Simon King
David J. Green
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Singular
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Cohomology of group number 926 of order 128
General information on the group
- The group has 3 minimal generators and exponent 16.
- It is non-abelian.
- It has p-Rank 3.
- Its center has rank 1.
- It has 3 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 3 and depth 1.
- The depth coincides with the Duflot bound.
- The Poincaré series is
( − 1) · (t5 + t4 − t3 + t2 + 1) |
| (t − 1)3 · (t2 + 1) · (t4 + 1) |
- The a-invariants are -∞,-5,-3,-3. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 10 minimal generators of maximal degree 8:
- a_1_0, a nilpotent element of degree 1
- b_1_1, an element of degree 1
- b_1_2, an element of degree 1
- b_2_4, an element of degree 2
- b_2_5, an element of degree 2
- b_5_14, an element of degree 5
- b_5_15, an element of degree 5
- b_5_16, an element of degree 5
- b_6_21, an element of degree 6
- c_8_35, a Duflot regular element of degree 8
Ring relations
There are 27 minimal relations of maximal degree 12:
- a_1_0·b_1_1
- a_1_0·b_1_2
- b_2_4·a_1_0
- b_2_5·b_1_2 + b_2_5·b_1_1 + a_1_03
- b_2_5·b_1_12 + b_2_4·b_1_1·b_1_2 + b_2_42
- b_2_5·a_1_03
- a_1_0·b_5_14
- a_1_0·b_5_15
- b_1_2·b_5_14 + b_1_1·b_5_15 + b_1_1·b_5_14
- b_1_2·b_5_16 + b_2_4·b_2_52 + b_2_42·b_1_1·b_1_2 + b_2_43
- b_1_1·b_5_16 + b_2_4·b_2_52 + b_2_42·b_1_1·b_1_2 + b_2_43
- b_2_5·b_5_15 + a_1_02·b_5_16
- b_2_53·b_1_1 + b_2_4·b_5_16 + b_2_4·b_2_52·b_1_1 + b_2_42·b_2_5·b_1_1
- b_1_1·b_1_2·b_5_15 + b_1_12·b_5_15 + b_1_12·b_5_14 + b_6_21·b_1_2 + b_2_4·b_5_15
+ b_2_4·b_5_14 + b_2_42·b_1_12·b_1_2 + b_2_43·b_1_1
- b_2_5·b_5_15 + b_6_21·a_1_0
- b_1_12·b_5_15 + b_1_12·b_5_14 + b_6_21·b_1_1 + b_2_4·b_5_14 + b_2_42·b_1_12·b_1_2
+ b_2_43·b_1_1
- b_2_5·b_1_1·b_5_14 + b_2_4·b_6_21 + b_2_43·b_1_1·b_1_2 + b_2_44
- b_5_15·b_5_16 + b_2_54·a_1_02
- b_5_14·b_5_16 + b_2_52·b_6_21 + b_2_4·b_2_5·b_6_21 + b_2_42·b_6_21 + b_2_43·b_2_52
+ b_2_44·b_1_1·b_1_2 + b_2_45 + b_2_52·a_1_0·b_5_16 + b_2_54·a_1_02
- b_5_152 + b_5_142 + b_1_25·b_5_15 + b_6_21·b_1_1·b_1_23 + b_6_21·b_1_12·b_1_22
+ b_6_21·b_1_13·b_1_2 + b_2_4·b_1_23·b_5_15 + b_2_42·b_1_1·b_5_14 + b_2_43·b_1_24 + b_2_43·b_1_13·b_1_2 + b_2_43·b_2_52 + b_2_44·b_1_12 + b_2_44·b_2_5 + c_8_35·b_1_22
- b_5_14·b_5_15 + b_5_142 + b_6_21·b_1_24 + b_6_21·b_1_1·b_1_23
+ b_6_21·b_1_12·b_1_22 + b_6_21·b_1_13·b_1_2 + b_6_21·b_1_14 + b_2_4·b_1_23·b_5_15 + b_2_4·b_1_13·b_5_14 + b_2_4·b_6_21·b_1_12 + b_2_42·b_1_2·b_5_15 + b_2_43·b_1_13·b_1_2 + b_2_43·b_2_52 + b_2_44·b_1_22 + b_2_44·b_1_12 + b_2_44·b_2_5 + c_8_35·b_1_1·b_1_2
- b_5_162 + b_2_55 + b_2_4·b_2_54 + b_2_44·b_1_1·b_1_2 + b_2_44·b_2_5 + b_2_45
+ b_2_52·a_1_0·b_5_16 + c_8_35·a_1_02
- b_5_142 + b_1_15·b_5_14 + b_6_21·b_1_1·b_1_23 + b_6_21·b_1_12·b_1_22
+ b_6_21·b_1_13·b_1_2 + b_6_21·b_1_14 + b_2_4·b_6_21·b_1_12 + b_2_42·b_1_2·b_5_15 + b_2_42·b_1_1·b_5_15 + b_2_42·b_1_15·b_1_2 + b_2_43·b_1_13·b_1_2 + b_2_43·b_1_14 + b_2_43·b_2_52 + b_2_44·b_1_1·b_1_2 + b_2_44·b_1_12 + b_2_44·b_2_5 + c_8_35·b_1_12
- b_6_21·b_5_16 + b_2_53·b_5_14 + b_2_42·b_6_21·b_1_1 + b_2_42·b_2_5·b_5_14
+ b_2_43·b_5_16 + b_2_43·b_2_52·b_1_1 + b_2_55·a_1_0 + c_8_35·a_1_03
- b_6_21·b_5_15 + b_6_21·b_5_14 + b_6_21·b_1_25 + b_6_21·b_1_1·b_1_24
+ b_6_21·b_1_12·b_1_23 + b_6_21·b_1_13·b_1_22 + b_6_21·b_1_14·b_1_2 + b_2_4·b_6_21·b_1_13 + b_2_43·b_5_16 + b_2_44·b_1_12·b_1_2 + b_2_44·b_2_5·b_1_1 + b_2_45·b_1_1 + c_8_35·b_1_1·b_1_22 + b_2_4·c_8_35·b_1_2
- b_6_21·b_5_14 + b_6_21·b_1_1·b_1_24 + b_6_21·b_1_12·b_1_23
+ b_6_21·b_1_13·b_1_22 + b_6_21·b_1_14·b_1_2 + b_6_21·b_1_15 + b_2_4·b_6_21·b_1_13 + b_2_43·b_5_16 + b_2_44·b_1_12·b_1_2 + b_2_44·b_2_5·b_1_1 + b_2_45·b_1_1 + c_8_35·b_1_12·b_1_2 + b_2_4·c_8_35·b_1_1
