Simon King
David J. Green
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Cohomology of group number 2061 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 3.
- Its center has rank 2.
- 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 2.
- The depth coincides with the Duflot bound.
- The Poincaré series is
( − 1) · (t8 + t6 + t5 + t4 + t3 + t2 + t + 1) |
| (t − 1)3 · (t2 + 1)2 · (t4 + 1) |
- The a-invariants are -∞,-∞,-4,-3. 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:
- a_1_2, a nilpotent element of degree 1
- b_1_0, an element of degree 1
- b_1_1, an element of degree 1
- b_1_3, an element of degree 1
- a_3_10, a nilpotent element of degree 3
- b_3_11, an element of degree 3
- c_4_16, a Duflot regular element of degree 4
- a_5_11, a nilpotent element of degree 5
- b_5_22, an element of degree 5
- a_7_27, a nilpotent element of degree 7
- c_8_46, a Duflot regular element of degree 8
Ring relations
There are 28 minimal relations of maximal degree 14:
- a_1_2·b_1_0
- b_1_1·b_1_3 + b_1_0·b_1_1 + b_1_02
- b_1_0·b_1_32 + b_1_0·b_1_12 + b_1_03 + a_1_23
- b_1_02·b_1_3 + b_1_02·b_1_1 + b_1_03
- b_1_0·a_3_10
- b_1_3·a_3_10 + b_1_1·a_3_10 + a_1_2·b_3_11 + a_1_2·b_1_13 + a_1_2·a_3_10
+ a_1_22·b_1_12 + a_1_23·b_1_1
- a_1_23·b_1_32 + a_1_23·b_1_12
- b_1_0·b_1_3·b_3_11 + b_1_0·b_1_1·b_3_11 + b_1_02·b_3_11 + a_1_22·a_3_10
+ a_1_23·b_1_12
- a_3_10·b_3_11 + a_1_2·b_1_12·b_3_11 + a_1_2·b_1_15 + a_3_102 + a_1_22·b_1_14
+ a_1_22·b_1_1·a_3_10 + c_4_16·a_1_2·b_1_3 + c_4_16·a_1_2·b_1_1
- a_3_102 + c_4_16·a_1_22
- b_1_02·b_1_1·b_3_11 + b_1_03·b_3_11 + b_1_05·b_1_1 + b_1_06 + b_1_0·a_5_11
- b_3_112 + b_1_16 + b_1_0·b_5_22 + b_1_05·b_1_1 + b_1_06 + a_3_102
+ a_1_22·b_1_14 + a_1_23·b_3_11 + c_4_16·b_1_32 + c_4_16·b_1_12 + c_4_16·b_1_02
- a_1_2·b_5_22 + a_1_2·b_1_32·b_3_11 + a_1_2·b_1_12·b_3_11 + a_1_22·b_1_3·b_3_11
- b_1_32·b_5_22 + b_1_34·b_3_11 + b_1_12·b_5_22 + b_1_14·b_3_11 + b_1_0·b_1_3·b_5_22
+ b_1_0·b_1_1·b_5_22 + b_1_04·b_3_11 + a_1_2·b_1_33·b_3_11 + a_1_22·b_1_35 + a_1_22·a_5_11 + c_4_16·a_1_23
- b_1_32·b_5_22 + b_1_34·b_3_11 + b_1_12·b_5_22 + b_1_14·b_3_11 + b_1_02·b_5_22
+ b_1_04·b_3_11 + a_1_2·b_1_33·b_3_11 + a_1_22·b_1_35 + a_1_22·a_5_11 + c_4_16·a_1_23
- a_3_10·b_5_22 + a_1_2·b_1_14·b_3_11 + a_1_2·b_1_17 + a_1_22·b_1_16
+ a_1_23·b_1_12·b_3_11 + c_4_16·a_1_2·b_1_33 + c_4_16·a_1_2·b_1_13 + c_4_16·a_1_22·b_1_12 + c_4_16·a_1_23·b_1_3
- b_3_11·a_5_11 + b_1_3·a_7_27 + b_1_1·a_7_27 + b_1_13·a_5_11 + a_1_2·b_1_34·b_3_11
+ a_1_2·b_1_14·b_3_11 + a_3_10·a_5_11 + a_1_2·b_1_32·a_5_11 + a_1_22·b_1_33·b_3_11 + a_1_22·b_1_36 + a_1_22·b_1_13·b_3_11 + a_1_22·b_1_3·a_5_11 + c_4_16·a_1_2·b_3_11 + c_4_16·a_1_2·a_3_10
- b_1_02·b_1_1·b_5_22 + b_1_03·b_5_22 + b_1_0·a_7_27 + a_1_23·b_1_12·b_3_11
- a_3_10·a_5_11 + a_1_2·a_7_27 + a_1_22·b_1_33·b_3_11 + a_1_22·b_1_13·b_3_11
+ a_1_22·b_1_3·a_5_11 + a_1_22·b_1_1·a_5_11 + c_4_16·a_1_2·a_3_10 + c_4_16·a_1_22·b_1_12 + c_4_16·a_1_23·b_1_1
