Simon King
David J. Green
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Singular
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Cohomology of group number 2045 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) · (t5 + t3 + 2·t2 + 2·t + 1) |
| (t + 1) · (t − 1)3 · (t2 + 1)2 |
- The a-invariants are -∞,-∞,-4,-3. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 12 minimal generators of maximal degree 8:
- a_1_0, a nilpotent 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
- c_4_13, a Duflot regular element of degree 4
- a_5_8, a nilpotent element of degree 5
- b_5_19, an element of degree 5
- b_5_20, an element of degree 5
- b_7_35, an element of degree 7
- c_8_40, a Duflot regular element of degree 8
Ring relations
There are 36 minimal relations of maximal degree 14:
- a_1_0·b_1_2
- b_1_1·b_1_3 + a_1_0·b_1_1 + a_1_02
- b_1_2·b_1_32 + b_1_23 + a_1_0·b_1_32 + a_1_0·b_1_12
- a_1_02·b_1_3 + a_1_02·b_1_1 + a_1_03
- a_1_0·b_3_10
- a_1_0·b_1_14
- b_1_32·b_3_10 + b_1_2·b_1_3·b_3_10 + b_1_22·b_3_10 + b_1_12·b_3_10 + b_1_13·b_1_22
+ b_4_12·b_1_2
- b_1_32·b_3_10 + b_1_22·b_3_10 + b_4_12·a_1_0
- b_3_102 + b_4_12·b_1_2·b_1_3 + b_4_12·a_1_0·b_1_3 + c_4_13·b_1_22
- b_1_12·b_1_2·b_3_10 + b_1_2·a_5_8
- b_3_102 + b_4_12·b_1_2·b_1_3 + b_1_3·a_5_8 + a_1_0·b_5_19 + b_4_12·a_1_0·b_1_1
+ a_1_0·a_5_8 + c_4_13·b_1_22 + c_4_13·a_1_0·b_1_3
- b_1_1·b_5_19 + b_1_12·b_1_2·b_3_10 + b_1_16 + b_4_12·b_1_1·b_1_2 + b_4_12·b_1_12
+ a_1_0·a_5_8 + c_4_13·a_1_0·b_1_1 + c_4_13·a_1_02
- b_3_102 + b_1_3·b_5_20 + b_1_2·b_5_19 + b_1_15·b_1_2 + b_4_12·b_1_32
+ b_4_12·b_1_2·b_1_3 + b_4_12·b_1_22 + b_4_12·b_1_1·b_1_2 + b_4_12·a_1_0·b_1_1 + c_4_13·b_1_22 + c_4_13·a_1_0·b_1_3 + c_4_13·a_1_0·b_1_1 + c_4_13·a_1_02
- b_3_102 + b_4_12·b_1_2·b_1_3 + a_1_0·b_5_20 + c_4_13·b_1_22 + c_4_13·a_1_0·b_1_1
+ c_4_13·a_1_02
- b_4_12·b_3_10 + b_4_12·b_1_23 + b_1_1·b_1_2·a_5_8 + c_4_13·b_1_22·b_1_3
+ c_4_13·b_1_12·b_1_2 + c_4_13·a_1_0·b_1_32 + c_4_13·a_1_0·b_1_12
- b_4_12·a_1_0·b_1_12 + a_1_0·b_1_1·a_5_8 + a_1_02·b_5_19 + c_4_13·a_1_02·b_1_1
+ c_4_13·a_1_03
- b_1_2·b_1_3·b_5_19 + b_1_22·b_5_20 + b_4_12·b_1_12·b_1_2 + b_1_12·a_5_8
+ a_1_0·b_1_3·b_5_19 + b_4_12·a_1_0·b_1_12 + a_1_0·b_1_1·a_5_8 + a_1_02·a_5_8 + c_4_13·b_1_1·b_1_22 + c_4_13·a_1_0·b_1_12
