Simon King
David J. Green
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Singular
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Cohomology of group number 1737 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 2.
- It has 3 conjugacy classes of maximal elementary abelian subgroups, which are of rank 3, 4 and 4, respectively.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 4 and depth 2.
- The depth coincides with the Duflot bound.
- The Poincaré series is
t5 + t4 − t3 + t2 + 1 |
| (t − 1)4 · (t2 + 1) · (t4 + 1) |
- The a-invariants are -∞,-∞,-6,-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:
- b_1_0, an element of degree 1
- b_1_1, an element of degree 1
- b_1_2, an element of degree 1
- b_1_3, an element of degree 1
- b_2_8, an element of degree 2
- c_2_9, a Duflot regular element of degree 2
- b_5_45, an element of degree 5
- b_5_46, an element of degree 5
- b_5_47, an element of degree 5
- b_6_69, an element of degree 6
- c_8_135, a Duflot regular element of degree 8
Ring relations
There are 27 minimal relations of maximal degree 12:
- b_1_0·b_1_1
- b_1_0·b_1_2
- b_1_1·b_1_22 + b_1_0·b_1_32
- b_2_8·b_1_0
- b_1_24 + b_2_8·b_1_1·b_1_2 + b_2_82 + c_2_9·b_1_12
- b_1_0·b_1_34
- b_1_2·b_5_45 + b_1_23·b_1_33 + b_2_83 + c_2_9·b_1_1·b_1_2·b_1_32
+ c_2_9·b_1_13·b_1_2 + b_2_8·c_2_9·b_1_12
- b_1_1·b_5_45 + b_2_8·b_1_1·b_1_2·b_1_32 + b_2_82·b_1_1·b_1_3
+ c_2_9·b_1_12·b_1_32 + c_2_9·b_1_13·b_1_3
- b_1_0·b_5_46 + b_1_0·b_5_45
- b_1_0·b_5_47
- b_1_2·b_5_46 + b_1_23·b_1_33 + b_1_1·b_5_47 + b_2_8·b_1_34 + b_2_83
+ c_2_9·b_1_1·b_1_2·b_1_32 + c_2_9·b_1_13·b_1_2 + b_2_8·c_2_9·b_1_12
- b_2_8·b_5_45 + b_2_8·b_1_22·b_1_33 + b_2_82·b_1_23
+ c_2_9·b_1_12·b_1_2·b_1_32 + c_2_9·b_1_13·b_1_2·b_1_3 + b_2_8·c_2_9·b_1_1·b_1_32
- b_1_32·b_5_45 + b_1_22·b_1_35 + b_1_1·b_1_2·b_5_47 + b_2_8·b_1_23·b_1_32
+ b_2_8·b_1_1·b_1_2·b_1_33 + c_2_9·b_1_1·b_1_34
- b_6_69·b_1_2 + b_2_8·b_5_47 + b_2_8·b_1_23·b_1_32 + b_2_82·b_1_22·b_1_3
+ c_2_9·b_1_1·b_1_34 + b_2_8·c_2_9·b_1_2·b_1_32 + b_2_82·c_2_9·b_1_3 + c_2_92·b_1_12·b_1_3
- b_6_69·b_1_0
- b_1_23·b_1_34 + b_6_69·b_1_1 + b_2_8·b_5_46 + b_2_8·b_1_22·b_1_33
+ b_2_8·b_1_1·b_1_34 + b_2_82·b_1_23 + c_2_9·b_1_13·b_1_2·b_1_3 + b_2_8·c_2_9·b_1_12·b_1_3
