7_17 + b_2_3⁴·a_1_1·a_5_7 − c_6_13·a_1_2·a_7_16
        + c_6_13·a_1_1·a_7_17 − c_6_13·a_1_1·a_7_16 − c_6_12·a_1_2·a_7_16 − c_6_12·a_1_1·a_7_16
        − b_2_3·c_6_12·a_1_2·a_5_9 + b_2_3·c_6_12·a_1_1·a_5_9 − b_2_3·c_6_12·a_1_1·a_5_7
        + b_2_3³·c_6_13·a_1_1·a_1_2 − c_6_13²·a_1_1·a_1_2 − c_6_12·c_6_13·a_1_1·a_1_2
        − c_6_12·c_6_13·a_1_0·a_1_2 + c_6_12·c_6_13·a_1_0·a_1_1 − c_6_12²·a_1_0·a_1_1
123. b_6_11·a_9_23 − b_2_3⁴·a_7_17 − b_2_3⁵·a_5_7 − b_2_3⁶·a_3_3
        − b_2_3³·a_1_1·a_1_2·a_7_17 + b_2_3⁴·a_1_1·a_1_2·a_5_9 + b_2_3·c_6_12·a_7_17
        − b_2_3²·c_6_13·a_5_7 + b_2_3²·c_6_13·a_5_6 − b_2_3²·c_6_12·a_5_9
        + b_2_3²·c_6_12·a_5_7 + b_2_3³·c_6_13·a_3_3 − b_2_3⁴·c_6_13·a_1_1
        + b_2_3⁴·c_6_12·a_1_2 + c_6_13·a_1_1·a_1_2·a_7_17 − c_6_12·a_1_1·a_1_2·a_7_17
        + b_2_3·c_6_12·a_1_1·a_1_2·a_5_9 − b_2_3·c_6_12·c_6_13·a_1_2
        − b_2_3·c_6_12·c_6_13·a_1_1 + b_2_3·c_6_12²·a_1_1
124.  − a_7_16·a_9_23 + b_2_3⁴·a_1_2·a_7_17 − b_2_3⁴·a_1_1·a_7_17 − b_2_3⁵·a_1_1·a_5_7
        + b_2_3⁶·a_1_1·a_3_4 + c_6_13·a_1_1·a_9_23 − c_6_12·a_1_2·a_9_23 + c_6_12·a_1_1·a_9_23
        − b_2_3·c_6_12·a_1_2·a_7_17 + b_2_3·c_6_12·a_1_1·a_7_17 + b_2_3²·c_6_12·a_1_2·a_5_9
        − b_2_3²·c_6_12·a_1_1·a_5_9 + b_2_3²·c_6_12·a_1_1·a_5_7
        + b_2_3³·c_6_12·a_1_1·a_3_4 − b_2_3⁴·c_6_13·a_1_1·a_1_2
        + b_2_3⁴·c_6_12·a_1_1·a_1_2 − c_6_12·c_6_13·a_1_2·a_3_5 + c_6_12·c_6_13·a_1_1·a_3_5
        − c_6_12²·a_1_2·a_3_5 − c_6_12²·a_1_1·a_3_5 + b_2_3·c_6_12·c_6_13·a_1_1·a_1_2
        + b_2_3·c_6_12²·a_1_1·a_1_2
125. a_7_17·a_9_23 − a_7_16·a_9_23 + b_2_3³·a_1_2·a_9_23 − b_2_3³·a_1_1·a_9_23
        + b_2_3⁴·a_1_2·a_7_17 + b_2_3⁶·a_1_1·a_3_4 − c_6_13·a_1_2·a_9_23
        − c_6_12·a_1_2·a_9_23 − c_6_12·a_1_1·a_9_23 − b_2_3·c_6_13·a_1_2·a_7_17
        + b_2_3·c_6_13·a_1_1·a_7_17 − b_2_3·c_6_12·a_1_2·a_7_17 − b_2_3·c_6_12·a_1_1·a_7_17
        − b_2_3²·c_6_13·a_1_1·a_5_7 + b_2_3²·c_6_12·a_1_1·a_5_9
        + b_2_3²·c_6_12·a_1_1·a_5_7 + b_2_3³·c_6_12·a_1_1·a_3_4
        − b_2_3⁴·c_6_13·a_1_1·a_1_2 − c_6_13²·a_1_0·a_3_4 − c_6_13²·a_1_0·a_3_3
        + c_6_12·c_6_13·a_1_0·a_3_3 + c_6_12²·a_1_0·a_3_4 + b_2_3·c_6_13²·a_1_1·a_1_2
        + b_2_3·c_6_12·c_6_13·a_1_1·a_1_2 + b_2_3·c_6_12²·a_1_1·a_1_2

