Simon King
David J. Green
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Singular
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Cohomology of group number 944 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 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 2 and 3, respectively.
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
t6 − t5 − t4 + t3 − 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_1, a nilpotent element of degree 1
- b_1_0, an element of degree 1
- b_1_2, an element of degree 1
- b_2_4, an element of degree 2
- b_4_5, an element of degree 4
- b_5_7, an element of degree 5
- b_5_8, an element of degree 5
- b_5_9, an element of degree 5
- b_8_16, an element of degree 8
- c_8_17, a Duflot regular element of degree 8
Ring relations
There are 29 minimal relations of maximal degree 16:
- a_1_1·b_1_0 + a_1_12
- b_1_0·b_1_2
- b_2_4·b_1_2 + a_1_13
- b_2_4·a_1_13
- b_4_5·b_1_2
- b_4_5·a_1_1
- a_1_1·b_5_7 + b_2_42·a_1_12
- b_1_2·b_5_7 + a_1_1·b_5_8
- b_1_0·b_5_8 + b_1_0·b_5_7 + b_2_42·a_1_12
- b_1_2·b_5_9
- b_1_0·b_5_9 + b_1_0·b_5_7 + b_2_4·b_4_5 + a_1_1·b_5_9 + b_2_42·a_1_12
- b_2_4·b_5_8 + b_2_4·b_5_7 + b_2_43·a_1_1 + a_1_12·b_5_9
- b_1_03·b_5_7 + b_4_5·b_1_04 + b_4_52 + b_2_4·b_1_06 + b_2_4·b_4_5·b_1_02
- b_4_5·b_5_8 + b_4_5·b_5_7
- b_4_5·b_5_9 + b_4_5·b_5_7 + b_2_4·b_1_02·b_5_7 + b_2_4·b_4_5·b_1_03
+ b_2_42·b_1_05 + b_2_42·b_4_5·b_1_0
- b_8_16·b_1_2 + a_1_1·b_1_23·b_5_8
- b_8_16·a_1_1 + b_2_44·a_1_1 + b_2_4·a_1_12·b_5_9
- b_8_16·b_1_0 + b_4_5·b_5_7 + b_4_5·b_1_05 + b_4_52·b_1_0 + b_2_4·b_1_07
+ b_2_42·b_4_5·b_1_0 + b_2_43·b_1_03 + b_2_44·b_1_0 + b_2_4·a_1_12·b_5_9
- b_5_8·b_5_9 + b_5_7·b_5_9 + b_2_42·a_1_1·b_5_9
- b_5_82 + b_5_72 + b_1_25·b_5_8 + a_1_1·b_1_24·b_5_8 + b_2_44·a_1_12
+ c_8_17·b_1_22
- b_5_7·b_5_8 + b_5_72 + a_1_1·b_1_24·b_5_8 + b_2_44·a_1_12 + c_8_17·a_1_1·b_1_2
- b_5_92 + b_5_72 + b_2_42·b_1_0·b_5_7 + b_2_42·b_4_5·b_1_02 + b_2_43·b_1_04
+ b_2_43·b_4_5 + b_2_42·a_1_1·b_5_9 + c_8_17·a_1_12
- b_5_92 + b_4_5·b_1_0·b_5_7 + b_4_52·b_1_02 + b_2_4·b_4_52 + b_2_42·b_1_0·b_5_7
+ b_2_42·b_1_06 + b_2_43·b_1_04 + b_2_43·b_4_5 + b_2_42·a_1_1·b_5_9 + b_2_44·a_1_12 + c_8_17·b_1_02
- b_5_8·b_5_9 + b_5_72 + b_2_4·b_8_16 + b_2_4·b_4_5·b_1_04 + b_2_4·b_4_52
+ b_2_42·b_1_06 + b_2_43·b_4_5 + b_2_44·b_1_02 + b_2_45 + b_2_42·a_1_1·b_5_9 + b_2_44·a_1_12
- b_4_5·b_8_16 + b_4_53 + b_2_4·b_4_5·b_1_0·b_5_7 + b_2_42·b_4_52
+ b_2_43·b_4_5·b_1_02 + b_2_44·b_4_5 + c_8_17·b_1_04
- b_8_16·b_5_8 + b_4_52·b_1_05 + b_2_4·b_4_5·b_1_02·b_5_7 + b_2_4·b_4_52·b_1_03
+ b_2_42·b_1_09 + b_2_42·b_4_5·b_5_7 + b_2_42·b_4_5·b_1_05 + b_2_43·b_1_02·b_5_7 + b_2_44·b_5_7 + a_1_1·b_1_27·b_5_8 + b_2_46·a_1_1 + b_2_43·a_1_12·b_5_9 + b_4_5·c_8_17·b_1_0 + c_8_17·a_1_1·b_1_24
- b_8_16·b_5_9 + b_4_52·b_1_05 + b_2_42·b_1_09 + b_2_42·b_4_52·b_1_0