- b_1_23·b_5_47 + b_2_8·b_1_1·b_5_47 + b_2_8·b_6_69 + b_2_82·b_1_22·b_1_32
+ b_2_83·b_1_2·b_1_3 + c_2_9·b_1_1·b_5_46 + b_2_8·c_2_9·b_1_23·b_1_3 + b_2_82·c_2_9·b_1_32 + b_2_82·c_2_9·b_1_1·b_1_3 + c_2_92·b_1_12·b_1_32
- b_5_45·b_5_47 + b_1_22·b_1_33·b_5_47 + b_2_82·b_6_69 + b_2_83·b_1_22·b_1_32
+ b_2_84·b_1_2·b_1_3 + c_2_9·b_1_1·b_1_32·b_5_47 + c_2_9·b_1_1·b_1_2·b_1_36 + c_2_9·b_1_12·b_1_2·b_1_35 + c_2_9·b_1_13·b_5_47 + c_2_9·b_1_13·b_1_2·b_1_34 + b_2_8·c_2_9·b_1_1·b_5_46 + b_2_82·c_2_9·b_1_23·b_1_3 + b_2_83·c_2_9·b_1_32 + c_2_92·b_1_14·b_1_2·b_1_3 + b_2_8·c_2_92·b_1_12·b_1_32 + b_2_8·c_2_92·b_1_13·b_1_3
- b_5_45·b_5_46 + b_5_452 + b_1_1·b_1_2·b_1_33·b_5_47 + b_2_8·b_1_2·b_1_37
+ b_2_8·b_1_1·b_1_32·b_5_47 + b_2_8·b_1_1·b_1_2·b_1_36 + b_2_8·b_6_69·b_1_1·b_1_3 + b_2_82·b_1_22·b_1_34 + b_2_82·b_1_1·b_1_35 + c_2_9·b_1_1·b_1_32·b_5_46 + c_2_9·b_1_12·b_1_3·b_5_46 + c_2_9·b_1_12·b_1_36 + b_2_8·c_2_9·b_1_1·b_1_2·b_1_34 + c_2_92·b_1_12·b_1_34 + c_2_92·b_1_14·b_1_32
- b_5_472 + b_5_45·b_5_46 + b_5_452 + b_1_2·b_1_34·b_5_47
+ b_1_1·b_1_2·b_1_33·b_5_47 + b_2_8·b_1_2·b_1_37 + b_2_8·b_1_22·b_1_3·b_5_47 + b_2_8·b_1_1·b_1_32·b_5_47 + b_2_82·b_1_3·b_5_46 + b_2_82·b_1_36 + b_2_82·b_1_22·b_1_34 + b_2_84·b_1_32 + c_8_135·b_1_22 + c_2_9·b_1_38 + c_2_9·b_1_22·b_1_36 + c_2_9·b_1_1·b_1_32·b_5_46 + c_2_9·b_1_1·b_1_2·b_1_3·b_5_47 + c_2_9·b_1_12·b_1_3·b_5_46 + b_2_8·c_2_9·b_1_2·b_1_35 + b_2_8·c_2_9·b_1_23·b_1_33 + b_2_8·c_2_9·b_1_1·b_5_47 + b_2_8·c_2_9·b_1_1·b_1_2·b_1_34 + b_2_8·c_2_9·b_6_69 + b_2_83·c_2_9·b_1_2·b_1_3 + b_2_84·c_2_9 + c_2_92·b_1_1·b_5_46 + c_2_92·b_1_12·b_1_34 + c_2_92·b_1_14·b_1_32 + b_2_82·c_2_92·b_1_32 + b_2_82·c_2_92·b_1_1·b_1_3 + c_2_93·b_1_12·b_1_32 + c_2_93·b_1_14 + c_2_93·b_1_0·b_1_33 + c_2_94·b_1_22
- b_5_46·b_5_47 + b_5_45·b_5_47 + b_1_1·b_1_34·b_5_47 + b_1_15·b_5_47 + b_6_69·b_1_34
+ b_2_8·b_1_22·b_1_36 + b_2_8·b_1_14·b_1_34 + b_2_82·b_1_2·b_1_35 + c_8_135·b_1_1·b_1_2 + c_2_9·b_1_12·b_1_3·b_5_47 + c_2_9·b_1_13·b_1_2·b_1_34 + c_2_9·b_1_16·b_1_2·b_1_3 + c_2_9·b_6_69·b_1_1·b_1_3 + b_2_8·c_2_9·b_1_3·b_5_46 + b_2_8·c_2_9·b_1_36 + b_2_8·c_2_9·b_1_22·b_1_34 + b_2_8·c_2_9·b_1_1·b_1_35 + b_2_82·c_2_9·b_1_23·b_1_3 + c_2_92·b_1_12·b_1_2·b_1_33 + c_2_92·b_1_13·b_1_2·b_1_32 + c_2_92·b_1_14·b_1_2·b_1_3 + b_2_8·c_2_92·b_1_12·b_1_32 + c_2_93·b_1_12·b_1_2·b_1_3 + c_2_93·b_1_13·b_1_2 + c_2_94·b_1_1·b_1_2
