Simon King
David J. Green
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Cohomology of group number 662 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 a unique conjugacy class of maximal elementary abelian subgroups, which is 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
- The a-invariants are -∞,-∞,-3,-3. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 14 minimal generators of maximal degree 8:
- a_1_0, a nilpotent element of degree 1
- a_1_1, a nilpotent element of degree 1
- a_1_2, a nilpotent element of degree 1
- b_2_3, an element of degree 2
- c_2_4, a Duflot regular element of degree 2
- a_3_6, a nilpotent element of degree 3
- a_4_5, a nilpotent element of degree 4
- a_4_6, a nilpotent element of degree 4
- a_5_11, a nilpotent element of degree 5
- a_5_12, a nilpotent element of degree 5
- b_5_13, an element of degree 5
- a_7_19, a nilpotent element of degree 7
- a_8_20, a nilpotent element of degree 8
- c_8_26, a Duflot regular element of degree 8
Ring relations
There are 61 minimal relations of maximal degree 16:
- a_1_12 + a_1_02
- a_1_0·a_1_2
- a_1_22 + a_1_0·a_1_1 + a_1_02
- b_2_3·a_1_2 + a_1_03
- a_1_1·a_3_6 + a_1_0·a_3_6
- a_1_2·a_3_6
- b_2_3·a_1_03
- a_4_5·a_1_1 + c_2_4·a_1_03
- a_4_5·a_1_0 + a_1_02·a_3_6 + c_2_4·a_1_03
- a_4_5·a_1_2
- b_2_3·a_3_6 + a_4_6·a_1_1 + a_1_02·a_3_6
- b_2_3·a_3_6 + a_4_6·a_1_0
- a_4_6·a_1_2 + a_1_02·a_3_6
- a_3_62
- a_1_2·a_5_11 + a_1_1·a_5_11 + b_2_32·a_1_02 + b_2_3·c_2_4·a_1_02
+ c_2_42·a_1_0·a_1_1
- a_1_1·a_5_12 + a_1_0·a_5_11 + b_2_32·a_1_02 + c_2_4·a_1_0·a_3_6
+ b_2_3·c_2_4·a_1_02 + c_2_42·a_1_02
- a_1_0·a_5_12 + a_1_0·a_5_11 + b_2_32·a_1_02 + a_4_6·a_1_02 + c_2_4·a_1_0·a_3_6
+ b_2_3·c_2_4·a_1_02 + c_2_42·a_1_02
- a_1_2·a_5_12 + a_1_1·a_5_11 + a_1_0·a_5_11 + a_4_6·a_1_02 + c_2_42·a_1_0·a_1_1
+ c_2_42·a_1_02
- a_1_1·b_5_13 + a_1_0·b_5_13 + b_2_3·a_4_5 + a_1_1·a_5_11 + a_4_6·a_1_02
+ c_2_42·a_1_02
- a_1_2·b_5_13 + a_1_0·a_5_11 + b_2_32·a_1_02 + a_4_6·a_1_02 + b_2_3·c_2_4·a_1_02
+ c_2_42·a_1_02
- a_4_5·a_3_6 + a_4_6·a_1_03 + c_2_4·a_1_02·a_3_6
- a_4_6·a_3_6
- b_2_3·a_5_12 + b_2_3·a_5_11 + b_2_33·a_1_1 + a_1_02·b_5_13 + b_2_3·a_4_6·a_1_0
+ b_2_32·c_2_4·a_1_1 + c_2_4·a_4_6·a_1_0 + c_2_4·a_1_02·a_3_6 + b_2_3·c_2_42·a_1_0 + c_2_42·a_1_03
- a_4_52
- a_4_62 + b_2_32·c_2_4·a_1_02