- b_3_10·a_5_8 + c_4_13·b_1_12·b_1_22
- b_4_12·b_1_23·b_1_3 + b_4_122 + b_1_12·b_1_2·a_5_8 + a_1_0·b_1_37 + a_1_03·a_5_8
+ c_4_13·b_1_34 + c_4_13·b_1_14
- b_3_10·b_5_19 + b_1_3·b_7_35 + b_1_23·b_5_19 + b_1_16·b_1_22 + b_4_12·b_1_34
+ b_4_12·b_1_23·b_1_3 + b_4_12·b_1_13·b_1_2 + b_1_12·b_1_2·a_5_8 + a_1_0·b_1_37 + a_1_03·b_5_19 + c_4_13·b_1_3·b_3_10 + c_4_13·b_1_12·b_1_22 + c_4_13·b_1_13·b_1_2 + c_4_13·a_1_0·b_1_13 + c_4_13·a_1_03·b_1_1
- b_3_10·b_5_20 + b_1_2·b_7_35 + b_1_23·b_5_20 + b_1_12·b_1_2·b_5_20 + b_1_16·b_1_22
+ b_1_17·b_1_2 + b_4_12·b_1_24 + b_4_12·b_1_13·b_1_2 + b_4_122 + b_1_12·b_1_2·a_5_8 + a_1_0·b_1_37 + b_4_12·a_1_0·b_1_33 + c_4_13·b_1_34 + c_4_13·b_1_2·b_3_10 + c_4_13·b_1_1·b_3_10 + c_4_13·b_1_14 + c_4_13·a_1_0·b_1_33 + c_4_13·a_1_0·b_1_13 + c_4_13·a_1_03·b_1_1
- a_1_0·b_7_35 + b_4_12·a_1_0·b_1_33 + c_4_13·a_1_0·b_1_13
- b_1_32·b_7_35 + b_1_22·b_7_35 + b_1_17·b_1_22 + b_4_12·b_1_35
+ b_4_12·b_1_14·b_1_2 + b_4_122·b_1_2 + b_1_13·b_1_2·a_5_8 + b_1_14·a_5_8 + a_1_0·b_1_38 + b_4_12·a_5_8 + c_4_13·b_1_25 + c_4_13·b_1_13·b_1_22
- b_1_2·b_1_3·b_7_35 + b_1_24·b_5_19 + b_1_12·b_7_35 + b_1_13·b_1_2·b_5_20
+ b_1_14·b_5_20 + b_1_17·b_1_22 + b_1_18·b_1_2 + b_1_19 + b_4_12·b_5_20 + b_4_12·b_1_15 + b_4_122·b_1_3 + b_4_122·b_1_2 + b_1_13·b_1_2·a_5_8 + b_4_12·a_5_8 + b_4_12·a_1_0·b_1_34 + c_4_13·b_1_24·b_1_3 + c_4_13·b_1_25 + c_4_13·b_1_13·b_1_22 + c_4_13·b_1_14·b_1_2 + b_4_12·c_4_13·b_1_2 + b_4_12·c_4_13·b_1_1 + c_4_13·a_1_0·b_1_34 + b_4_12·c_4_13·a_1_0
- b_3_10·b_7_35 + b_1_18·b_1_22 + b_4_122·b_1_2·b_1_3 + b_4_122·b_1_22
+ a_5_8·b_5_20 + b_1_15·a_5_8 + b_4_12·a_1_0·b_5_19 + c_4_13·b_1_2·b_5_20 + c_4_13·b_1_15·b_1_2 + b_4_12·c_4_13·b_1_2·b_1_3 + b_4_12·c_4_13·b_1_1·b_1_2 + c_4_13·b_1_1·a_5_8 + c_4_13·a_1_0·b_1_35 + b_4_12·c_4_13·a_1_0·b_1_3 + b_4_12·c_4_13·a_1_0·b_1_1 + c_4_13·a_1_0·a_5_8 + c_4_132·b_1_22 + c_4_132·b_1_1·b_1_2
- b_1_12·b_1_2·b_7_35 + b_1_14·b_1_2·b_5_20 + b_1_18·b_1_22 + b_1_19·b_1_2
+ a_5_8·b_5_20 + b_1_15·a_5_8 + b_4_12·a_1_0·b_5_19 + c_4_13·b_1_2·a_5_8 + c_4_13·b_1_1·a_5_8 + c_4_13·a_1_0·b_1_35 + b_4_12·c_4_13·a_1_0·b_1_3 + c_4_13·a_1_0·a_5_8
- b_5_192 + b_1_35·b_5_19 + b_1_25·b_5_20 + b_1_110 + b_4_122·b_1_22