- b_1_3·b_3_11·b_5_22 + b_1_1·b_3_11·b_5_22 + b_1_19 + b_1_0·b_3_11·b_5_22
+ b_1_08·b_1_1 + a_1_22·b_1_34·b_3_11 + a_1_22·b_1_17 + a_1_22·a_7_27 + a_1_22·b_1_12·a_5_11 + c_4_16·b_1_35 + c_4_16·b_1_15 + c_4_16·b_1_04·b_1_1 + c_4_16·a_1_2·b_1_34 + c_4_16·a_1_22·b_1_33 + c_4_16·a_1_22·b_1_13
- b_1_0·b_1_1·b_3_11·b_5_22 + b_1_02·b_3_11·b_5_22 + a_5_11·b_5_22 + b_1_33·a_7_27
+ b_1_13·a_7_27 + b_1_15·a_5_11 + b_1_05·a_5_11 + a_1_2·b_1_36·b_3_11 + a_1_2·b_1_16·b_3_11 + a_1_2·b_1_34·a_5_11 + a_1_2·b_1_12·a_7_27 + a_1_22·b_1_35·b_3_11 + a_1_22·b_1_38 + a_1_22·b_1_3·a_7_27 + a_1_22·b_1_13·a_5_11 + c_4_16·a_1_2·b_1_32·b_3_11 + c_4_16·a_1_2·b_1_12·b_3_11 + c_4_16·a_1_22·b_1_3·b_3_11 + c_4_16·a_1_22·b_1_14
- b_1_0·b_1_1·b_3_11·b_5_22 + b_1_02·b_3_11·b_5_22 + b_3_11·a_7_27 + b_1_13·a_7_27
+ b_1_03·a_7_27 + a_1_2·b_1_16·b_3_11 + a_1_2·b_1_19 + a_1_2·b_1_32·a_7_27 + a_1_22·b_1_15·b_3_11 + a_1_22·b_1_18 + a_1_22·b_1_33·a_5_11 + a_1_22·b_1_1·a_7_27 + a_1_22·b_1_13·a_5_11 + c_4_16·b_1_3·a_5_11 + c_4_16·b_1_1·a_5_11 + c_4_16·b_1_0·a_5_11 + c_4_16·a_1_2·b_1_35 + c_4_16·a_1_2·b_1_12·b_3_11 + c_4_16·a_1_2·a_5_11 + c_4_16·a_1_22·b_1_3·b_3_11 + c_4_16·a_1_22·b_1_34 + c_4_16·a_1_22·b_1_14 + c_4_16·a_1_22·b_1_1·a_3_10 + c_4_16·a_1_23·b_3_11 + c_4_162·a_1_2·b_1_3 + c_4_162·a_1_2·b_1_1 + c_4_162·a_1_22
- a_3_10·a_7_27 + a_1_22·b_1_15·b_3_11 + a_1_22·b_1_18 + a_1_22·b_1_3·a_7_27
+ a_1_22·b_1_1·a_7_27 + c_4_16·a_1_2·a_5_11 + c_4_16·a_1_22·b_1_34 + c_4_16·a_1_22·b_1_1·b_3_11 + c_4_16·a_1_23·b_3_11 + c_4_162·a_1_22
- b_5_222 + b_1_110 + b_1_07·b_3_11 + b_1_010 + b_1_05·a_5_11 + a_1_22·b_1_18
+ c_8_46·b_1_02 + c_4_16·b_1_36 + c_4_16·b_1_16 + c_4_16·b_1_03·b_3_11 + c_4_16·a_1_22·b_1_14 + c_4_162·b_1_02
- a_5_112 + a_1_2·b_1_34·a_5_11 + a_1_2·b_1_14·a_5_11 + a_1_22·b_1_38
+ a_1_22·b_1_15·b_3_11 + a_1_22·b_1_3·a_7_27 + c_8_46·a_1_22 + c_4_16·a_1_22·b_1_3·b_3_11 + c_4_16·a_1_22·b_1_34 + c_4_16·a_1_22·b_1_1·b_3_11 + c_4_16·a_1_23·b_3_11
- b_1_011·b_1_1 + b_1_012 + b_5_22·a_7_27 + b_1_15·a_7_27 + b_1_05·a_7_27
+ b_1_07·a_5_11 + a_1_2·b_1_18·b_3_11 + a_1_2·b_1_111 + a_1_2·b_1_34·a_7_27 + a_1_22·b_1_17·b_3_11 + a_1_22·b_1_110 + a_1_22·b_1_35·a_5_11 + a_1_22·b_1_13·a_7_27 + a_1_22·b_1_15·a_5_11 + c_8_46·b_1_03·b_1_1 + c_8_46·b_1_04 + c_4_16·b_1_33·a_5_11 + c_4_16·b_1_13·a_5_11 + c_4_16·a_1_2·b_1_37 + c_4_16·a_1_2·b_1_14·b_3_11 + c_4_16·a_1_2·b_1_12·a_5_11 + c_4_16·a_1_22·b_1_33·b_3_11 + c_4_16·a_1_22·b_1_16 + c_4_16·a_1_22·b_1_3·a_5_11 + c_4_162·b_1_03·b_1_1 + c_4_162·b_1_04 + c_4_162·a_1_2·b_1_33 + c_4_162·a_1_2·b_1_13 + c_4_162·a_1_22·b_1_12 + c_4_162·a_1_23·b_1_3
- a_5_11·a_7_27 + a_1_2·b_1_16·a_5_11 + a_1_22·b_1_37·b_3_11 + a_1_22·b_1_17·b_3_11
+ a_1_22·b_1_110 + a_1_22·b_1_15·a_5_11 + c_8_46·a_1_2·a_3_10 + c_4_16·a_1_2·a_7_27 + c_4_16·a_1_2·b_1_12·a_5_11 + c_8_46·a_1_23·b_1_3 + c_8_46·a_1_23·b_1_1 + c_4_162·a_1_2·a_3_10 + c_4_162·a_1_22·b_1_32 + c_4_162·a_1_23·b_1_3 + c_4_162·a_1_23·b_1_1
- a_7_272 + a_1_22·b_1_112 + c_4_16·a_1_2·b_1_34·a_5_11
