Simon King
David J. Green
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Singular
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Cohomology of group number 1956 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, 3 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
( − 1) · (t5 − 2·t4 + t3 − t − 1) |
| (t + 1) · (t − 1)4 · (t2 + 1)2 |
- The a-invariants are -∞,-∞,-5,-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 6:
- 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_3_10, an element of degree 3
- b_4_12, an element of degree 4
- b_4_13, an element of degree 4
- b_4_14, an element of degree 4
- c_4_15, a Duflot regular element of degree 4
- c_4_16, a Duflot regular element of degree 4
- b_6_39, an element of degree 6
Ring relations
There are 29 minimal relations of maximal degree 12:
- b_1_0·b_1_2
- b_1_1·b_1_2 + b_1_12 + b_1_0·b_1_3 + b_1_0·b_1_1
- b_1_0·b_1_32
- b_1_1·b_1_32 + b_1_13 + b_1_0·b_1_1·b_1_3
- b_1_0·b_3_10
- b_4_12·b_1_0
- b_1_12·b_3_10 + b_1_15 + b_4_12·b_1_1
- b_1_2·b_1_3·b_3_10 + b_1_1·b_1_3·b_3_10 + b_1_12·b_3_10 + b_1_14·b_1_3 + b_4_13·b_1_2
+ b_4_12·b_1_2
- b_1_32·b_3_10 + b_1_1·b_1_3·b_3_10 + b_1_12·b_3_10 + b_1_14·b_1_3 + b_4_13·b_1_3
+ b_4_13·b_1_1 + b_4_12·b_1_3
- b_1_32·b_3_10 + b_1_12·b_3_10 + b_1_14·b_1_3 + b_1_15 + b_4_14·b_1_2 + b_4_12·b_1_3
- b_1_32·b_3_10 + b_1_1·b_1_3·b_3_10 + b_1_12·b_3_10 + b_1_15 + b_4_14·b_1_0
+ b_4_13·b_1_3 + b_4_13·b_1_0 + b_4_12·b_1_3
- b_1_32·b_3_10 + b_1_12·b_3_10 + b_4_14·b_1_1 + b_4_13·b_1_3 + b_4_12·b_1_3
- b_3_102 + b_1_23·b_3_10 + b_4_13·b_1_2·b_1_3 + b_4_13·b_1_22 + b_4_12·b_1_2·b_1_3
+ b_4_12·b_1_22 + b_4_12·b_1_12 + c_4_15·b_1_22
- b_4_13·b_3_10 + b_4_13·b_1_2·b_1_32 + b_4_13·b_1_22·b_1_3 + b_4_13·b_1_23
+ b_4_12·b_3_10 + b_4_12·b_1_2·b_1_32 + b_4_12·b_1_22·b_1_3 + b_4_12·b_1_23 + b_4_12·b_1_12·b_1_3 + b_4_12·b_1_13 + c_4_15·b_1_22·b_1_3 + c_4_15·b_1_12·b_1_3 + c_4_15·b_1_13 + c_4_15·b_1_02·b_1_3 + c_4_15·b_1_02·b_1_1
- b_6_39·b_1_2 + b_4_13·b_1_33 + b_4_13·b_1_23 + b_4_12·b_3_10 + b_4_12·b_1_2·b_1_32
+ b_4_12·b_1_12·b_1_3 + c_4_16·b_1_2·b_1_32 + c_4_16·b_1_22·b_1_3 + c_4_16·b_1_12·b_1_3 + c_4_16·b_1_13 + c_4_16·b_1_02·b_1_3 + c_4_16·b_1_02·b_1_1 + c_4_15·b_1_22·b_1_3 + c_4_15·b_1_23 + c_4_15·b_1_12·b_1_3 + c_4_15·b_1_13 + c_4_15·b_1_02·b_1_3 + c_4_15·b_1_02·b_1_1
- b_6_39·b_1_3 + b_4_14·b_3_10 + b_4_14·b_1_33 + b_4_13·b_3_10 + b_4_13·b_1_33
+ b_4_13·b_1_22·b_1_3 + b_4_13·b_1_23 + b_4_12·b_3_10 + b_4_12·b_1_23 + b_4_12·b_1_13 + c_4_16·b_1_33 + c_4_16·b_1_2·b_1_32 + c_4_16·b_1_12·b_1_3 + c_4_16·b_1_13 + c_4_16·b_1_02·b_1_3 + c_4_15·b_1_13 + c_4_15·b_1_0·b_1_1·b_1_3 + c_4_15·b_1_02·b_1_3