+ b_1_14·b_1_2·a_5_8 + a_1_0·b_1_34·b_5_19 + b_4_12·a_1_0·b_1_35 + a_5_82 + c_8_40·b_1_32 + c_4_13·b_1_25·b_1_3 + c_4_13·b_1_14·b_1_22 + c_4_13·b_1_16 + b_4_12·c_4_13·b_1_2·b_1_3 + b_4_12·c_4_13·b_1_22 + c_4_13·b_1_2·a_5_8 + c_4_13·a_1_0·b_1_35
- b_5_19·b_5_20 + b_1_15·b_5_20 + b_4_12·b_1_3·b_5_19 + b_4_12·b_1_2·b_5_20
+ b_4_12·b_1_1·b_5_20 + b_4_122·b_1_2·b_1_3 + a_1_0·b_1_34·b_5_19 + b_4_12·b_1_1·a_5_8 + b_4_12·a_1_0·b_5_19 + c_8_40·b_1_2·b_1_3 + c_4_13·b_1_26 + c_4_13·b_1_15·b_1_2 + b_4_12·c_4_13·b_1_2·b_1_3 + b_4_12·c_4_13·b_1_22 + c_4_13·a_1_0·b_5_19 + b_4_12·c_4_13·a_1_0·b_1_3 + c_4_13·a_1_0·a_5_8 + c_4_132·a_1_0·b_1_1 + c_4_132·a_1_02
- a_5_8·b_5_19 + b_1_15·a_5_8 + a_1_0·b_1_34·b_5_19 + b_4_12·b_1_1·a_5_8
+ b_4_12·a_1_0·b_5_19 + a_5_82 + c_8_40·a_1_0·b_1_3 + c_4_13·a_1_0·b_5_19 + b_4_12·c_4_13·a_1_0·b_1_1 + c_4_13·a_1_0·a_5_8
- b_5_202 + b_1_14·b_1_2·b_5_20 + b_4_12·b_1_15·b_1_2 + b_4_122·b_1_22
+ b_1_14·b_1_2·a_5_8 + b_1_15·a_5_8 + a_1_0·b_1_39 + c_8_40·b_1_22 + c_4_13·b_1_36 + c_4_13·b_1_25·b_1_3 + c_4_13·b_1_26 + c_4_13·b_1_14·b_1_22 + c_4_13·b_1_15·b_1_2 + b_4_12·c_4_13·b_1_2·b_1_3 + b_4_12·c_4_13·b_1_22 + c_4_13·b_1_2·a_5_8 + b_4_12·c_4_13·a_1_0·b_1_3 + b_4_12·c_4_13·a_1_0·b_1_1 + c_4_132·b_1_12 + c_4_132·a_1_02
- a_5_82 + c_4_13·b_1_14·b_1_22 + c_8_40·a_1_02 + c_4_132·a_1_02
- b_4_12·b_7_35 + b_4_12·b_1_12·b_5_20 + b_4_12·b_1_16·b_1_2 + b_4_12·b_1_17
+ b_4_122·b_1_22·b_1_3 + b_4_122·b_1_23 + b_1_1·a_5_8·b_5_20 + a_1_0·b_1_310 + b_4_12·b_1_12·a_5_8 + b_4_12·a_1_0·b_1_36 + c_4_13·b_1_37 + c_4_13·b_1_22·b_5_19 + c_4_13·b_1_26·b_1_3 + c_4_13·b_1_12·b_5_20 + c_4_13·b_1_16·b_1_2 + c_4_13·b_1_17 + b_4_12·c_4_13·b_1_23 + b_4_12·c_4_13·b_1_12·b_1_2 + c_4_13·a_1_0·b_1_3·b_5_19 + c_4_13·a_1_0·b_1_1·a_5_8 + c_4_13·a_1_02·b_5_19 + c_4_13·a_1_02·a_5_8 + c_4_132·b_1_22·b_1_3 + c_4_132·b_1_12·b_1_2 + c_4_132·b_1_13 + c_4_132·a_1_0·b_1_32 + c_4_132·a_1_0·b_1_12 + c_4_132·a_1_02·b_1_1 + c_4_132·a_1_03
- a_5_8·b_7_35 + b_1_12·a_5_8·b_5_20 + b_1_16·b_1_2·a_5_8 + b_1_17·a_5_8
+ b_4_12·a_1_0·b_1_32·b_5_19 + c_4_13·b_1_12·b_1_2·b_5_20 + c_4_13·b_1_17·b_1_2 + c_4_13·a_1_0·b_1_37 + b_4_12·c_4_13·a_1_0·b_1_33 + c_4_13·a_1_03·a_5_8 + c_4_132·b_1_12·b_1_22 + c_4_132·b_1_13·b_1_2
- b_5_19·b_7_35 + b_1_16·b_1_2·b_5_20 + b_1_17·b_5_20 + b_1_110·b_1_22
