Simon King
David J. Green
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Cohomology of group number 784 of order 128
General information on the group
- The group has 3 minimal generators and exponent 8.
- It is non-abelian.
- It has p-Rank 3.
- Its center has rank 2.
- It has 2 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) · (t6 − t5 + 2·t4 − t3 + t2 + 1) |
| (t − 1)3 · (t2 + 1) · (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 12 minimal generators of maximal degree 8:
- a_1_1, a nilpotent element of degree 1
- a_1_2, a nilpotent element of degree 1
- b_1_0, an element of degree 1
- a_2_3, a nilpotent element of degree 2
- b_2_5, an element of degree 2
- c_2_4, a Duflot regular element of degree 2
- a_4_8, a nilpotent element of degree 4
- a_5_8, a nilpotent element of degree 5
- b_5_16, an element of degree 5
- b_5_17, an element of degree 5
- a_8_26, a nilpotent element of degree 8
- c_8_35, a Duflot regular element of degree 8
Ring relations
There are 38 minimal relations of maximal degree 16:
- a_1_1·a_1_2
- a_1_1·b_1_0 + a_1_22
- a_1_2·b_1_0
- a_2_3·a_1_2
- b_2_5·b_1_0 + b_2_5·a_1_2 + a_1_13
- a_2_32 + c_2_4·a_1_12
- b_2_5·a_1_13
- a_4_8·a_1_1 + a_2_3·a_1_13 + a_2_3·c_2_4·a_1_1 + c_2_4·a_1_13
- a_4_8·b_1_0 + a_2_3·a_1_13 + a_2_3·c_2_4·b_1_0
- a_4_8·a_1_2 + a_2_3·a_1_13
- a_2_3·a_4_8 + a_2_3·b_2_5·a_1_12 + a_2_3·c_2_4·a_1_12 + c_2_42·a_1_12
- a_1_1·a_5_8 + b_2_52·a_1_12 + a_2_3·b_2_5·a_1_12 + a_2_3·c_2_4·a_1_12
- b_1_0·a_5_8 + a_2_3·b_1_04 + a_2_3·b_2_5·a_1_12
- a_1_1·b_5_16 + a_1_2·a_5_8 + b_2_52·a_1_12 + b_2_5·c_2_4·a_1_12 + c_2_42·a_1_12
- a_1_2·b_5_16 + a_1_2·a_5_8 + a_2_3·b_2_5·a_1_12
- b_1_0·b_5_17 + b_1_0·b_5_16 + b_2_5·a_4_8 + a_2_3·b_1_04 + a_1_2·a_5_8
+ b_2_52·a_1_12 + c_2_4·b_1_04 + a_2_3·c_2_4·b_1_02 + a_2_3·b_2_5·c_2_4 + b_2_5·c_2_4·a_1_12 + c_2_42·b_1_02
- a_1_2·b_5_17 + b_2_5·a_4_8 + a_1_2·a_5_8 + b_2_52·a_1_12 + a_2_3·b_2_5·c_2_4
+ b_2_5·c_2_4·a_1_12
- a_2_3·a_5_8 + a_2_3·b_2_52·a_1_1 + c_2_42·a_1_13
- b_2_5·b_5_16 + b_2_53·a_1_2 + b_2_53·a_1_1 + a_1_12·b_5_17 + a_2_3·b_2_52·a_1_1
+ b_2_52·c_2_4·a_1_2 + b_2_52·c_2_4·a_1_1 + a_2_3·c_2_4·a_1_13 + b_2_5·c_2_42·a_1_1 + c_2_42·a_1_13
- a_4_82 + c_2_43·a_1_12
- a_4_8·b_5_16 + a_2_3·c_2_4·b_5_16 + a_2_3·c_2_42·a_1_13 + c_2_43·a_1_13
- a_4_8·a_5_8 + a_2_3·a_1_12·b_5_17 + a_2_3·b_2_52·c_2_4·a_1_1 + c_2_43·a_1_13
- a_4_8·b_5_17 + b_2_54·a_1_2 + a_4_8·a_5_8 + b_2_5·a_1_12·b_5_17 + a_2_3·c_2_4·b_5_17
+ c_2_4·a_1_12·b_5_17 + a_2_3·b_2_52·c_2_4·a_1_1 + a_2_3·c_2_42·a_1_13 + c_2_43·a_1_13