- b_6_21·b_1_1·b_1_25 + b_6_21·b_1_12·b_1_24 + b_6_21·b_1_13·b_1_23
+ b_6_21·b_1_14·b_1_22 + b_6_21·b_1_15·b_1_2 + b_6_212 + b_2_43·b_6_21 + b_2_43·b_2_53 + b_2_44·b_1_13·b_1_2 + b_2_45·b_1_1·b_1_2 + b_2_45·b_1_12 + b_2_46 + b_2_55·a_1_02 + c_8_35·b_1_12·b_1_22 + b_2_42·c_8_35
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_8_35, a Duflot regular element of degree 8
- b_1_22 + b_1_1·b_1_2 + b_1_12 + b_2_5, an element of degree 2
- b_1_12, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, 3, 7, 9].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 1
- a_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_2_4 → 0, an element of degree 2
- b_2_5 → 0, an element of degree 2
- b_5_14 → 0, an element of degree 5
- b_5_15 → 0, an element of degree 5
- b_5_16 → 0, an element of degree 5
- b_6_21 → 0, an element of degree 6
- c_8_35 → c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- b_1_1 → c_1_1, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- b_2_4 → 0, an element of degree 2
- b_2_5 → 0, an element of degree 2
- b_5_14 → c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_1·c_1_22
+ c_1_02·c_1_12·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
- b_5_15 → c_1_0·c_1_1·c_1_23 + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_23 + c_1_02·c_1_13
+ c_1_04·c_1_2 + c_1_04·c_1_1, an element of degree 5
- b_5_16 → 0, an element of degree 5
- b_6_21 → c_1_0·c_1_12·c_1_23 + c_1_0·c_1_13·c_1_22 + c_1_02·c_1_1·c_1_23
+ c_1_02·c_1_12·c_1_22 + c_1_02·c_1_13·c_1_2 + c_1_04·c_1_1·c_1_2, an element of degree 6
- c_8_35 → c_1_0·c_1_1·c_1_26 + c_1_0·c_1_16·c_1_2 + c_1_02·c_1_26
+ c_1_02·c_1_13·c_1_23 + c_1_02·c_1_16 + c_1_04·c_1_1·c_1_23 + c_1_04·c_1_13·c_1_2 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- b_1_1 → c_1_1, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- b_2_4 → c_1_1·c_1_2, an element of degree 2
- b_2_5 → 0, an element of degree 2
- b_5_14 → c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_1·c_1_22
+ c_1_02·c_1_12·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
- b_5_15 → c_1_0·c_1_1·c_1_23 + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_23 + c_1_02·c_1_13
+ c_1_04·c_1_2 + c_1_04·c_1_1, an element of degree 5
- b_5_16 → 0, an element of degree 5
- b_6_21 → 0, an element of degree 6
- c_8_35 → c_1_13·c_1_25 + c_1_0·c_1_1·c_1_26 + c_1_0·c_1_12·c_1_25
+ c_1_0·c_1_13·c_1_24 + c_1_0·c_1_14·c_1_23 + c_1_0·c_1_15·c_1_22 + c_1_0·c_1_16·c_1_2 + c_1_02·c_1_26 + c_1_02·c_1_1·c_1_25 + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_13·c_1_23 + c_1_02·c_1_14·c_1_22 + c_1_02·c_1_15·c_1_2 + c_1_02·c_1_16 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_0 → 0, an element of degree 1
- b_1_1 → c_1_1, an element of degree 1
- b_1_2 → c_1_1, an element of degree 1
- b_2_4 → c_1_1·c_1_2, an element of degree 2
- b_2_5 → c_1_22 + c_1_1·c_1_2, an element of degree 2
- b_5_14 → c_1_13·c_1_22 + c_1_14·c_1_2 + c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2
+ c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
- b_5_15 → 0, an element of degree 5
- b_5_16 → c_1_25 + c_1_13·c_1_22, an element of degree 5
- b_6_21 → c_1_14·c_1_22 + c_1_15·c_1_2 + c_1_0·c_1_12·c_1_23 + c_1_0·c_1_14·c_1_2
+ c_1_02·c_1_1·c_1_23 + c_1_02·c_1_14 + c_1_04·c_1_1·c_1_2 + c_1_04·c_1_12, an element of degree 6
- c_8_35 → c_1_1·c_1_27 + c_1_12·c_1_26 + c_1_14·c_1_24 + c_1_15·c_1_23 + c_1_17·c_1_2
+ c_1_0·c_1_13·c_1_24 + c_1_0·c_1_16·c_1_2 + c_1_02·c_1_16 + c_1_04·c_1_24 + c_1_04·c_1_13·c_1_2 + c_1_08, an element of degree 8
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