Simon King
David J. Green
Cohomology
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Cohomology of group number 2282 of order 128
General information on the group
- The group has 5 minimal generators and exponent 4.
- It is non-abelian.
- It has p-Rank 3.
- Its center has rank 2.
- It has 5 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 + 2·t7 + 2·t6 + t4 + 2·t3 + 3·t2 + 2·t + 1) |
| (t − 1)3 · (t2 + 1)2 · (t4 + 1) |
- The a-invariants are -∞,-∞,-5,-3. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 9 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_1_4, an element of degree 1
- c_4_31, a Duflot regular element of degree 4
- b_5_39, an element of degree 5
- b_6_50, an element of degree 6
- c_8_83, a Duflot regular element of degree 8
Ring relations
There are 11 minimal relations of maximal degree 12:
- b_1_22 + b_1_1·b_1_2 + a_1_0·b_1_3 + a_1_0·b_1_2
- b_1_42 + b_1_32 + b_1_1·b_1_3 + b_1_12 + a_1_0·b_1_4 + a_1_0·b_1_1 + a_1_02
- a_1_0·b_1_32 + a_1_0·b_1_1·b_1_3
- b_1_1·b_1_32 + b_1_12·b_1_3 + a_1_03
- a_1_03·b_1_12
- b_1_32·b_5_39 + b_1_1·b_1_3·b_5_39 + c_4_31·b_1_33 + c_4_31·b_1_12·b_1_3
+ c_4_31·a_1_03
- a_1_0·b_1_4·b_5_39 + a_1_0·b_1_3·b_5_39 + a_1_0·b_1_2·b_5_39 + a_1_0·b_1_1·b_5_39
+ a_1_0·b_1_13·b_1_2·b_1_3·b_1_4 + a_1_0·b_1_14·b_1_3·b_1_4 + a_1_0·b_1_15·b_1_3 + b_6_50·a_1_0 + a_1_02·b_5_39 + a_1_02·b_1_13·b_1_3·b_1_4 + c_4_31·a_1_0·b_1_3·b_1_4 + c_4_31·a_1_0·b_1_12 + c_4_31·a_1_02·b_1_4 + c_4_31·a_1_02·b_1_2
- b_1_32·b_5_39 + b_1_1·b_1_4·b_5_39 + b_1_1·b_1_2·b_5_39 + b_1_12·b_5_39
+ b_1_14·b_1_2·b_1_3·b_1_4 + b_1_15·b_1_3·b_1_4 + b_1_16·b_1_3 + b_6_50·b_1_1 + a_1_0·b_1_1·b_5_39 + a_1_0·b_1_14·b_1_3·b_1_4 + a_1_02·b_5_39 + a_1_02·b_1_13·b_1_3·b_1_4 + a_1_02·b_1_14·b_1_3 + a_1_03·b_1_1·b_1_2·b_1_3·b_1_4 + c_4_31·b_1_33 + c_4_31·b_1_1·b_1_3·b_1_4 + c_4_31·b_1_12·b_1_3 + c_4_31·b_1_13 + c_4_31·a_1_0·b_1_1·b_1_4 + c_4_31·a_1_0·b_1_1·b_1_2 + c_4_31·a_1_02·b_1_3
- b_5_392 + b_1_18·b_1_3·b_1_4 + b_1_18·b_1_2·b_1_3 + a_1_0·b_1_14·b_5_39
+ a_1_02·b_1_16·b_1_3·b_1_4 + a_1_02·b_1_16·b_1_2·b_1_3 + a_1_02·b_1_17·b_1_3 + b_6_50·a_1_02·b_1_3·b_1_4 + b_6_50·a_1_02·b_1_12 + c_4_31·b_1_16 + c_4_31·a_1_0·b_1_14·b_1_3 + c_8_83·a_1_02 + c_4_31·a_1_02·b_1_12·b_1_3·b_1_4 + c_4_31·a_1_02·b_1_13·b_1_4 + c_4_31·a_1_03·b_1_1·b_1_2·b_1_4 + c_4_312·b_1_32
- b_1_15·b_1_2·b_5_39 + b_1_19·b_1_2·b_1_3 + b_1_110·b_1_3 + b_6_50·b_5_39