- b_5_452 + b_1_04·b_1_3·b_5_45 + b_1_05·b_5_45 + b_2_8·b_1_1·b_1_2·b_1_36
+ b_2_82·b_1_36 + b_2_84·b_1_22 + c_8_135·b_1_02 + c_2_9·b_1_12·b_1_36 + c_2_9·b_1_07·b_1_3 + c_2_92·b_1_12·b_1_34 + c_2_92·b_1_05·b_1_3 + c_2_93·b_1_03·b_1_3 + c_2_94·b_1_02
- b_5_462 + b_5_45·b_5_46 + b_1_22·b_1_38 + b_1_1·b_1_34·b_5_46
+ b_1_1·b_1_2·b_1_33·b_5_47 + b_1_15·b_5_46 + b_2_8·b_1_2·b_1_37 + b_2_8·b_1_1·b_1_32·b_5_47 + b_2_82·b_1_22·b_1_34 + b_2_82·b_1_1·b_5_46 + b_2_82·b_1_1·b_1_35 + b_2_82·b_1_13·b_1_33 + b_2_82·b_1_15·b_1_3 + c_8_135·b_1_12 + c_2_9·b_1_1·b_1_32·b_5_46 + c_2_9·b_1_12·b_1_3·b_5_46 + c_2_9·b_1_12·b_1_36 + c_2_9·b_1_13·b_5_46 + c_2_9·b_1_14·b_1_34 + c_2_9·b_1_15·b_1_33 + c_2_9·b_1_16·b_1_32 + b_2_82·c_2_9·b_1_1·b_1_33 + b_2_82·c_2_9·b_1_12·b_1_32 + b_2_82·c_2_9·b_1_14 + c_2_92·b_1_12·b_1_34 + c_2_92·b_1_13·b_1_33 + c_2_92·b_1_14·b_1_32 + c_2_92·b_1_15·b_1_3 + c_2_92·b_1_16 + b_2_82·c_2_92·b_1_1·b_1_3 + b_2_82·c_2_92·b_1_12 + c_2_94·b_1_12
- b_6_69·b_5_45 + b_2_8·b_1_2·b_1_33·b_5_47 + b_2_82·b_1_22·b_5_47 + b_2_83·b_5_46
+ b_2_83·b_1_35 + b_2_83·b_1_22·b_1_33 + b_2_84·b_1_2·b_1_32 + b_2_84·b_1_23 + b_2_85·b_1_3 + c_2_9·b_1_1·b_1_2·b_1_37 + c_2_9·b_1_12·b_1_32·b_5_47 + c_2_9·b_1_12·b_1_2·b_1_36 + c_2_9·b_1_13·b_1_3·b_5_47 + c_2_9·b_1_14·b_5_47 + c_2_9·b_6_69·b_1_1·b_1_32 + b_2_8·c_2_9·b_1_22·b_1_35 + b_2_8·c_2_9·b_1_1·b_1_36 + b_2_8·c_2_9·b_1_12·b_5_46 + b_2_8·c_2_9·b_1_13·b_1_34 + b_2_82·c_2_9·b_1_2·b_1_34 + b_2_82·c_2_9·b_1_23·b_1_32 + b_2_83·c_2_9·b_1_22·b_1_3 + c_2_92·b_1_14·b_1_2·b_1_32 + b_2_8·c_2_92·b_1_14·b_1_3
- b_6_69·b_5_47 + b_2_8·b_1_34·b_5_47 + b_2_8·b_1_22·b_1_32·b_5_47
+ b_2_8·b_6_69·b_1_1·b_1_32 + b_2_82·b_1_32·b_5_46 + b_2_82·b_1_1·b_1_36 + b_2_82·b_1_12·b_5_46 + b_2_83·b_1_2·b_1_34 + b_2_84·b_1_22·b_1_3 + c_2_9·b_1_34·b_5_46 + c_2_9·b_1_22·b_1_37 + c_2_9·b_1_14·b_5_46 + b_2_8·c_8_135·b_1_2 + b_2_8·c_2_9·b_1_32·b_5_47 + b_2_8·c_2_9·b_1_2·b_1_36 + b_2_8·c_2_9·b_1_22·b_5_47 + b_2_8·c_2_9·b_1_1·b_1_3·b_5_47 + b_2_8·c_2_9·b_6_69·b_1_3 + b_2_8·c_2_9·b_6_69·b_1_1 + b_2_82·c_2_9·b_5_46 + b_2_82·c_2_9·b_1_35 + b_2_82·c_2_9·b_1_22·b_1_33 + b_2_82·c_2_9·b_1_1·b_1_34 + b_2_82·c_2_9·b_1_13·b_1_32 + b_2_82·c_2_9·b_1_14·b_1_3 + c_2_92·b_1_1·b_1_3·b_5_46 + c_2_92·b_1_1·b_1_36 + c_2_92·b_1_14·b_1_33 + c_2_92·b_1_15·b_1_32 + c_2_92·b_1_16·b_1_3 + b_2_8·c_2_92·b_1_23·b_1_32 + b_2_8·c_2_92·b_1_1·b_1_2·b_1_33 + b_2_82·c_2_92·b_1_33 + b_2_82·c_2_92·b_1_22·b_1_3 + b_2_82·c_2_92·b_1_12·b_1_3 + c_2_93·b_1_12·b_1_33 + c_2_93·b_1_13·b_1_32 + b_2_8·c_2_93·b_1_1·b_1_2·b_1_3 + b_2_82·c_2_93·b_1_1 + c_2_94·b_1_13 + b_2_8·c_2_94·b_1_2
- b_1_22·b_1_34·b_5_47 + b_6_69·b_5_46 + b_6_69·b_5_45 + b_2_8·b_1_14·b_5_46
+ b_2_82·b_1_2·b_1_36 + b_2_83·b_1_35 + c_2_9·b_1_12·b_1_32·b_5_47 + c_2_9·b_1_12·b_1_2·b_1_36 + c_2_9·b_1_13·b_1_3·b_5_47 + c_2_9·b_1_13·b_1_2·b_1_35 + c_2_9·b_1_14·b_5_47 + c_2_9·b_1_15·b_1_2·b_1_33 + c_2_9·b_1_17·b_1_2·b_1_3 + b_2_8·c_8_135·b_1_1 + b_2_8·c_2_9·b_1_32·b_5_46 + b_2_8·c_2_9·b_1_12·b_1_35 + b_2_8·c_2_9·b_1_15·b_1_32 + b_2_8·c_2_9·b_1_16·b_1_3 + b_2_82·c_2_9·b_1_23·b_1_32 + c_2_92·b_1_14·b_1_2·b_1_32 + c_2_92·b_1_16·b_1_2 + b_2_8·c_2_92·b_1_1·b_1_34 + b_2_8·c_2_92·b_1_12·b_1_33 + b_2_8·c_2_92·b_1_14·b_1_3 + c_2_93·b_1_13·b_1_2·b_1_3 + c_2_93·b_1_14·b_1_2 + b_2_8·c_2_93·b_1_12·b_1_3 + b_2_8·c_2_93·b_1_13 + b_2_8·c_2_94·b_1_1
- b_6_692 + b_2_8·b_6_69·b_1_34 + b_2_82·b_1_13·b_5_46 + b_2_83·b_1_3·b_5_47
+ b_2_85·b_1_2·b_1_3 + c_2_9·b_1_22·b_1_38 + b_2_8·c_2_9·b_1_1·b_1_2·b_1_36 + b_2_8·c_2_9·b_6_69·b_1_1·b_1_3 + b_2_82·c_8_135 + b_2_82·c_2_9·b_1_2·b_5_47 + b_2_82·c_2_9·b_1_1·b_5_46 + b_2_82·c_2_9·b_1_1·b_1_35 + b_2_82·c_2_9·b_1_12·b_1_34 + b_2_82·c_2_9·b_1_13·b_1_33 + b_2_82·c_2_9·b_1_14·b_1_32 + b_2_83·c_2_9·b_1_2·b_1_33 + b_2_84·c_2_9·b_1_32 + b_2_84·c_2_9·b_1_22 + c_2_92·b_1_13·b_5_46 + c_2_92·b_1_13·b_1_35 + c_2_92·b_1_15·b_1_33 + c_2_92·b_1_17·b_1_3 + b_2_82·c_2_92·b_1_34 + b_2_82·c_2_92·b_1_22·b_1_32 + b_2_82·c_2_92·b_1_1·b_1_33 + b_2_82·c_2_92·b_1_14 + b_2_83·c_2_92·b_1_2·b_1_3 + c_2_93·b_1_13·b_1_33 + c_2_93·b_1_14·b_1_32 + c_2_93·b_1_15·b_1_3 + c_2_93·b_1_16 + b_2_82·c_2_93·b_1_1·b_1_3 + c_2_94·b_1_14 + b_2_82·c_2_94
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_2_9, a Duflot regular element of degree 2
- c_8_135, a Duflot regular element of degree 8
- b_1_32 + b_1_2·b_1_3 + b_1_22 + b_1_1·b_1_3 + b_1_12 + b_1_0·b_1_3 + b_1_02, an element of degree 2
- b_1_32, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 8, 10].
