Simon King
David J. Green
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Cohomology of group number 1922 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 2 conjugacy classes of maximal elementary abelian subgroups, which are of rank 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 − t4 − 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:
- 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_5, a nilpotent element of degree 3
- b_3_10, an element of degree 3
- a_4_6, a nilpotent element of degree 4
- b_4_16, an element of degree 4
- c_4_17, a Duflot regular element of degree 4
- c_4_18, a Duflot regular element of degree 4
- a_6_23, a nilpotent element of degree 6
Ring relations
There are 29 minimal relations of maximal degree 12:
- a_1_2·b_1_0
- b_1_0·b_1_3 + b_1_0·b_1_1 + a_1_2·b_1_1
- b_1_0·b_1_12 + a_1_2·b_1_12 + a_1_23
- a_1_2·b_1_1·b_1_3
- b_1_0·a_3_5
- b_1_3·b_3_10 + b_1_1·b_3_10 + b_1_12·b_1_32 + b_1_13·b_1_3 + b_1_1·a_3_5
+ a_1_2·b_1_13
- a_1_2·b_3_10
- b_1_32·a_3_5 + a_4_6·b_1_3 + a_1_2·b_1_3·a_3_5 + a_1_22·b_1_33 + a_1_22·a_3_5
- b_1_0·b_1_1·b_3_10 + b_1_04·b_1_1 + a_4_6·b_1_0
- a_1_2·b_1_3·a_3_5 + a_4_6·a_1_2 + a_1_22·a_3_5
- b_4_16·b_1_0 + b_1_1·b_1_3·a_3_5 + a_1_2·b_1_14 + a_4_6·b_1_1
- b_1_1·b_1_3·a_3_5 + a_1_2·b_1_14 + b_4_16·a_1_2 + a_4_6·b_1_1
- a_3_52 + a_1_22·b_1_34 + a_4_6·a_1_2·b_1_3 + c_4_17·a_1_22
- b_3_102 + b_1_14·b_1_32 + b_1_03·b_3_10 + b_1_05·b_1_1 + a_4_6·b_1_02
+ c_4_17·b_1_02
- a_3_5·b_3_10 + a_1_2·b_1_15 + b_4_16·a_1_2·b_1_1 + a_4_6·b_1_12
- a_4_6·b_3_10 + a_4_6·b_1_12·b_1_3 + c_4_17·b_1_02·b_1_1
- a_1_22·b_1_35 + a_4_6·a_3_5 + a_4_6·a_1_2·b_1_32 + c_4_17·a_1_22·b_1_3
+ c_4_17·a_1_23
- b_4_16·b_3_10 + b_4_16·b_1_12·b_1_3 + b_1_14·a_3_5 + a_6_23·b_1_1
+ a_4_6·b_1_12·b_1_3 + a_4_6·b_1_13 + c_4_18·a_1_2·b_1_12 + c_4_18·a_1_23 + c_4_17·a_1_23
- b_4_16·b_3_10 + b_4_16·b_1_12·b_1_3 + a_1_2·b_1_16 + a_6_23·b_1_3 + b_4_16·a_3_5
+ a_4_6·b_1_1·b_1_32 + a_4_6·b_1_12·b_1_3 + a_4_6·b_1_13 + a_1_22·b_1_35 + a_4_6·a_1_22·b_1_3 + c_4_18·a_1_2·b_1_32 + c_4_17·a_1_22·b_1_3 + c_4_18·a_1_23
- b_1_06·b_1_1 + a_6_23·b_1_0 + a_4_6·b_1_03 + c_4_18·b_1_02·b_1_1
- a_6_23·a_1_2 + c_4_18·a_1_22·b_1_3 + c_4_17·a_1_23
- a_4_6·b_1_3·a_3_5 + a_4_62
- b_4_16·b_1_3·a_3_5 + a_4_6·b_4_16 + c_4_18·a_1_2·b_1_13
- b_1_13·b_1_35 + b_1_14·b_1_34 + b_1_16·b_1_32 + b_4_16·b_1_13·b_1_3
+ b_4_16·b_1_14 + b_4_162 + a_6_23·b_1_1·b_1_3 + a_6_23·b_1_12 + b_4_16·b_1_1·a_3_5 + b_4_16·a_1_2·b_1_13 + a_4_6·b_1_14 + c_4_18·b_1_12·b_1_32 + c_4_18·b_1_14 + c_4_17·b_1_12·b_1_32 + c_4_18·a_1_2·b_1_13