+ b_6_50·b_1_13·b_1_3·b_1_4 + b_6_50·b_1_13·b_1_2·b_1_4 + b_6_50·b_1_13·b_1_2·b_1_3 + b_6_50·b_1_14·b_1_3 + a_1_0·b_1_14·b_1_2·b_5_39 + a_1_0·b_1_18·b_1_3·b_1_4 + a_1_0·b_1_19·b_1_3 + b_6_50·a_1_0·b_1_12·b_1_3·b_1_4 + b_6_50·a_1_0·b_1_12·b_1_2·b_1_3 + b_6_50·a_1_0·b_1_13·b_1_2 + b_6_50·a_1_0·b_1_14 + a_1_02·b_1_13·b_1_2·b_5_39 + a_1_02·b_1_17·b_1_2·b_1_3 + a_1_02·b_1_18·b_1_3 + b_6_50·a_1_02·b_1_1·b_1_2·b_1_3 + b_6_50·a_1_02·b_1_12·b_1_4 + b_6_50·a_1_02·b_1_13 + c_4_31·b_1_2·b_1_3·b_5_39 + c_4_31·b_1_12·b_5_39 + c_4_31·b_1_15·b_1_3·b_1_4 + c_4_31·b_1_15·b_1_2·b_1_4 + c_4_31·b_1_16·b_1_4 + c_4_31·b_1_16·b_1_2 + c_4_31·b_1_17 + c_4_31·b_6_50·b_1_3 + c_4_31·a_1_0·b_1_1·b_5_39 + c_4_31·a_1_0·b_1_14·b_1_3·b_1_4 + c_4_31·a_1_0·b_1_14·b_1_2·b_1_4 + c_4_31·a_1_0·b_1_14·b_1_2·b_1_3 + c_4_31·b_6_50·a_1_0 + c_8_83·a_1_02·b_1_4 + c_8_83·a_1_02·b_1_3 + c_8_83·a_1_02·b_1_2 + c_8_83·a_1_02·b_1_1 + c_4_31·a_1_02·b_5_39 + c_4_31·a_1_02·b_1_12·b_1_2·b_1_3·b_1_4 + c_4_31·a_1_02·b_1_13·b_1_3·b_1_4 + c_4_31·a_1_02·b_1_14·b_1_3 + c_8_83·a_1_03 + c_4_31·a_1_03·b_1_1·b_1_2·b_1_3·b_1_4 + c_4_312·b_1_2·b_1_32 + c_4_312·b_1_12·b_1_3 + c_4_312·a_1_0·b_1_2·b_1_3 + c_4_312·a_1_0·b_1_1·b_1_3 + c_4_312·a_1_0·b_1_12 + c_4_312·a_1_02·b_1_4 + c_4_312·a_1_02·b_1_2
- b_6_50·b_1_4·b_5_39 + b_6_50·b_1_35·b_1_4 + b_6_50·b_1_36 + b_6_50·b_1_2·b_5_39
+ b_6_50·b_1_14·b_1_3·b_1_4 + b_6_50·b_1_14·b_1_2·b_1_4 + b_6_50·b_1_14·b_1_2·b_1_3 + b_6_50·b_1_15·b_1_4 + b_6_50·b_1_15·b_1_3 + b_6_50·b_1_15·b_1_2 + b_6_50·b_1_16 + b_6_502 + b_6_50·a_1_0·b_5_39 + b_6_50·a_1_0·b_1_14·b_1_4 + a_1_02·b_1_15·b_5_39 + a_1_02·b_1_18·b_1_3·b_1_4 + a_1_02·b_1_18·b_1_2·b_1_3 + b_6_50·a_1_02·b_1_12·b_1_2·b_1_4 + b_6_50·a_1_02·b_1_13·b_1_4 + c_8_83·b_1_34 + c_8_83·b_1_13·b_1_3 + c_4_31·b_1_37·b_1_4 + c_4_31·b_1_38 + c_4_31·b_1_2·b_1_3·b_1_4·b_5_39 + c_4_31·b_1_2·b_1_37 + c_4_31·b_1_1·b_1_2·b_1_3·b_5_39 + c_4_31·b_1_12·b_1_2·b_5_39 + c_4_31·b_1_13·b_5_39 + c_4_31·b_1_15·b_1_2·b_1_3·b_1_4 + c_4_31·b_1_16·b_1_2·b_1_4 + c_4_31·b_6_50·b_1_3·b_1_4 + c_4_31·b_6_50·b_1_2·b_1_3 + c_4_31·b_6_50·b_1_1·b_1_4 + c_4_31·b_6_50·b_1_1·b_1_3 + c_4_31·b_6_50·b_1_1·b_1_2 + c_4_31·a_1_0·b_1_12·b_5_39 + c_4_31·a_1_0·b_1_15·b_1_3·b_1_4 + c_4_31·a_1_0·b_1_15·b_1_2·b_1_4 + c_4_31·a_1_0·b_1_16·b_1_2 + c_4_31·b_6_50·a_1_0·b_1_4 + c_4_31·b_6_50·a_1_0·b_1_3 + c_4_31·b_6_50·a_1_0·b_1_2 + c_4_31·b_6_50·a_1_0·b_1_1 + c_8_83·a_1_02·b_1_3·b_1_4 + c_8_83·a_1_02·b_1_2·b_1_3 + c_8_83·a_1_02·b_1_1·b_1_4 + c_8_83·a_1_02·b_1_1·b_1_3 + c_8_83·a_1_02·b_1_1·b_1_2 + c_8_83·a_1_02·b_1_12 + c_4_31·a_1_02·b_1_3·b_5_39 + c_4_31·a_1_02·b_1_1·b_5_39 + c_4_31·a_1_02·b_1_14·b_1_3·b_1_4 + c_4_31·a_1_02·b_1_14·b_1_2·b_1_4 + c_4_31·a_1_02·b_1_14·b_1_2·b_1_3 + c_4_31·a_1_02·b_1_15·b_1_4 + c_4_31·b_6_50·a_1_02 + c_8_83·a_1_03·b_1_3 + c_4_312·b_1_2·b_1_32·b_1_4 + c_4_312·b_1_1·b_1_2·b_1_3·b_1_4 + c_4_312·b_1_12·b_1_2·b_1_3 + c_4_312·b_1_13·b_1_4 + c_4_312·b_1_13·b_1_2 + c_4_312·b_1_14 + c_4_312·a_1_0·b_1_2·b_1_3·b_1_4 + c_4_312·a_1_0·b_1_12·b_1_4 + c_4_312·a_1_02·b_1_3·b_1_4 + c_4_312·a_1_02·b_1_2·b_1_3 + c_4_312·a_1_03·b_1_4 + c_4_312·a_1_03·b_1_3 + c_4_312·a_1_03·b_1_1