+ b_1_111·b_1_2 + b_1_112 + b_4_12·b_1_33·b_5_19 + b_4_12·b_1_23·b_5_19 + b_1_16·b_1_2·a_5_8 + a_1_0·b_1_36·b_5_19 + b_4_12·a_1_0·b_1_32·b_5_19 + c_8_40·b_1_3·b_3_10 + c_8_40·b_1_23·b_1_3 + c_4_13·b_1_3·b_7_35 + c_4_13·b_1_23·b_5_19 + c_4_13·b_1_28 + c_4_13·b_1_12·b_1_2·b_5_20 + c_4_13·b_1_13·b_5_20 + c_4_13·b_1_16·b_1_22 + c_4_13·b_1_18 + b_4_12·c_4_13·b_1_34 + b_4_12·c_4_13·b_1_24 + b_4_12·c_4_13·b_1_14 + c_4_13·a_1_0·b_1_37 + c_8_40·a_1_03·b_1_1 + c_4_13·a_1_03·a_5_8 + c_4_132·b_1_3·b_3_10 + c_4_132·b_1_23·b_1_3 + c_4_132·b_1_24 + c_4_132·b_1_12·b_1_22 + c_4_132·b_1_14 + c_4_132·a_1_0·b_1_13 + c_4_132·a_1_03·b_1_1
- b_5_20·b_7_35 + b_1_17·b_5_20 + b_4_12·b_1_23·b_5_20 + b_4_12·b_1_23·b_5_19
+ b_4_12·b_1_13·b_5_20 + b_4_12·b_1_17·b_1_2 + b_1_12·a_5_8·b_5_20 + b_1_16·b_1_2·a_5_8 + b_1_17·a_5_8 + a_1_0·b_1_311 + b_4_12·a_1_0·b_1_32·b_5_19 + b_4_12·a_1_0·b_1_37 + c_8_40·b_1_2·b_3_10 + c_8_40·b_1_24 + c_8_40·b_1_12·b_1_22 + c_4_13·b_1_38 + c_4_13·b_1_2·b_7_35 + c_4_13·b_1_23·b_5_20 + c_4_13·b_1_27·b_1_3 + c_4_13·b_1_28 + c_4_13·b_1_1·b_7_35 + c_4_13·b_1_12·b_1_2·b_5_20 + c_4_13·b_1_13·b_5_20 + c_4_13·b_1_17·b_1_2 + c_4_13·b_1_18 + b_4_12·c_4_13·b_1_13·b_1_2 + b_4_12·c_4_13·b_1_14 + c_4_13·b_1_13·a_5_8 + c_4_13·a_1_0·b_1_32·b_5_19 + c_4_13·a_1_0·b_1_37 + c_4_13·a_1_03·b_5_19 + c_4_132·b_1_2·b_3_10 + c_4_132·b_1_23·b_1_3 + c_4_132·b_1_24 + c_4_132·b_1_14 + c_4_132·a_1_0·b_1_33 + c_4_132·a_1_03·b_1_1
- b_7_352 + b_1_18·b_1_2·b_5_20 + b_1_112·b_1_22 + b_1_114 + b_4_12·b_1_19·b_1_2
+ b_1_19·a_5_8 + a_1_0·b_1_313 + c_8_40·b_1_26 + c_8_40·b_1_14·b_1_22 + c_4_13·b_1_310 + c_4_13·b_1_29·b_1_3 + c_4_13·b_1_210 + c_4_13·b_1_14·b_1_2·b_5_20 + c_4_13·b_1_19·b_1_2 + c_4_13·b_1_110 + b_4_12·c_8_40·b_1_2·b_1_3 + b_4_12·c_4_13·b_1_15·b_1_2 + b_4_122·c_4_13·b_1_2·b_1_3 + c_4_13·b_1_14·b_1_2·a_5_8 + c_4_13·b_1_15·a_5_8 + b_4_12·c_8_40·a_1_0·b_1_3 + c_4_13·c_8_40·b_1_22 + c_4_132·b_1_25·b_1_3 + c_4_132·b_1_15·b_1_2 + c_4_132·b_1_16 + b_4_12·c_4_132·b_1_22 + c_4_132·b_1_2·a_5_8 + c_4_132·a_1_0·b_1_35 + b_4_12·c_4_132·a_1_0·b_1_1 + c_4_133·b_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_13, a Duflot regular element of degree 4
- c_8_40, 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_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
