Simon King
David J. Green
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Cohomology of group number 1626 of order 128
General information on the group
- The group has 4 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
t5 + t2 + 1 |
| (t − 1)4 · (t2 + 1) · (t4 + 1) |
- The a-invariants are -∞,-∞,-∞,-4,-4. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 11 minimal generators of maximal degree 8:
- a_1_0, a nilpotent element of degree 1
- a_1_1, a nilpotent element of degree 1
- b_1_2, an element of degree 1
- c_1_3, a Duflot regular element of degree 1
- a_2_8, a nilpotent element of degree 2
- c_2_9, a Duflot regular element of degree 2
- b_3_19, an element of degree 3
- a_5_36, a nilpotent element of degree 5
- a_5_48, a nilpotent element of degree 5
- a_6_73, a nilpotent element of degree 6
- c_8_146, a Duflot regular element of degree 8
Ring relations
There are 27 minimal relations of maximal degree 12:
- a_1_02
- a_1_0·a_1_1
- a_1_0·b_1_22 + a_1_13
- a_2_8·a_1_0
- a_2_82 + c_2_9·a_1_12
- a_1_1·b_3_19 + a_2_8·b_1_22 + a_1_13·b_1_2 + a_2_8·a_1_12
- a_1_0·b_3_19 + a_2_8·a_1_12
- a_1_13·b_1_22
- a_2_8·b_3_19 + a_2_8·a_1_12·b_1_2 + c_2_9·a_1_1·b_1_22 + c_2_9·a_1_13
- b_3_192 + a_2_8·a_1_12·b_1_22 + c_2_9·b_1_24
- a_1_1·a_5_36 + a_2_8·a_1_1·b_1_23 + c_2_9·a_1_12·b_1_22 + c_2_9·a_1_13·b_1_2
+ a_2_8·c_2_9·a_1_12
- a_1_0·a_5_36 + a_2_8·a_1_12·b_1_22
- a_1_0·a_5_48
- a_2_8·a_5_36 + a_2_8·a_1_12·b_1_23 + c_2_9·a_1_12·b_1_23
+ a_2_8·c_2_9·a_1_1·b_1_22 + a_2_8·c_2_9·a_1_12·b_1_2 + c_2_92·a_1_13
- b_1_22·a_5_36 + a_2_8·b_1_25 + a_1_12·b_1_25 + a_1_12·a_5_48
+ a_2_8·a_1_12·b_1_23 + c_2_9·a_1_1·b_1_24 + c_2_9·a_1_12·b_1_23 + a_2_8·c_2_9·a_1_1·b_1_22
- a_6_73·a_1_1 + a_2_8·a_5_48 + a_2_8·a_1_1·b_1_24 + a_2_8·a_1_12·b_1_23
+ c_2_9·a_1_12·b_1_23 + a_2_8·c_2_9·a_1_1·b_1_22 + a_2_8·c_2_9·a_1_12·b_1_2 + c_2_92·a_1_13
- a_6_73·a_1_0
- b_3_19·a_5_36 + a_2_8·a_1_1·b_1_25 + a_2_8·a_1_1·a_5_48 + c_2_9·a_1_1·b_1_25
+ a_2_8·c_2_9·b_1_24 + a_2_8·c_2_9·a_1_1·b_1_23 + a_2_8·c_2_9·a_1_12·b_1_22 + c_2_92·a_1_12·b_1_22
- b_3_19·a_5_48 + b_3_19·a_5_36 + a_6_73·b_1_22 + a_2_8·b_1_26 + a_1_12·b_1_2·a_5_48
+ a_2_8·c_2_9·a_1_12·b_1_22
- a_2_8·a_6_73 + c_2_9·a_1_1·a_5_48 + c_2_9·a_1_12·b_1_24
+ a_2_8·c_2_9·a_1_1·b_1_23 + c_2_92·a_1_12·b_1_22 + c_2_92·a_1_13·b_1_2 + a_2_8·c_2_92·a_1_12
- a_6_73·b_3_19 + a_2_8·a_1_1·b_1_2·a_5_48 + c_2_9·b_1_22·a_5_48 + c_2_9·a_1_1·b_1_26