- b_1_06·b_1_1 + b_6_39·b_1_0 + c_4_16·b_1_0·b_1_1·b_1_3 + c_4_16·b_1_03
+ c_4_15·b_1_0·b_1_1·b_1_3 + c_4_15·b_1_02·b_1_3 + c_4_15·b_1_03
- b_6_39·b_1_1 + b_4_12·b_1_13 + c_4_16·b_1_0·b_1_1·b_1_3 + c_4_16·b_1_02·b_1_3
+ c_4_15·b_1_13 + c_4_15·b_1_0·b_1_1·b_1_3 + c_4_15·b_1_02·b_1_3
- b_1_25·b_3_10 + b_1_18 + b_4_13·b_1_2·b_1_33 + b_4_13·b_1_24
+ b_4_12·b_1_2·b_1_33 + b_4_12·b_1_22·b_1_32 + b_4_12·b_1_24 + b_4_122 + c_4_16·b_1_24 + c_4_16·b_1_14 + c_4_15·b_1_14
- b_4_12·b_1_3·b_3_10 + b_4_12·b_1_14 + b_4_12·b_4_13 + b_4_122 + c_4_15·b_1_13·b_1_3
+ c_4_15·b_1_14
- b_4_13·b_1_24 + b_4_13·b_1_14 + b_4_13·b_4_14 + b_4_132 + b_4_12·b_1_2·b_1_33
+ b_4_12·b_1_24 + b_4_12·b_1_13·b_1_3 + b_4_12·b_1_14 + b_4_122 + c_4_16·b_1_23·b_1_3 + c_4_16·b_1_13·b_1_3 + c_4_16·b_1_03·b_1_1 + c_4_15·b_1_2·b_1_33 + c_4_15·b_1_22·b_1_32 + c_4_15·b_1_13·b_1_3 + c_4_15·b_1_03·b_1_1
- b_1_25·b_3_10 + b_1_18 + b_4_14·b_1_3·b_3_10 + b_4_13·b_1_22·b_1_32
+ b_4_13·b_1_23·b_1_3 + b_4_13·b_1_14 + b_4_12·b_1_2·b_1_33 + b_4_12·b_1_23·b_1_3 + b_4_12·b_1_13·b_1_3 + b_4_12·b_1_14 + b_4_12·b_4_14 + b_4_122 + c_4_16·b_1_23·b_1_3 + c_4_16·b_1_24 + c_4_16·b_1_13·b_1_3 + c_4_16·b_1_14 + c_4_15·b_1_2·b_1_33
- b_4_142 + b_4_13·b_1_34 + b_4_13·b_1_14 + b_4_132 + b_4_122
+ c_4_16·b_1_22·b_1_32 + c_4_16·b_1_14 + c_4_15·b_1_34 + c_4_15·b_1_22·b_1_32
- b_1_25·b_3_10 + b_1_18 + b_6_39·b_1_02 + b_4_13·b_1_22·b_1_32
+ b_4_13·b_1_23·b_1_3 + b_4_13·b_1_24 + b_4_13·b_1_14 + b_4_132 + b_4_12·b_1_23·b_1_3 + b_4_12·b_1_24 + b_4_12·b_1_13·b_1_3 + c_4_16·b_1_24 + c_4_16·b_1_14 + c_4_15·b_1_22·b_1_32 + c_4_15·b_1_14 + c_4_15·b_1_03·b_1_1
- b_6_39·b_3_10 + b_4_13·b_1_23·b_1_32 + b_4_13·b_1_15 + b_4_132·b_1_2
+ b_4_12·b_1_2·b_1_34 + b_4_12·b_1_22·b_1_33 + b_4_12·b_1_23·b_1_32 + b_4_12·b_4_14·b_1_3 + b_4_12·b_4_13·b_1_2 + b_4_122·b_1_3 + c_4_16·b_1_23·b_1_32 + c_4_16·b_1_24·b_1_3 + c_4_15·b_1_2·b_1_34 + c_4_15·b_1_22·b_3_10 + c_4_15·b_1_23·b_1_32 + c_4_15·b_1_24·b_1_3 + b_4_14·c_4_16·b_1_1 + b_4_14·c_4_15·b_1_0 + b_4_13·c_4_16·b_1_3 + b_4_13·c_4_16·b_1_2 + b_4_13·c_4_15·b_1_2 + b_4_13·c_4_15·b_1_1 + b_4_13·c_4_15·b_1_0 + b_4_12·c_4_16·b_1_3 + b_4_12·c_4_16·b_1_2 + b_4_12·c_4_16·b_1_1
- b_4_13·b_1_05·b_1_1 + b_4_13·b_6_39 + b_4_13·b_4_14·b_1_32 + b_4_132·b_1_22