- c_4_13 → c_1_04, an element of degree 4
- a_5_8 → 0, an element of degree 5
- b_5_19 → 0, an element of degree 5
- b_5_20 → 0, an element of degree 5
- b_7_35 → 0, an element of degree 7
- c_8_40 → c_1_18 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- a_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 → 0, an element of degree 1
- b_3_10 → 0, an element of degree 3
- b_4_12 → c_1_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
- c_4_13 → c_1_02·c_1_22 + c_1_04, an element of degree 4
- a_5_8 → 0, an element of degree 5
- b_5_19 → c_1_25 + c_1_0·c_1_24 + c_1_02·c_1_23, an element of degree 5
- b_5_20 → c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
- b_7_35 → c_1_27 + c_1_0·c_1_26 + c_1_04·c_1_23, an element of degree 7
- c_8_40 → c_1_28 + c_1_14·c_1_24 + 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 3
- a_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_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_0·c_1_23 + c_1_02·c_1_22, an element of degree 4
- c_4_13 → c_1_0·c_1_23 + c_1_04, an element of degree 4
- a_5_8 → 0, an element of degree 5
- b_5_19 → 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
- b_5_20 → 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
- b_7_35 → c_1_12·c_1_25 + c_1_14·c_1_23 + c_1_0·c_1_12·c_1_24 + c_1_0·c_1_14·c_1_22
+ c_1_02·c_1_12·c_1_23 + c_1_02·c_1_14·c_1_2, an element of degree 7
- c_8_40 → c_1_14·c_1_24 + c_1_18 + c_1_0·c_1_27 + c_1_02·c_1_26 + c_1_04·c_1_24
+ c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- a_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 → c_1_2, an element of degree 1
- b_3_10 → 0, 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
- c_4_13 → c_1_24 + c_1_02·c_1_22 + c_1_04, an element of degree 4
- a_5_8 → 0, an element of degree 5
- b_5_19 → c_1_25 + 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
- b_5_20 → c_1_25 + c_1_0·c_1_24 + c_1_02·c_1_23, an element of degree 5
- b_7_35 → c_1_27 + c_1_0·c_1_26 + c_1_02·c_1_25, an element of degree 7
- c_8_40 → c_1_12·c_1_26 + c_1_18 + c_1_0·c_1_27 + c_1_02·c_1_26 + c_1_04·c_1_24
+ c_1_08, an element of degree 8
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