+ a_2_8·c_2_9·b_1_25 + c_2_9·a_1_12·b_1_25 + c_2_9·a_1_12·a_5_48 + c_2_92·a_1_1·b_1_24 + c_2_92·a_1_12·b_1_23 + a_2_8·c_2_92·a_1_1·b_1_22
- a_5_362 + c_2_9·a_1_12·b_1_26 + c_2_92·a_1_12·b_1_24
- a_5_36·a_5_48 + a_2_8·b_1_23·a_5_48 + a_1_12·b_1_23·a_5_48
+ c_2_9·a_1_1·b_1_22·a_5_48 + c_2_9·a_1_12·b_1_2·a_5_48 + a_2_8·c_2_9·a_1_1·a_5_48
- a_5_482 + a_1_1·b_1_24·a_5_48 + a_1_12·b_1_28 + a_2_8·a_1_1·b_1_27
+ c_8_146·a_1_12 + c_2_92·a_1_12·b_1_24 + a_2_8·c_2_92·a_1_12·b_1_22 + c_2_93·a_1_12·b_1_22
- a_6_73·a_5_36 + a_2_8·a_1_1·b_1_23·a_5_48 + c_2_9·a_1_1·b_1_23·a_5_48
+ c_2_9·a_1_12·b_1_27 + a_2_8·c_2_9·b_1_22·a_5_48 + a_2_8·c_2_9·a_1_1·b_1_2·a_5_48 + a_2_8·c_2_92·a_1_1·b_1_24 + c_2_92·a_1_12·a_5_48
- a_6_73·a_5_48 + a_2_8·a_1_1·b_1_28 + a_2_8·a_1_1·b_1_23·a_5_48
+ c_2_9·a_1_1·b_1_23·a_5_48 + c_2_9·a_1_12·b_1_27 + a_2_8·c_8_146·a_1_1 + a_2_8·c_2_9·b_1_22·a_5_48 + a_2_8·c_2_9·a_1_1·b_1_2·a_5_48 + a_2_8·c_2_92·a_1_1·b_1_24 + c_2_92·a_1_12·a_5_48 + a_2_8·c_2_92·a_1_12·b_1_23 + a_2_8·c_2_93·a_1_1·b_1_22
- a_6_732 + c_2_9·a_1_1·b_1_24·a_5_48 + a_2_8·c_2_9·a_1_1·b_1_27
+ c_2_9·c_8_146·a_1_12 + c_2_92·a_1_12·b_1_26 + a_2_8·c_2_93·a_1_12·b_1_22 + c_2_94·a_1_12·b_1_22
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_3, a Duflot regular element of degree 1
- c_2_9, a Duflot regular element of degree 2
- c_8_146, a Duflot regular element of degree 8
- b_1_22, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 7, 9].
- 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
- b_1_2 → 0, an element of degree 1
- c_1_3 → c_1_0, an element of degree 1
- a_2_8 → 0, an element of degree 2
- c_2_9 → c_1_12, an element of degree 2
- b_3_19 → 0, an element of degree 3
- a_5_36 → 0, an element of degree 5
- a_5_48 → 0, an element of degree 5
- a_6_73 → 0, an element of degree 6
- c_8_146 → c_1_28, an element of degree 8
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
- b_1_2 → c_1_3, an element of degree 1
- c_1_3 → c_1_0, an element of degree 1
- a_2_8 → 0, an element of degree 2
- c_2_9 → c_1_12, an element of degree 2
- b_3_19 → c_1_1·c_1_32, an element of degree 3
- a_5_36 → 0, an element of degree 5
- a_5_48 → 0, an element of degree 5
- a_6_73 → 0, an element of degree 6
- c_8_146 → c_1_24·c_1_34 + c_1_28 + c_1_12·c_1_36 + c_1_14·c_1_34 + c_1_16·c_1_32, an element of degree 8
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