Simon King
David J. Green
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Singular
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Cohomology of group number 1929 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
- The a-invariants are -∞,-∞,-4,-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_3_11, an element of degree 3
- a_4_4, a nilpotent element of degree 4
- c_4_16, a Duflot regular element of degree 4
- c_4_17, a Duflot regular element of degree 4
- b_5_27, an element of degree 5
- a_6_21, a nilpotent 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_0·b_1_3 + b_1_0·b_1_1
- b_1_2·b_1_32 + b_1_0·b_1_12
- b_1_0·b_1_32 + b_1_0·b_1_1·b_1_3
- b_1_2·b_3_10
- b_1_0·b_3_11
- b_1_3·b_3_10 + b_1_1·b_3_11 + b_1_1·b_3_10
- a_4_4·b_1_2
- b_1_32·b_3_11 + b_1_35 + b_1_1·b_1_34 + b_1_12·b_3_11 + b_1_12·b_1_33
+ b_1_13·b_1_32 + a_4_4·b_1_3
- b_1_0·b_1_1·b_3_10 + b_1_04·b_1_1 + a_4_4·b_1_0
- b_3_10·b_3_11 + b_1_1·b_1_35 + b_1_12·b_1_34
- b_3_112 + b_1_36 + b_1_12·b_1_34 + c_4_16·b_1_22
- b_3_102 + b_1_12·b_1_34 + a_4_4·b_1_02 + c_4_16·b_1_02
- b_3_112 + b_1_36 + b_1_2·b_5_27 + b_1_22·b_1_3·b_3_11 + b_1_12·b_1_34
+ c_4_17·b_1_2·b_1_3 + c_4_17·b_1_22 + c_4_17·b_1_0·b_1_3 + c_4_17·b_1_0·b_1_1 + c_4_16·b_1_2·b_1_3
- b_1_0·b_5_27 + b_1_03·b_3_10 + b_1_05·b_1_1 + c_4_17·b_1_0·b_1_3 + c_4_17·b_1_0·b_1_1
+ c_4_17·b_1_02 + c_4_16·b_1_0·b_1_3 + c_4_16·b_1_02
- b_1_06·b_1_1 + a_4_4·b_3_10 + a_4_4·b_1_1·b_1_32 + a_4_4·b_1_03
+ c_4_16·b_1_02·b_1_1
- a_4_4·b_3_11 + a_4_4·b_1_33 + a_4_4·b_1_1·b_1_32
- b_1_23·b_1_3·b_3_11 + a_6_21·b_1_2 + c_4_17·b_1_22·b_1_3 + c_4_16·b_1_22·b_1_3
- b_1_06·b_1_1 + a_6_21·b_1_0 + a_4_4·b_1_03
- a_4_42
- b_3_11·b_5_27 + b_3_10·b_5_27 + b_1_33·b_5_27 + b_1_14·b_1_3·b_3_11 + b_1_14·b_1_34
+ b_1_15·b_3_10 + b_1_15·b_1_33 + b_1_16·b_1_32 + a_6_21·b_1_32 + a_4_4·b_1_1·b_1_33 + a_4_4·b_1_13·b_1_3 + a_4_4·b_1_0·b_3_10 + a_4_4·b_1_04 + c_4_17·b_1_3·b_3_11 + c_4_17·b_1_34 + c_4_17·b_1_2·b_3_11 + c_4_17·b_1_1·b_1_33 + c_4_17·b_1_0·b_3_10 + c_4_17·b_1_0·b_1_12·b_1_3 + c_4_16·b_1_3·b_3_11 + c_4_16·b_1_34 + c_4_16·b_1_2·b_3_11 + c_4_16·b_1_23·b_1_3 + c_4_16·b_1_1·b_3_11 + c_4_16·b_1_1·b_3_10 + c_4_16·b_1_0·b_3_10 + c_4_16·b_1_0·b_1_12·b_1_3 + c_4_16·b_1_03·b_1_1 + c_4_16·b_1_04
- b_3_10·b_5_27 + b_1_1·b_1_32·b_5_27 + b_1_14·b_1_3·b_3_11 + b_1_14·b_1_34
+ b_1_15·b_3_11 + b_1_15·b_3_10 + a_6_21·b_1_1·b_1_3 + a_4_4·b_1_12·b_1_32 + a_4_4·b_1_0·b_3_10 + a_4_4·b_1_04 + c_4_17·b_1_1·b_3_11 + c_4_17·b_1_1·b_1_33 + c_4_17·b_1_12·b_1_32 + c_4_17·b_1_0·b_3_10 + c_4_17·b_1_0·b_1_12·b_1_3 + c_4_16·b_1_1·b_3_11 + c_4_16·b_1_1·b_3_10 + c_4_16·b_1_1·b_1_33 + c_4_16·b_1_0·b_3_10 + c_4_16·b_1_0·b_1_12·b_1_3 + c_4_16·b_1_03·b_1_1 + c_4_16·b_1_04
- a_6_21·b_1_33 + a_6_21·b_1_1·b_1_32 + a_6_21·b_1_12·b_1_3 + a_6_21·b_1_13
+ a_4_4·b_5_27 + a_4_4·b_1_1·b_1_34 + a_4_4·b_1_12·b_1_33 + a_4_4·b_1_14·b_1_3 + a_4_4·b_1_02·b_3_10 + a_4_4·c_4_17·b_1_3 + a_4_4·c_4_17·b_1_1 + a_4_4·c_4_17·b_1_0 + a_4_4·c_4_16·b_1_3 + a_4_4·c_4_16·b_1_0
- a_6_21·b_3_10 + a_6_21·b_1_1·b_1_32 + c_4_16·b_1_04·b_1_1
- a_6_21·b_3_11 + a_6_21·b_1_33 + a_6_21·b_1_1·b_1_32 + c_4_17·b_1_2·b_1_3·b_3_11
+ c_4_16·b_1_2·b_1_3·b_3_11 + c_4_16·b_1_24·b_1_3
- b_5_272 + b_1_310 + b_1_1·b_1_34·b_5_27 + b_1_12·b_1_38 + b_1_14·b_1_3·b_5_27
+ b_1_14·b_1_36 + b_1_15·b_1_35 + b_1_18·b_1_32 + a_4_4·b_1_36 + a_4_4·b_1_1·b_1_35 + a_4_4·b_1_12·b_1_34 + a_4_4·b_1_13·b_1_33 + a_4_4·b_1_15·b_1_3 + a_4_4·b_1_06 + c_4_17·b_1_1·b_1_35 + c_4_17·b_1_15·b_1_3 + c_4_16·b_1_36 + c_4_16·b_1_1·b_1_35 + c_4_16·b_1_16 + c_4_16·b_1_06 + c_4_172·b_1_32 + c_4_172·b_1_22 + c_4_172·b_1_12 + c_4_172·b_1_02 + c_4_162·b_1_32 + c_4_162·b_1_22 + c_4_162·b_1_02
