Simon King
David J. Green
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Cohomology of group number 922 of order 128
General information on the group
- The group has 3 minimal generators and exponent 16.
- It is non-abelian.
- It has p-Rank 3.
- Its center has rank 1.
- 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 exceeds the Duflot bound, which is 1.
- The Poincaré series is
( − 1) · (t7 + t6 + t5 + t3 + t2 + t + 1) |
| (t + 1) · (t − 1)3 · (t2 + 1) · (t4 + 1) |
- The a-invariants are -∞,-∞,-6,-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:
- b_1_0, an element of degree 1
- b_1_1, an element of degree 1
- b_1_2, an element of degree 1
- a_2_4, a nilpotent element of degree 2
- b_2_5, an element of degree 2
- b_3_9, an element of degree 3
- b_5_16, an element of degree 5
- b_5_18, an element of degree 5
- a_6_20, a nilpotent element of degree 6
- b_7_31, an element of degree 7
- c_8_39, a Duflot regular element of degree 8
Ring relations
There are 35 minimal relations of maximal degree 14:
- b_1_0·b_1_1
- b_1_0·b_1_2
- a_2_4·b_1_0
- b_2_5·b_1_1 + a_2_4·b_1_2
- a_2_42
- b_1_2·b_3_9 + a_2_4·b_2_5
- b_1_1·b_3_9
- a_2_4·b_2_5·b_1_2
- a_2_4·b_3_9
- b_1_2·b_5_16
- b_3_92 + b_1_0·b_5_16 + b_2_5·b_1_0·b_3_9
- b_1_1·b_5_16
- b_3_92 + b_1_0·b_5_18
- a_2_4·b_5_16
- b_2_5·b_5_18 + b_2_5·b_5_16 + b_2_52·b_3_9 + a_6_20·b_1_2 + a_2_4·b_1_25
+ a_2_4·b_1_1·b_1_24
- a_6_20·b_1_0
- a_6_20·b_1_1 + a_2_4·b_5_18 + a_2_4·b_1_1·b_1_24 + a_2_4·b_1_12·b_1_23
- b_3_9·b_5_18 + b_3_9·b_5_16 + b_2_5·b_1_0·b_5_16 + b_2_52·b_1_0·b_3_9
- a_2_4·a_6_20
- b_1_2·b_7_31 + b_1_1·b_1_22·b_5_18 + b_1_12·b_1_2·b_5_18 + b_2_5·a_6_20
+ a_2_4·b_1_2·b_5_18 + a_2_4·b_2_53
- b_3_9·b_5_16 + b_1_0·b_7_31 + b_1_03·b_5_16 + b_2_52·b_1_0·b_3_9
- b_1_1·b_7_31 + b_1_12·b_1_2·b_5_18 + b_1_13·b_5_18 + a_2_4·b_1_1·b_5_18
- a_6_20·b_3_9
- a_2_4·b_7_31 + a_2_4·b_1_1·b_1_2·b_5_18 + a_2_4·b_1_12·b_5_18
- b_5_16·b_5_18 + b_5_162 + b_2_5·b_1_0·b_7_31 + b_2_5·b_1_03·b_5_16
+ b_2_53·b_1_0·b_3_9
- b_5_16·b_5_18 + b_3_9·b_7_31 + b_1_03·b_7_31 + b_1_05·b_5_16 + b_2_52·b_1_0·b_5_16
+ b_2_52·b_1_03·b_3_9 + b_2_53·b_1_0·b_3_9
- b_5_162 + b_2_5·b_1_03·b_5_16 + b_2_5·b_1_05·b_3_9 + c_8_39·b_1_02
- b_5_182 + b_5_162 + b_1_1·b_1_24·b_5_18 + b_1_13·b_1_22·b_5_18
+ b_2_52·b_1_0·b_5_16 + b_2_53·b_1_0·b_3_9 + a_2_4·b_1_28 + a_2_4·b_1_1·b_1_22·b_5_18 + a_2_4·b_1_12·b_1_2·b_5_18 + a_2_4·b_1_13·b_1_25 + a_2_4·b_1_14·b_1_24 + a_2_4·b_1_15·b_1_23 + c_8_39·b_1_12
- a_6_20·b_5_16
- a_6_20·b_5_18 + a_2_4·b_1_1·b_1_23·b_5_18 + a_2_4·b_1_12·b_1_22·b_5_18
+ a_2_4·c_8_39·b_1_1
- a_6_202
- b_5_18·b_7_31 + b_5_16·b_7_31 + b_1_12·b_1_25·b_5_18 + b_1_13·b_1_24·b_5_18
+ b_1_14·b_1_23·b_5_18 + b_1_15·b_1_22·b_5_18 + b_2_5·b_1_03·b_7_31 + b_2_5·b_1_05·b_5_16 + b_2_52·b_1_0·b_7_31 + b_2_52·b_1_05·b_3_9 + b_2_53·b_1_0·b_5_16 + b_2_53·b_1_03·b_3_9 + a_2_4·b_1_1·b_1_24·b_5_18 + a_2_4·b_1_1·b_1_29 + a_2_4·b_1_12·b_1_23·b_5_18 + a_2_4·b_1_12·b_1_28 + a_2_4·b_1_13·b_1_22·b_5_18 + a_2_4·b_1_14·b_1_2·b_5_18 + a_2_4·b_1_14·b_1_26 + a_2_4·b_1_17·b_1_23 + c_8_39·b_1_13·b_1_2 + c_8_39·b_1_14 + b_2_5·c_8_39·b_1_02 + a_2_4·c_8_39·b_1_12
- b_5_16·b_7_31 + b_2_5·b_1_03·b_7_31 + b_2_5·b_1_05·b_5_16 + b_2_5·b_1_07·b_3_9
+ b_2_52·b_1_0·b_7_31 + b_2_52·b_1_03·b_5_16 + b_2_52·b_1_05·b_3_9 + b_2_53·b_1_03·b_3_9 + b_2_54·b_1_0·b_3_9 + c_8_39·b_1_0·b_3_9 + c_8_39·b_1_04
- a_6_20·b_7_31 + a_2_4·b_1_12·b_1_24·b_5_18 + a_2_4·b_1_14·b_1_22·b_5_18
+ a_2_4·c_8_39·b_1_12·b_1_2 + a_2_4·c_8_39·b_1_13