- a_6_23·b_3_10 + a_6_23·b_1_12·b_1_3 + a_6_23·b_1_03 + c_4_17·b_1_04·b_1_1
+ a_4_6·c_4_18·b_1_0
- a_6_23·a_3_5 + a_4_62·a_1_2 + a_4_6·c_4_18·a_1_2 + c_4_18·a_1_22·a_3_5
+ c_4_17·a_1_22·a_3_5
- a_4_6·a_6_23 + a_4_62·a_1_2·b_1_3 + a_4_6·c_4_18·a_1_2·b_1_3 + a_4_6·c_4_17·a_1_22
- b_1_17·b_3_10 + b_1_19·b_1_3 + a_6_23·b_1_12·b_1_32 + a_6_23·b_1_14
+ b_4_16·b_1_13·a_3_5 + b_4_16·a_6_23 + b_4_162·a_1_2·b_1_1 + a_4_6·b_1_14·b_1_32 + a_4_6·b_4_16·b_1_1·b_1_3 + c_4_17·b_1_13·b_3_10 + c_4_17·b_1_15·b_1_3 + c_4_18·b_1_13·a_3_5 + c_4_18·a_1_2·b_1_15 + c_4_17·b_1_13·a_3_5 + b_4_16·c_4_17·a_1_2·b_1_1 + a_4_6·c_4_18·b_1_12 + a_4_6·c_4_17·b_1_12
- a_6_232 + c_4_182·a_1_22·b_1_32
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_17, a Duflot regular element of degree 4
- c_4_18, a Duflot regular element of degree 4
- b_1_32 + b_1_1·b_1_3 + b_1_12 + 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
- 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_5 → 0, an element of degree 3
- b_3_10 → 0, an element of degree 3
- a_4_6 → 0, an element of degree 4
- b_4_16 → 0, an element of degree 4
- c_4_17 → c_1_04, an element of degree 4
- c_4_18 → c_1_14 + c_1_04, an element of degree 4
- a_6_23 → 0, an element of degree 6
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 → 0, an element of degree 1
- b_1_3 → 0, an element of degree 1
- a_3_5 → 0, an element of degree 3
- b_3_10 → c_1_0·c_1_22 + c_1_02·c_1_2, an element of degree 3
- a_4_6 → 0, an element of degree 4
- b_4_16 → 0, an element of degree 4
- c_4_17 → c_1_0·c_1_23 + c_1_04, an element of degree 4
- c_4_18 → c_1_12·c_1_22 + c_1_14 + c_1_02·c_1_22 + c_1_04, an element of degree 4
- a_6_23 → 0, an element of degree 6
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_2 → 0, an element of degree 1
- b_1_0 → 0, an element of degree 1
- b_1_1 → c_1_3, an element of degree 1
- b_1_3 → c_1_2, an element of degree 1
- a_3_5 → 0, an element of degree 3
- b_3_10 → c_1_2·c_1_32, an element of degree 3
- a_4_6 → 0, an element of degree 4
- b_4_16 → c_1_12·c_1_32 + c_1_12·c_1_2·c_1_3 + c_1_0·c_1_2·c_1_32 + c_1_02·c_1_32, an element of degree 4
- c_4_17 → c_1_2·c_1_33 + 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_18 → c_1_22·c_1_32 + c_1_23·c_1_3 + c_1_12·c_1_32 + c_1_14 + 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
- a_6_23 → 0, an element of degree 6
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