+ b_2_43·b_4_5·b_1_03 + b_2_44·b_5_9 + b_2_44·b_1_05 + b_4_5·c_8_17·b_1_0 + b_2_4·c_8_17·b_1_03
- b_8_16·b_5_7 + b_4_52·b_1_05 + b_2_4·b_4_5·b_1_02·b_5_7 + b_2_4·b_4_52·b_1_03
+ b_2_42·b_1_09 + b_2_42·b_4_5·b_5_7 + b_2_42·b_4_5·b_1_05 + b_2_43·b_1_02·b_5_7 + b_2_44·b_5_7 + b_2_43·a_1_12·b_5_9 + b_4_5·c_8_17·b_1_0
- b_8_162 + b_4_54 + b_2_4·b_4_5·b_1_010 + b_2_4·b_4_52·b_1_06
+ b_2_4·b_4_53·b_1_02 + b_2_42·b_1_012 + b_2_42·b_4_5·b_1_08 + b_2_42·b_4_52·b_1_04 + b_2_44·b_4_52 + b_2_46·b_1_04 + b_2_48 + c_8_17·b_1_08 + b_4_5·c_8_17·b_1_04 + b_4_52·c_8_17
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:
- c_8_17, a Duflot regular element of degree 8
- b_1_22 + b_1_02 + b_2_4, an element of degree 2
- b_1_02, 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_1 → 0, an element of degree 1
- b_1_0 → 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_4_5 → 0, an element of degree 4
- b_5_7 → 0, an element of degree 5
- b_5_8 → 0, an element of degree 5
- b_5_9 → 0, an element of degree 5
- b_8_16 → 0, an element of degree 8
- c_8_17 → c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 2
- a_1_1 → 0, an element of degree 1
- b_1_0 → 0, an element of degree 1
- b_1_2 → c_1_1, an element of degree 1
- b_2_4 → 0, an element of degree 2
- b_4_5 → 0, an element of degree 4
- b_5_7 → 0, an element of degree 5
- b_5_8 → c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
- b_5_9 → 0, an element of degree 5
- b_8_16 → 0, an element of degree 8
- c_8_17 → 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_1 → 0, an element of degree 1
- b_1_0 → c_1_1, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_2_4 → c_1_22 + c_1_1·c_1_2, an element of degree 2
- b_4_5 → c_1_0·c_1_13 + c_1_02·c_1_12, an element of degree 4
- b_5_7 → 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_0·c_1_14 + c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_2 + c_1_04·c_1_1, an element of degree 5
- b_5_8 → 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_0·c_1_14 + c_1_02·c_1_1·c_1_22 + c_1_02·c_1_12·c_1_2 + c_1_04·c_1_1, an element of degree 5
- b_5_9 → c_1_13·c_1_22 + c_1_14·c_1_2 + c_1_0·c_1_14 + c_1_04·c_1_1, an element of degree 5
- b_8_16 → c_1_28 + c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_15·c_1_23 + c_1_16·c_1_22
+ 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_0·c_1_17 + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22 + c_1_02·c_1_16 + c_1_03·c_1_15 + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_13·c_1_2 + c_1_04·c_1_14 + c_1_05·c_1_13 + c_1_06·c_1_12, an element of degree 8
- c_8_17 → c_1_0·c_1_13·c_1_24 + c_1_0·c_1_16·c_1_2 + 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_03·c_1_15 + c_1_04·c_1_24 + c_1_04·c_1_12·c_1_22 + c_1_04·c_1_14 + c_1_05·c_1_13 + c_1_06·c_1_12 + c_1_08, an element of degree 8
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