+ c_4_16·a_1_2·b_1_14·a_5_11 + c_4_16·a_1_22·b_1_15·b_3_11 + c_4_16·a_1_22·b_1_18 + c_4_16·a_1_22·b_1_3·a_7_27 + c_4_16·c_8_46·a_1_22 + c_4_162·a_1_22·b_1_3·b_3_11 + c_4_162·a_1_22·b_1_34 + c_4_162·a_1_22·b_1_1·b_3_11 + c_4_162·a_1_22·b_1_14 + c_4_162·a_1_23·b_3_11 + c_4_163·a_1_22
Data used for Benson′s test
- Benson′s completion test succeeded in degree 14.
- 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_8_46, a Duflot regular element of degree 8
- b_1_32 + b_1_12, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 8, 11].
- 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 2
- a_1_2 → 0, an element of degree 1
- b_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_3 → 0, an element of degree 1
- a_3_10 → 0, an element of degree 3
- b_3_11 → 0, an element of degree 3
- c_4_16 → c_1_04, an element of degree 4
- a_5_11 → 0, an element of degree 5
- b_5_22 → 0, an element of degree 5
- a_7_27 → 0, an element of degree 7
- c_8_46 → c_1_18 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_2 → 0, an element of degree 1
- b_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_3 → c_1_2, an element of degree 1
- a_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
- c_4_16 → c_1_02·c_1_22 + c_1_04, an element of degree 4
- a_5_11 → 0, an element of degree 5
- b_5_22 → c_1_0·c_1_24 + c_1_02·c_1_23, an element of degree 5
- a_7_27 → 0, an element of degree 7
- c_8_46 → c_1_14·c_1_24 + c_1_18 + c_1_0·c_1_27 + c_1_03·c_1_25 + c_1_04·c_1_24
+ c_1_05·c_1_23 + c_1_06·c_1_22 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_2 → 0, an element of degree 1
- b_1_0 → c_1_2, an element of degree 1
- b_1_1 → c_1_2, an element of degree 1
- b_1_3 → 0, an element of degree 1
- a_3_10 → 0, an element of degree 3
- b_3_11 → c_1_1·c_1_22 + c_1_12·c_1_2, an element of degree 3
- c_4_16 → c_1_02·c_1_22 + c_1_04, an element of degree 4
- a_5_11 → 0, an element of degree 5
- b_5_22 → c_1_25 + c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
- a_7_27 → 0, an element of degree 7
- c_8_46 → c_1_28 + c_1_1·c_1_27 + c_1_12·c_1_26 + c_1_14·c_1_24 + c_1_18
+ 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_04·c_1_1·c_1_23 + c_1_04·c_1_12·c_1_22 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_2 → 0, an element of degree 1
- b_1_0 → 0, an element of degree 1
- b_1_1 → c_1_2, an element of degree 1
- b_1_3 → 0, an element of degree 1
- a_3_10 → 0, an element of degree 3
- b_3_11 → c_1_23 + c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
- c_4_16 → c_1_02·c_1_22 + c_1_04, an element of degree 4
- a_5_11 → 0, an element of degree 5
- b_5_22 → c_1_25 + c_1_0·c_1_24 + c_1_02·c_1_23, an element of degree 5
- a_7_27 → 0, an element of degree 7
- c_8_46 → c_1_14·c_1_24 + c_1_18 + c_1_02·c_1_26 + c_1_03·c_1_25 + c_1_04·c_1_24
+ c_1_05·c_1_23 + c_1_06·c_1_22 + c_1_08, an element of degree 8
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