Simon King
David J. Green
Cohomology
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Cohomology of group number 231 of order 128
General information on the group
- The group has 3 minimal generators and exponent 8.
- It is non-abelian.
- It has p-Rank 4.
- Its center has rank 3.
- 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 3.
- The depth coincides with the Duflot bound.
- The Poincaré series is
(2) · (t4 + 1/2·t3 + 1/2·t2 + 1/2·t + 1/2) |
| (t + 1)2 · (t − 1)4 · (t2 + 1)2 |
- The a-invariants are -∞,-∞,-∞,-4,-4. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 15 minimal generators of maximal degree 6:
- a_1_0, a nilpotent element of degree 1
- a_1_1, a nilpotent element of degree 1
- c_1_2, a Duflot regular element of degree 1
- a_2_5, a nilpotent element of degree 2
- b_2_4, an element of degree 2
- a_3_7, a nilpotent element of degree 3
- a_3_8, a nilpotent element of degree 3
- a_3_9, a nilpotent element of degree 3
- a_4_14, a nilpotent element of degree 4
- a_4_15, a nilpotent element of degree 4
- b_4_13, an element of degree 4
- c_4_16, a Duflot regular element of degree 4
- c_4_17, a Duflot regular element of degree 4
- a_5_27, a nilpotent element of degree 5
- a_6_37, a nilpotent element of degree 6
Ring relations
There are 65 minimal relations of maximal degree 12:
- a_1_02
- a_1_0·a_1_1
- a_1_13
- a_2_5·a_1_1
- a_2_5·a_1_0
- b_2_4·a_1_1
- a_2_52
- a_1_1·a_3_7
- a_1_0·a_3_7
- a_1_1·a_3_8
- a_1_0·a_3_8
- a_1_0·a_3_9
- a_2_5·a_3_9 + a_2_5·a_3_8
- a_2_5·a_3_7 + a_1_12·a_3_9
- b_2_4·a_3_9 + b_2_4·a_3_8 + a_2_5·a_3_8 + a_2_5·a_3_7
- a_4_14·a_1_1 + a_2_5·a_3_7
- a_4_14·a_1_0 + a_2_5·a_3_7
- a_4_15·a_1_1
- a_4_15·a_1_0 + a_2_5·a_3_8 + a_2_5·a_3_7
- b_4_13·a_1_1
- b_4_13·a_1_0 + b_2_4·a_3_7
- a_3_72
- a_3_82
- a_3_7·a_3_8
- a_3_8·a_3_9
- a_3_7·a_3_9
- a_3_92 + c_4_16·a_1_12
- a_2_5·a_4_14
- a_2_5·a_4_15
- b_2_4·a_4_14 + a_2_5·b_4_13
- a_3_92 + a_1_1·a_5_27
- a_1_0·a_5_27
- a_4_14·a_3_9 + a_4_14·a_3_8
- a_4_14·a_3_7
- a_4_15·a_3_9 + a_4_14·a_3_9 + a_2_5·b_2_4·a_3_8
- a_4_15·a_3_8 + a_4_14·a_3_9 + a_2_5·b_2_4·a_3_8
- a_4_15·a_3_7 + a_4_14·a_3_9
- b_4_13·a_3_9 + b_4_13·a_3_8 + a_4_14·a_3_9
- b_4_13·a_3_7 + b_2_42·a_3_7 + b_2_4·c_4_17·a_1_0
- a_4_14·a_3_9 + a_2_5·a_5_27
- b_4_13·a_3_9 + b_2_4·a_5_27 + a_4_14·a_3_9 + a_2_5·b_2_4·a_3_8 + b_2_4·c_4_16·a_1_0
- a_6_37·a_1_1
- a_6_37·a_1_0 + a_4_14·a_3_9 + a_2_5·b_2_4·a_3_8
- a_4_142
- a_4_152
- a_4_14·a_4_15
- b_4_132 + b_2_42·b_4_13 + b_2_42·c_4_17
- a_4_14·b_4_13 + a_2_5·b_2_4·b_4_13 + a_2_5·b_2_4·c_4_17
- a_3_9·a_5_27 + c_4_16·a_1_1·a_3_9
- a_3_8·a_5_27
- a_3_7·a_5_27
- a_2_5·a_6_37
- a_4_15·b_4_13 + b_2_4·a_6_37 + b_2_42·a_4_15 + a_2_5·b_2_4·b_4_13 + a_2_5·b_2_43
+ a_2_5·b_2_4·c_4_16
- a_4_15·a_5_27 + a_2_5·c_4_17·a_3_8 + a_2_5·c_4_16·a_3_8
- b_4_13·a_5_27 + b_2_42·a_5_27 + a_2_5·b_2_4·a_5_27 + a_2_5·b_2_42·a_3_8
+ b_2_4·c_4_17·a_3_8 + b_2_4·c_4_16·a_3_7 + b_2_42·c_4_16·a_1_0
- a_4_14·a_5_27 + a_2_5·b_2_4·a_5_27 + a_2_5·c_4_17·a_3_8
- a_6_37·a_3_9 + a_2_5·c_4_17·a_3_8 + a_2_5·c_4_16·a_3_8
- a_6_37·a_3_8 + a_2_5·c_4_17·a_3_8 + a_2_5·c_4_16·a_3_8 + c_4_16·a_1_12·a_3_9
- a_6_37·a_3_7 + a_2_5·c_4_17·a_3_8 + c_4_17·a_1_12·a_3_9 + c_4_16·a_1_12·a_3_9
- a_5_272 + c_4_162·a_1_12
- a_4_15·a_6_37
- b_4_13·a_6_37 + b_2_4·a_4_15·c_4_17 + a_2_5·b_4_13·c_4_16 + a_2_5·b_2_42·c_4_17
- a_4_14·a_6_37
- a_6_37·a_5_27 + a_2_5·c_4_17·a_5_27 + a_2_5·b_2_4·c_4_16·a_3_8
- a_6_372
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:
- c_1_2, a Duflot regular element of degree 1
- c_4_16, a Duflot regular element of degree 4
- c_4_17, a Duflot regular element of degree 4
- b_2_4, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 5, 7].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 3
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- c_1_2 → c_1_0, an element of degree 1
- a_2_5 → 0, an element of degree 2
- b_2_4 → 0, an element of degree 2
- a_3_7 → 0, an element of degree 3
- a_3_8 → 0, an element of degree 3
- a_3_9 → 0, an element of degree 3
- a_4_14 → 0, an element of degree 4
- a_4_15 → 0, an element of degree 4
- b_4_13 → 0, an element of degree 4
- c_4_16 → c_1_14, an element of degree 4
- c_4_17 → c_1_24 + c_1_14, an element of degree 4
- a_5_27 → 0, an element of degree 5
- a_6_37 → 0, an element of degree 6
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- c_1_2 → c_1_0, an element of degree 1
- a_2_5 → 0, an element of degree 2
- b_2_4 → c_1_32, an element of degree 2
- a_3_7 → 0, an element of degree 3
- a_3_8 → 0, an element of degree 3
- a_3_9 → 0, an element of degree 3
- a_4_14 → 0, an element of degree 4
- a_4_15 → 0, an element of degree 4
- b_4_13 → c_1_22·c_1_32 + c_1_12·c_1_32, an element of degree 4
- c_4_16 → c_1_22·c_1_32 + c_1_14, an element of degree 4
- c_4_17 → c_1_22·c_1_32 + c_1_24 + c_1_12·c_1_32 + c_1_14, an element of degree 4
- a_5_27 → 0, an element of degree 5
- a_6_37 → 0, an element of degree 6
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