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_31, a Duflot regular element of degree 4
- c_8_83, a Duflot regular element of degree 8
- b_1_42, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 7, 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_1_4 → 0, an element of degree 1
- c_4_31 → c_1_14, an element of degree 4
- b_5_39 → 0, an element of degree 5
- b_6_50 → 0, an element of degree 6
- c_8_83 → 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_1_4 → c_1_2, an element of degree 1
- c_4_31 → c_1_12·c_1_22 + c_1_14, an element of degree 4
- b_5_39 → c_1_1·c_1_24 + c_1_12·c_1_23, an element of degree 5
- b_6_50 → c_1_12·c_1_24 + c_1_14·c_1_22, an element of degree 6
- c_8_83 → 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_1_4 → c_1_2, an element of degree 1
- c_4_31 → c_1_12·c_1_22 + c_1_14, an element of degree 4
- b_5_39 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
- b_6_50 → c_1_02·c_1_24 + c_1_04·c_1_22, an element of degree 6
- c_8_83 → 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 → c_1_2, 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_1_4 → c_1_2, an element of degree 1
- c_4_31 → c_1_24 + c_1_12·c_1_22 + c_1_14, an element of degree 4
- b_5_39 → c_1_25 + c_1_1·c_1_24 + c_1_12·c_1_23, an element of degree 5
- b_6_50 → c_1_1·c_1_25 + c_1_14·c_1_22, an element of degree 6
- c_8_83 → c_1_28 + c_1_1·c_1_27 + c_1_12·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 → c_1_2, 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_1_4 → c_1_2, an element of degree 1
- c_4_31 → c_1_24 + c_1_12·c_1_22 + c_1_14, an element of degree 4
- b_5_39 → c_1_25 + c_1_1·c_1_24 + c_1_14·c_1_2, an element of degree 5
- b_6_50 → c_1_26 + c_1_1·c_1_25 + c_1_14·c_1_22, an element of degree 6
- c_8_83 → c_1_28 + 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 → 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_1_4 → c_1_2, an element of degree 1
- c_4_31 → c_1_12·c_1_22 + c_1_14, an element of degree 4
- b_5_39 → c_1_1·c_1_24 + c_1_14·c_1_2, an element of degree 5
- b_6_50 → c_1_26, an element of degree 6
- c_8_83 → c_1_04·c_1_24 + c_1_08, an element of degree 8
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