- 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 2
- b_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_1_3 → 0, an element of degree 1
- b_2_8 → 0, an element of degree 2
- c_2_9 → c_1_02, an element of degree 2
- b_5_45 → 0, an element of degree 5
- b_5_46 → 0, an element of degree 5
- b_5_47 → 0, an element of degree 5
- b_6_69 → 0, an element of degree 6
- c_8_135 → c_1_18 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- b_1_0 → c_1_2, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_1_3 → 0, an element of degree 1
- b_2_8 → 0, an element of degree 2
- c_2_9 → c_1_0·c_1_2 + c_1_02, an element of degree 2
- b_5_45 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
- b_5_46 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
- b_5_47 → 0, an element of degree 5
- b_6_69 → 0, an element of degree 6
- c_8_135 → c_1_12·c_1_26 + c_1_18 + c_1_04·c_1_24 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 4
- b_1_0 → 0, an element of degree 1
- b_1_1 → c_1_2, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_1_3 → c_1_3, an element of degree 1
- b_2_8 → c_1_0·c_1_2, an element of degree 2
- c_2_9 → c_1_02, an element of degree 2
- b_5_45 → c_1_02·c_1_2·c_1_32, an element of degree 5
- b_5_46 → c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_02·c_1_23, an element of degree 5
- b_5_47 → c_1_0·c_1_34, an element of degree 5
- b_6_69 → c_1_0·c_1_2·c_1_34 + c_1_0·c_1_12·c_1_23 + c_1_0·c_1_14·c_1_2
+ c_1_03·c_1_22·c_1_3 + c_1_03·c_1_23, an element of degree 6
- c_8_135 → c_1_12·c_1_22·c_1_34 + c_1_12·c_1_26 + c_1_14·c_1_34 + c_1_18
+ c_1_02·c_1_2·c_1_35 + c_1_02·c_1_24·c_1_32 + c_1_02·c_1_25·c_1_3 + c_1_02·c_1_26 + c_1_02·c_1_12·c_1_23·c_1_3 + c_1_02·c_1_14·c_1_2·c_1_3 + c_1_04·c_1_34 + c_1_04·c_1_24 + c_1_06·c_1_2·c_1_3 + c_1_06·c_1_22 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 4
- b_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → c_1_3, an element of degree 1
- b_1_3 → c_1_2, an element of degree 1
- b_2_8 → c_1_32, an element of degree 2
- c_2_9 → c_1_0·c_1_3 + c_1_02, an element of degree 2
- b_5_45 → c_1_35 + c_1_23·c_1_32, an element of degree 5
- b_5_46 → c_1_35 + c_1_23·c_1_32 + c_1_24·c_1_3, an element of degree 5
- b_5_47 → c_1_2·c_1_34 + c_1_22·c_1_33 + c_1_12·c_1_33 + c_1_14·c_1_3 + c_1_0·c_1_34
+ c_1_0·c_1_24 + c_1_02·c_1_33 + c_1_02·c_1_22·c_1_3, an element of degree 5
- b_6_69 → c_1_12·c_1_34 + c_1_14·c_1_32 + c_1_0·c_1_35 + c_1_0·c_1_2·c_1_34
+ c_1_0·c_1_22·c_1_33 + c_1_0·c_1_24·c_1_3 + c_1_02·c_1_34 + c_1_02·c_1_2·c_1_33, an element of degree 6
- c_8_135 → c_1_2·c_1_37 + c_1_22·c_1_36 + c_1_23·c_1_35 + c_1_12·c_1_2·c_1_35
+ c_1_12·c_1_24·c_1_32 + c_1_14·c_1_34 + c_1_14·c_1_2·c_1_33 + c_1_14·c_1_24 + c_1_18 + c_1_0·c_1_37 + c_1_0·c_1_23·c_1_34 + c_1_0·c_1_24·c_1_33 + c_1_0·c_1_26·c_1_3 + c_1_0·c_1_12·c_1_35 + c_1_0·c_1_14·c_1_33 + c_1_02·c_1_36 + c_1_02·c_1_2·c_1_35 + c_1_02·c_1_25·c_1_3 + c_1_02·c_1_12·c_1_34 + c_1_02·c_1_14·c_1_32 + c_1_03·c_1_22·c_1_33 + c_1_03·c_1_24·c_1_3 + c_1_04·c_1_34 + c_1_04·c_1_2·c_1_33 + c_1_04·c_1_22·c_1_32 + c_1_04·c_1_24 + c_1_08, an element of degree 8
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