- b_7_312 + b_1_13·b_1_26·b_5_18 + b_1_17·b_1_22·b_5_18 + b_2_5·b_1_05·b_7_31
+ b_2_5·b_1_09·b_3_9 + b_2_52·b_1_03·b_7_31 + b_2_52·b_1_05·b_5_16 + b_2_52·b_1_07·b_3_9 + b_2_54·b_1_0·b_5_16 + b_2_54·b_1_03·b_3_9 + b_2_55·b_1_0·b_3_9 + a_2_4·b_1_12·b_1_210 + a_2_4·b_1_13·b_1_24·b_5_18 + a_2_4·b_1_14·b_1_23·b_5_18 + a_2_4·b_1_14·b_1_28 + a_2_4·b_1_15·b_1_22·b_5_18 + a_2_4·b_1_15·b_1_27 + a_2_4·b_1_16·b_1_2·b_5_18 + a_2_4·b_1_16·b_1_26 + a_2_4·b_1_18·b_1_24 + a_2_4·b_1_19·b_1_23 + c_8_39·b_1_14·b_1_22 + c_8_39·b_1_16 + c_8_39·b_1_0·b_5_16 + c_8_39·b_1_06 + b_2_5·c_8_39·b_1_0·b_3_9 + b_2_5·c_8_39·b_1_04
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_8_39, a Duflot regular element of degree 8
- b_1_22 + b_1_1·b_1_2 + b_1_12 + b_1_02 + b_2_5, an element of degree 2
- b_1_1·b_1_22 + b_1_12·b_1_2 + b_2_5·b_1_2 + b_2_5·b_1_0, an element of degree 3
- The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 10].
- 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 1
- b_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
- a_2_4 → 0, an element of degree 2
- b_2_5 → 0, an element of degree 2
- b_3_9 → 0, an element of degree 3
- b_5_16 → 0, an element of degree 5
- b_5_18 → 0, an element of degree 5
- a_6_20 → 0, an element of degree 6
- b_7_31 → 0, an element of degree 7
- c_8_39 → c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- b_1_0 → c_1_1, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- a_2_4 → 0, an element of degree 2
- b_2_5 → c_1_22 + c_1_1·c_1_2, an element of degree 2
- b_3_9 → c_1_0·c_1_12 + c_1_02·c_1_1, an element of degree 3
- b_5_16 → c_1_0·c_1_12·c_1_22 + c_1_0·c_1_13·c_1_2 + c_1_02·c_1_1·c_1_22
+ c_1_02·c_1_12·c_1_2 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
- b_5_18 → c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
- a_6_20 → 0, an element of degree 6
- b_7_31 → c_1_0·c_1_12·c_1_24 + c_1_0·c_1_15·c_1_2 + c_1_02·c_1_1·c_1_24
+ c_1_02·c_1_13·c_1_22 + c_1_02·c_1_15 + c_1_03·c_1_14 + c_1_04·c_1_1·c_1_22 + c_1_04·c_1_12·c_1_2 + c_1_05·c_1_12 + c_1_06·c_1_1, an element of degree 7
- c_8_39 → c_1_0·c_1_13·c_1_24 + c_1_0·c_1_16·c_1_2 + c_1_04·c_1_24
+ c_1_04·c_1_13·c_1_2 + c_1_04·c_1_14 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- b_1_0 → 0, an element of degree 1
- b_1_1 → c_1_1, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- a_2_4 → 0, an element of degree 2
- b_2_5 → 0, an element of degree 2
- b_3_9 → 0, an element of degree 3
- b_5_16 → 0, an element of degree 5
- b_5_18 → c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
- a_6_20 → 0, an element of degree 6
- b_7_31 → c_1_02·c_1_14·c_1_2 + c_1_02·c_1_15 + c_1_04·c_1_12·c_1_2 + c_1_04·c_1_13, an element of degree 7
- c_8_39 → c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22 + c_1_04·c_1_24
+ c_1_04·c_1_12·c_1_22 + c_1_04·c_1_14 + c_1_08, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- b_1_0 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → c_1_1, an element of degree 1
- a_2_4 → 0, an element of degree 2
- b_2_5 → c_1_22 + c_1_1·c_1_2, an element of degree 2
- b_3_9 → 0, an element of degree 3
- b_5_16 → 0, an element of degree 5
- b_5_18 → 0, an element of degree 5
- a_6_20 → 0, an element of degree 6
- b_7_31 → 0, an element of degree 7
- c_8_39 → c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22 + c_1_04·c_1_24
+ c_1_04·c_1_12·c_1_22 + c_1_04·c_1_14 + c_1_08, an element of degree 8
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