- a_4_4·a_6_21
- b_1_17·b_1_3·b_3_11 + b_1_17·b_1_34 + b_1_18·b_3_11 + b_1_19·b_1_32
+ a_6_21·b_5_27 + a_6_21·b_1_13·b_1_32 + a_6_21·b_1_14·b_1_3 + a_6_21·b_1_15 + a_4_4·b_1_37 + a_4_4·b_1_1·b_1_36 + a_4_4·b_1_12·b_5_27 + a_4_4·b_1_12·b_1_35 + a_4_4·b_1_13·b_1_34 + a_4_4·b_1_14·b_1_33 + c_4_17·b_1_13·b_1_3·b_3_11 + c_4_17·b_1_13·b_1_34 + c_4_17·b_1_14·b_3_11 + c_4_17·b_1_15·b_1_32 + c_4_16·b_1_13·b_1_3·b_3_11 + c_4_16·b_1_13·b_1_34 + c_4_16·b_1_14·b_3_10 + c_4_16·b_1_14·b_1_33 + c_4_16·b_1_15·b_1_32 + c_4_17·a_6_21·b_1_3 + c_4_17·a_6_21·b_1_2 + c_4_17·a_6_21·b_1_1 + c_4_16·a_6_21·b_1_3 + c_4_16·a_6_21·b_1_2 + a_4_4·c_4_17·b_3_10 + a_4_4·c_4_17·b_1_1·b_1_32 + a_4_4·c_4_17·b_1_13 + a_4_4·c_4_16·b_1_33 + a_4_4·c_4_16·b_1_1·b_1_32 + a_4_4·c_4_16·b_1_12·b_1_3 + a_4_4·c_4_16·b_1_13 + a_4_4·c_4_16·b_1_03 + c_4_16·c_4_17·b_1_02·b_1_1
- a_6_212
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_16, a Duflot regular element of degree 4
- c_4_17, a Duflot regular element of degree 4
- b_1_32 + b_1_22 + b_1_1·b_1_3 + b_1_12 + b_1_02, an element of degree 2
- b_1_12, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 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_3_11 → 0, an element of degree 3
- a_4_4 → 0, an element of degree 4
- c_4_16 → c_1_04, an element of degree 4
- c_4_17 → c_1_14, an element of degree 4
- b_5_27 → 0, an element of degree 5
- a_6_21 → 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 → c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
- b_3_11 → 0, an element of degree 3
- a_4_4 → 0, an element of degree 4
- c_4_16 → c_1_02·c_1_22 + c_1_04, an element of degree 4
- c_4_17 → c_1_12·c_1_22 + c_1_14, an element of degree 4
- b_5_27 → c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_0·c_1_24 + c_1_04·c_1_2, an element of degree 5
- a_6_21 → 0, 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 → 0, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- b_1_3 → 0, an element of degree 1
- b_3_10 → 0, an element of degree 3
- b_3_11 → c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
- a_4_4 → 0, an element of degree 4
- c_4_16 → c_1_02·c_1_22 + c_1_04, an element of degree 4
- c_4_17 → c_1_12·c_1_22 + c_1_14, an element of degree 4
- b_5_27 → c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
- a_6_21 → 0, 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 → 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_3_10 → c_1_2·c_1_32, an element of degree 3
- b_3_11 → c_1_33 + c_1_2·c_1_32, an element of degree 3
- a_4_4 → 0, an element of degree 4
- c_4_16 → c_1_34 + c_1_22·c_1_32 + c_1_0·c_1_2·c_1_32 + c_1_0·c_1_22·c_1_3
+ c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_22 + c_1_04, an element of degree 4
- c_4_17 → c_1_2·c_1_33 + c_1_22·c_1_32 + c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3
+ 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_23 + c_1_02·c_1_32 + c_1_02·c_1_2·c_1_3 + c_1_02·c_1_22, an element of degree 4
- b_5_27 → c_1_35 + c_1_2·c_1_34 + c_1_23·c_1_32 + c_1_12·c_1_33 + c_1_12·c_1_2·c_1_32
+ c_1_12·c_1_22·c_1_3 + c_1_12·c_1_23 + c_1_14·c_1_3 + c_1_14·c_1_2 + c_1_0·c_1_2·c_1_33 + c_1_0·c_1_22·c_1_32 + c_1_0·c_1_23·c_1_3 + c_1_02·c_1_33 + c_1_02·c_1_2·c_1_32 + c_1_04·c_1_3, an element of degree 5
- a_6_21 → 0, an element of degree 6
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