- a_4_8·b_5_17 + b_2_54·a_1_2 + a_8_26·a_1_1 + b_2_5·a_1_12·b_5_17 + a_2_3·b_2_53·a_1_1
+ a_2_3·c_2_4·b_5_17 + a_2_3·b_2_5·c_2_42·a_1_1 + c_2_43·a_1_13
- a_8_26·b_1_0 + a_4_8·a_5_8 + a_2_3·b_2_52·c_2_4·a_1_1 + a_2_3·c_2_42·b_1_03
+ a_2_3·c_2_42·a_1_13 + c_2_43·a_1_13
- a_8_26·a_1_2 + a_4_8·a_5_8 + a_2_3·b_2_52·c_2_4·a_1_1 + c_2_43·a_1_13
- a_5_8·b_5_16 + a_2_3·b_1_03·b_5_16 + a_5_82 + a_2_3·b_2_53·a_1_12
+ b_2_53·c_2_4·a_1_12 + a_2_3·b_2_52·c_2_4·a_1_12 + b_2_52·c_2_42·a_1_12 + a_2_3·c_2_43·a_1_12
- b_5_16·b_5_17 + b_5_162 + a_5_8·b_5_16 + b_2_52·a_1_1·b_5_17 + b_2_53·a_4_8
+ a_2_3·b_2_5·a_1_1·b_5_17 + c_2_4·b_1_03·b_5_16 + b_2_5·c_2_4·a_1_1·b_5_17 + b_2_52·c_2_4·a_4_8 + a_2_3·c_2_4·b_1_0·b_5_16 + a_2_3·b_2_53·c_2_4 + b_2_53·c_2_4·a_1_12 + c_2_42·b_1_0·b_5_16 + c_2_42·a_1_1·b_5_17 + a_2_3·b_2_52·c_2_42 + c_2_42·a_1_2·a_5_8 + b_2_52·c_2_42·a_1_12 + a_2_3·b_2_5·c_2_42·a_1_12 + a_2_3·c_2_43·a_1_12 + c_2_44·a_1_22 + c_2_44·a_1_12
- b_5_172 + b_5_162 + b_2_55 + b_2_52·a_1_1·b_5_17 + b_2_53·a_4_8
+ a_2_3·b_2_53·a_1_12 + a_2_3·b_2_53·c_2_4 + c_8_35·a_1_12 + c_2_42·b_1_06 + b_2_52·c_2_42·a_1_12 + a_2_3·c_2_43·a_1_12 + c_2_44·b_1_02
- b_5_162 + a_5_8·b_5_16 + a_2_3·b_2_53·a_1_12 + c_8_35·b_1_02
+ c_2_4·b_1_03·b_5_16 + c_2_4·b_1_08 + a_2_3·c_2_4·b_1_06 + b_2_53·c_2_4·a_1_12 + a_2_3·b_2_52·c_2_4·a_1_12 + c_2_43·b_1_04 + a_2_3·c_2_43·b_1_02 + a_2_3·c_2_43·a_1_12 + c_2_44·a_1_12
- a_5_82 + b_2_54·a_1_12 + c_8_35·a_1_22
- a_5_8·b_5_17 + a_5_8·b_5_16 + b_2_5·a_8_26 + b_2_52·a_1_1·b_5_17 + a_2_3·b_2_54
+ a_2_3·b_2_53·a_1_12 + b_2_5·c_2_4·a_1_1·b_5_17 + b_2_52·c_2_4·a_4_8 + a_2_3·c_2_4·b_1_06 + b_2_53·c_2_4·a_1_12 + a_2_3·c_2_4·a_1_1·b_5_17 + b_2_5·c_2_42·a_4_8 + a_2_3·c_2_42·b_1_04 + a_2_3·b_2_5·c_2_42·a_1_12 + a_2_3·b_2_5·c_2_43 + a_2_3·c_2_43·a_1_12
- a_2_3·a_8_26 + a_2_3·b_2_53·a_1_12 + b_2_53·c_2_4·a_1_12
+ a_2_3·c_2_4·a_1_1·b_5_17 + a_2_3·b_2_52·c_2_4·a_1_12 + a_2_3·b_2_5·c_2_42·a_1_12 + b_2_5·c_2_43·a_1_12 + a_2_3·c_2_43·a_1_12
- a_4_8·a_8_26 + b_2_53·c_2_42·a_1_12 + a_2_3·c_2_42·a_1_1·b_5_17
+ b_2_5·c_2_44·a_1_12 + a_2_3·c_2_44·a_1_12
- a_8_26·a_5_8 + a_2_3·b_2_55·a_1_1 + a_2_3·b_2_52·a_1_12·b_5_17
+ b_2_52·c_2_4·a_1_12·b_5_17 + a_2_3·c_8_35·a_1_13 + a_2_3·b_2_53·c_2_42·a_1_1
- a_8_26·b_5_16 + a_2_3·b_2_55·a_1_1 + a_2_3·b_2_52·a_1_12·b_5_17
+ b_2_52·c_2_4·a_1_12·b_5_17 + a_2_3·b_2_54·c_2_4·a_1_1 + a_2_3·b_2_5·c_2_4·a_1_12·b_5_17 + a_2_3·c_2_42·b_1_02·b_5_16 + b_2_5·c_2_42·a_1_12·b_5_17 + c_2_43·a_1_12·b_5_17 + a_2_3·b_2_52·c_2_43·a_1_1 + a_2_3·b_2_5·c_2_44·a_1_1 + c_2_45·a_1_13