+ b_4_12·b_1_22·b_1_34 + b_4_12·b_1_23·b_1_33 + b_4_12·b_1_24·b_1_32 + b_4_12·b_1_16 + b_4_12·b_4_13·b_1_22 + b_4_122·b_1_2·b_1_3 + b_4_122·b_1_12 + c_4_16·b_1_23·b_3_10 + c_4_16·b_1_24·b_1_32 + c_4_16·b_1_25·b_1_3 + c_4_16·b_1_26 + c_4_15·b_1_22·b_1_34 + c_4_15·b_1_25·b_1_3 + c_4_15·b_1_26 + c_4_15·b_1_16 + b_4_13·c_4_16·b_1_32 + b_4_13·c_4_16·b_1_2·b_1_3 + b_4_13·c_4_16·b_1_12 + b_4_13·c_4_16·b_1_02 + b_4_13·c_4_15·b_1_2·b_1_3 + b_4_13·c_4_15·b_1_22 + b_4_13·c_4_15·b_1_12 + b_4_13·c_4_15·b_1_0·b_1_1 + b_4_13·c_4_15·b_1_02 + b_4_12·c_4_16·b_1_12 + b_4_12·c_4_15·b_1_2·b_1_3 + b_4_12·c_4_15·b_1_1·b_1_3
- b_4_12·b_1_22·b_1_34 + b_4_12·b_1_23·b_1_33 + b_4_12·b_1_24·b_1_32
+ b_4_12·b_1_16 + b_4_12·b_6_39 + b_4_12·b_4_14·b_1_32 + b_4_12·b_4_13·b_1_2·b_1_3 + b_4_12·b_4_13·b_1_22 + b_4_122·b_1_22 + b_4_122·b_1_12 + c_4_16·b_1_23·b_3_10 + c_4_16·b_1_24·b_1_32 + c_4_16·b_1_25·b_1_3 + c_4_16·b_1_26 + c_4_15·b_1_22·b_1_34 + c_4_15·b_1_25·b_1_3 + c_4_15·b_1_26 + c_4_15·b_1_16 + b_4_13·c_4_16·b_1_12 + b_4_12·c_4_16·b_1_32 + b_4_12·c_4_16·b_1_2·b_1_3 + b_4_12·c_4_16·b_1_1·b_1_3 + b_4_12·c_4_15·b_1_2·b_1_3 + b_4_12·c_4_15·b_1_22 + b_4_12·c_4_15·b_1_1·b_1_3
- b_4_14·b_6_39 + b_4_13·b_1_36 + b_4_13·b_1_16 + b_4_13·b_1_05·b_1_1
+ b_4_132·b_1_32 + b_4_132·b_1_2·b_1_3 + b_4_12·b_1_2·b_1_35 + b_4_12·b_1_22·b_1_34 + b_4_12·b_1_23·b_1_33 + b_4_12·b_1_16 + b_4_12·b_4_13·b_1_2·b_1_3 + b_4_122·b_1_12 + c_4_16·b_1_22·b_1_34 + c_4_16·b_1_23·b_1_33 + c_4_16·b_1_24·b_1_32 + c_4_16·b_1_25·b_1_3 + c_4_15·b_1_36 + c_4_15·b_1_2·b_1_35 + c_4_15·b_1_22·b_1_34 + c_4_15·b_1_24·b_1_32 + c_4_15·b_1_25·b_1_3 + c_4_15·b_1_16 + b_4_14·c_4_16·b_1_32 + b_4_13·c_4_16·b_1_32 + b_4_13·c_4_16·b_1_22 + b_4_13·c_4_16·b_1_12 + b_4_13·c_4_16·b_1_0·b_1_1 + b_4_13·c_4_16·b_1_02 + b_4_13·c_4_15·b_1_32 + b_4_13·c_4_15·b_1_2·b_1_3 + b_4_13·c_4_15·b_1_02 + b_4_12·c_4_16·b_1_22 + b_4_12·c_4_16·b_1_1·b_1_3 + b_4_12·c_4_16·b_1_12 + b_4_12·c_4_15·b_1_32 + b_4_12·c_4_15·b_1_1·b_1_3 + b_4_12·c_4_15·b_1_12
- b_6_392 + b_4_13·b_1_38 + b_4_13·b_1_18 + b_4_132·b_1_23·b_1_3
+ b_4_12·b_4_13·b_1_22·b_1_32 + b_4_12·b_4_13·b_1_23·b_1_3 + b_4_12·b_4_132 + b_4_122·b_1_34 + b_4_122·b_1_2·b_3_10 + b_4_122·b_1_22·b_1_32 + b_4_122·b_1_24 + b_4_122·b_1_14 + b_4_122·b_4_13 + c_4_16·b_1_22·b_1_36 + c_4_16·b_1_18 + c_4_15·b_1_38 + c_4_15·b_1_22·b_1_36 + b_4_13·c_4_16·b_1_23·b_1_3 + b_4_13·c_4_16·b_1_14 + b_4_12·c_4_16·b_1_23·b_1_3 + b_4_12·c_4_16·b_1_13·b_1_3 + b_4_12·c_4_15·b_1_22·b_1_32 + b_4_12·c_4_15·b_1_13·b_1_3 + b_4_12·c_4_15·b_1_14 + b_4_122·c_4_15 + c_4_162·b_1_34 + c_4_162·b_1_22·b_1_32 + c_4_162·b_1_04 + c_4_152·b_1_22·b_1_32 + c_4_152·b_1_24 + c_4_152·b_1_04
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_4_15, a Duflot regular element of degree 4
- c_4_16, a Duflot regular element of degree 4
- b_1_32 + b_1_2·b_1_3 + b_1_22 + 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, 3, 6, 8].