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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_12, a Duflot regular element of degree 6
  2. c_6_13, a Duflot regular element of degree 6
  3. b_2_3, 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].

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Ring generators

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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_1_2 → 0, an element of degree 1
4. b_2_3 → 0, an element of degree 2
5. a_3_2 → 0, an element of degree 3
6. a_3_3 → 0, an element of degree 3
7. a_3_4 → 0, an element of degree 3
8. a_3_5 → 0, an element of degree 3
9. b_4_7 → 0, an element of degree 4
10. a_5_5 → 0, an element of degree 5
11. a_5_6 → 0, an element of degree 5
12. a_5_7 → 0, an element of degree 5
13. a_5_8 → 0, an element of degree 5
14. a_5_9 → 0, an element of degree 5
15. b_6_11 → 0, an element of degree 6
16. c_6_12 →  − c_2_1³, an element of degree 6
17. c_6_13 →  − c_2_2³, an element of degree 6
18. a_7_16 → 0, an element of degree 7
19. a_7_17 → 0, an element of degree 7
20. a_9_23 → 0, an element of degree 9

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_1_2 → a_1_2, an element of degree 1
4. b_2_3 → c_2_5, an element of degree 2
5. a_3_2 → 0, an element of degree 3
6. a_3_3 → 0, an element of degree 3
7. a_3_4 → c_2_3·a_1_2, an element of degree 3
8. a_3_5 → 0, an element of degree 3
9. b_4_7 → c_2_3·c_2_5, an element of degree 4
10. a_5_5 → c_2_3·c_2_5·a_1_2, an element of degree 5
11. a_5_6 →  − c_2_3·c_2_5·a_1_2, an element of degree 5
12. a_5_7 →  − c_2_3²·a_1_2, an element of degree 5
13. a_5_8 → c_2_3·c_2_5·a_1_2 − c_2_3²·a_1_2, an element of degree 5
14. a_5_9 → c_2_5²·a_1_1 − c_2_4·c_2_5·a_1_2, an element of degree 5
15. b_6_11 →  − c_2_3·c_2_5² − c_2_3²·c_2_5, an element of degree 6
16. c_6_12 →  − c_2_3·c_2_5² − c_2_3³, an element of degree 6
17. c_6_13 → c_2_5²·a_1_1·a_1_2 + c_2_4·c_2_5² − c_2_4³, an element of degree 6
18. a_7_16 →  − c_2_3²·c_2_5·a_1_2 + c_2_3³·a_1_2, an element of degree 7
19. a_7_17 →  − c_2_3·c_2_5²·a_1_2 + c_2_3·c_2_5²·a_1_1 − c_2_3·c_2_4·c_2_5·a_1_2
       − c_2_3²·c_2_5·a_1_2 + c_2_3³·a_1_2, an element of degree 7
20. a_9_23 →  − c_2_3·c_2_5³·a_1_1 − c_2_3·c_2_4·c_2_5²·a_1_2 − c_2_3·c_2_4³·a_1_2
       − c_2_3²·c_2_5²·a_1_1 + c_2_3²·c_2_4·c_2_5·a_1_2 − c_2_3⁴·a_1_2, an element of degree 9

About the group

Ring generators

Ring relations

Completion information

Restriction maps

Back to groups of order 243