- a_8_26·b_5_17 + b_2_54·a_5_8 + b_2_56·a_1_1 + a_2_3·b_2_53·b_5_17
+ b_2_53·a_1_12·b_5_17 + b_2_55·c_2_4·a_1_2 + b_2_55·c_2_4·a_1_1 + a_2_3·b_2_54·c_2_4·a_1_1 + a_2_3·c_8_35·a_1_13 + b_2_54·c_2_42·a_1_2 + a_2_3·c_2_42·b_1_02·b_5_16 + a_2_3·b_2_5·c_2_42·b_5_17 + b_2_5·c_2_42·a_1_12·b_5_17 + c_2_4·c_8_35·a_1_13 + a_2_3·c_2_43·b_1_05 + c_2_43·a_1_12·b_5_17 + a_2_3·c_2_44·b_1_03 + c_2_45·a_1_13
- a_8_262 + b_2_56·c_2_4·a_1_12 + b_2_55·c_2_42·a_1_12
+ b_2_52·c_2_45·a_1_12
Data used for Benson′s test
- Benson′s completion test succeeded in degree 16.
- 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_2_4, a Duflot regular element of degree 2
- c_8_35, a Duflot regular element of degree 8
- b_1_02 + b_2_5, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 6, 9].
- 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_1 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- b_1_0 → 0, an element of degree 1
- a_2_3 → 0, an element of degree 2
- b_2_5 → 0, an element of degree 2
- c_2_4 → c_1_12, an element of degree 2
- a_4_8 → 0, an element of degree 4
- a_5_8 → 0, an element of degree 5
- b_5_16 → 0, an element of degree 5
- b_5_17 → 0, an element of degree 5
- a_8_26 → 0, an element of degree 8
- c_8_35 → c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_1 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- b_1_0 → c_1_2, an element of degree 1
- a_2_3 → 0, an element of degree 2
- b_2_5 → 0, an element of degree 2
- c_2_4 → c_1_1·c_1_2 + c_1_12, an element of degree 2
- a_4_8 → 0, an element of degree 4
- a_5_8 → 0, an element of degree 5
- b_5_16 → c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
- b_5_17 → c_1_1·c_1_24 + c_1_14·c_1_2 + c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
- a_8_26 → 0, an element of degree 8
- c_8_35 → c_1_1·c_1_27 + c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_14·c_1_24
+ c_1_15·c_1_23 + c_1_16·c_1_22 + c_1_02·c_1_1·c_1_25 + c_1_02·c_1_12·c_1_24 + c_1_04·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_1 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- b_1_0 → 0, an element of degree 1
- a_2_3 → 0, an element of degree 2
- b_2_5 → c_1_22, an element of degree 2
- c_2_4 → c_1_12, an element of degree 2
- a_4_8 → 0, an element of degree 4
- a_5_8 → 0, an element of degree 5
- b_5_16 → 0, an element of degree 5
- b_5_17 → c_1_25, an element of degree 5
- a_8_26 → 0, an element of degree 8
- c_8_35 → c_1_28 + c_1_12·c_1_26 + c_1_14·c_1_24 + c_1_04·c_1_24 + c_1_08, an element of degree 8
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