- 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_3_10 → 0, an element of degree 3
- b_4_12 → 0, an element of degree 4
- b_4_13 → 0, an element of degree 4
- b_4_14 → 0, an element of degree 4
- c_4_15 → c_1_04, an element of degree 4
- c_4_16 → c_1_14 + c_1_04, an element of degree 4
- b_6_39 → 0, an element of degree 6
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_3_10 → 0, an element of degree 3
- b_4_12 → 0, an element of degree 4
- b_4_13 → c_1_1·c_1_23 + c_1_12·c_1_22, an element of degree 4
- b_4_14 → c_1_1·c_1_23 + c_1_12·c_1_22, an element of degree 4
- c_4_15 → c_1_02·c_1_22 + c_1_04, an element of degree 4
- c_4_16 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
- b_6_39 → c_1_12·c_1_24 + c_1_14·c_1_22, an element of degree 6
Restriction map to a maximal el. ab. subgp. of rank 3
- b_1_0 → 0, an element of degree 1
- b_1_1 → c_1_2, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- b_1_3 → c_1_2, an element of degree 1
- b_3_10 → c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
- b_4_12 → c_1_24 + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
- b_4_13 → 0, an element of degree 4
- b_4_14 → c_1_24 + c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
- c_4_15 → c_1_24 + c_1_02·c_1_22 + c_1_04, an element of degree 4
- c_4_16 → c_1_24 + c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
- b_6_39 → c_1_0·c_1_25 + c_1_04·c_1_22, an element of degree 6
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_2, an element of degree 1
- b_1_3 → c_1_3, an element of degree 1
- b_3_10 → c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
- b_4_12 → c_1_1·c_1_23 + c_1_12·c_1_22 + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3
+ c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
- b_4_13 → c_1_1·c_1_23 + c_1_12·c_1_22 + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_23
+ c_1_02·c_1_2·c_1_3 + c_1_02·c_1_22, an element of degree 4
- b_4_14 → c_1_1·c_1_22·c_1_3 + c_1_12·c_1_2·c_1_3 + c_1_0·c_1_33 + c_1_0·c_1_22·c_1_3
+ c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3, an element of degree 4
- c_4_15 → c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_0·c_1_23 + c_1_02·c_1_32
+ c_1_02·c_1_2·c_1_3 + c_1_04, an element of degree 4
- c_4_16 → c_1_1·c_1_2·c_1_32 + c_1_12·c_1_32 + c_1_12·c_1_22 + c_1_14 + c_1_0·c_1_33
+ c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3 + c_1_0·c_1_23 + c_1_02·c_1_2·c_1_3 + c_1_04, an element of degree 4
- b_6_39 → c_1_1·c_1_2·c_1_34 + c_1_1·c_1_23·c_1_32 + c_1_1·c_1_25 + c_1_12·c_1_34
+ c_1_12·c_1_23·c_1_3 + c_1_12·c_1_24 + c_1_14·c_1_32 + c_1_14·c_1_2·c_1_3 + 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_0·c_1_1·c_1_24 + c_1_0·c_1_12·c_1_23 + c_1_02·c_1_34 + c_1_02·c_1_2·c_1_33 + c_1_02·c_1_22·c_1_32 + c_1_02·c_1_23·c_1_3 + c_1_02·c_1_1·c_1_23 + c_1_02·c_1_12·c_1_22 + c_1_03·c_1_2·c_1_32 + c_1_03·c_1_22·c_1_3 + c_1_04·c_1_32, an element of degree 6
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