- a_4_5·a_4_6 + b_2_3·a_4_6·a_1_02 + c_2_4·a_4_6·a_1_02
- a_3_6·a_5_11 + a_4_5·a_4_6 + c_2_42·a_1_0·a_3_6
- a_3_6·a_5_12
- a_3_6·b_5_13 + a_1_1·a_7_19 + a_4_5·a_4_6 + c_2_4·a_1_1·a_5_11 + b_2_32·c_2_4·a_1_02
+ c_2_42·a_1_0·a_3_6 + c_2_43·a_1_0·a_1_1
- a_3_6·b_5_13 + a_1_0·a_7_19 + a_4_5·a_4_6 + c_2_4·a_1_0·a_5_11 + b_2_32·c_2_4·a_1_02
+ c_2_42·a_1_0·a_3_6 + c_2_43·a_1_02
- a_1_2·a_7_19 + c_2_4·a_1_1·a_5_11 + b_2_32·c_2_4·a_1_02 + b_2_3·c_2_42·a_1_02
+ c_2_43·a_1_0·a_1_1
- a_4_6·a_5_11 + b_2_32·a_4_6·a_1_0 + b_2_3·c_2_4·a_4_6·a_1_0 + c_2_42·a_4_6·a_1_0
- a_4_6·a_5_12 + a_4_5·a_5_11 + c_2_4·a_4_6·a_1_03 + c_2_42·a_1_02·a_3_6
+ c_2_43·a_1_03
- a_4_5·a_5_12 + a_4_5·a_5_11 + c_2_43·a_1_03
- a_4_5·b_5_13 + b_2_34·a_1_1 + b_2_34·a_1_0 + a_4_5·a_5_11 + b_2_3·a_1_02·b_5_13
+ c_2_4·a_1_02·b_5_13 + c_2_4·a_4_6·a_1_03 + c_2_43·a_1_03
- a_4_6·b_5_13 + b_2_3·a_7_19 + a_4_5·a_5_11 + b_2_3·a_1_02·b_5_13 + b_2_32·a_4_6·a_1_0
+ b_2_3·c_2_4·a_5_11 + b_2_33·c_2_4·a_1_1 + b_2_3·c_2_4·a_4_6·a_1_0 + b_2_32·c_2_42·a_1_1 + b_2_32·c_2_42·a_1_0 + c_2_42·a_4_6·a_1_0 + c_2_42·a_1_02·a_3_6 + b_2_3·c_2_43·a_1_0 + c_2_43·a_1_03
- a_4_5·a_5_11 + a_1_02·a_7_19 + c_2_4·a_4_6·a_1_03 + c_2_42·a_1_02·a_3_6
+ c_2_43·a_1_03
- a_4_5·b_5_13 + b_2_34·a_1_1 + b_2_34·a_1_0 + a_8_20·a_1_1 + a_4_5·a_5_11
+ b_2_32·a_4_6·a_1_0 + b_2_3·c_2_4·a_4_6·a_1_0 + c_2_4·a_4_6·a_1_03 + c_2_42·a_4_6·a_1_0 + c_2_42·a_1_02·a_3_6 + c_2_43·a_1_03
- a_4_5·b_5_13 + b_2_34·a_1_1 + b_2_34·a_1_0 + a_8_20·a_1_0 + b_2_32·a_4_6·a_1_0
+ b_2_3·c_2_4·a_4_6·a_1_0 + c_2_4·a_4_6·a_1_03 + c_2_42·a_4_6·a_1_0
- a_8_20·a_1_2 + c_2_4·a_4_6·a_1_03 + c_2_42·a_1_02·a_3_6
- a_5_122
- a_5_12·b_5_13 + a_5_11·b_5_13 + b_2_32·a_1_0·b_5_13 + b_2_33·a_4_5 + a_5_112
+ b_2_3·a_1_0·a_7_19 + b_2_34·a_1_02 + b_2_3·c_2_4·a_1_0·b_5_13 + b_2_32·c_2_4·a_4_5 + c_2_4·a_1_0·a_7_19 + c_2_42·a_1_0·b_5_13 + c_2_42·a_1_0·a_5_11 + c_2_43·a_1_0·a_3_6
- a_3_6·a_7_19 + c_2_42·a_4_6·a_1_02
- b_5_132 + b_2_35 + b_2_32·a_1_0·b_5_13 + c_8_26·a_1_0·a_1_1
+ c_2_42·a_4_6·a_1_02 + b_2_3·c_2_43·a_1_02 + c_2_44·a_1_02
- b_5_132 + b_2_35 + b_2_32·a_1_0·b_5_13 + a_5_112 + b_2_34·a_1_02
+ c_8_26·a_1_02 + b_2_32·c_2_42·a_1_02 + c_2_42·a_4_6·a_1_02 + b_2_3·c_2_43·a_1_02
- a_5_11·a_5_12 + b_2_32·a_4_6·a_1_02 + c_8_26·a_1_1·a_1_2 + c_2_42·a_1_0·a_5_11
+ b_2_32·c_2_42·a_1_02 + c_2_43·a_1_0·a_3_6 + b_2_3·c_2_43·a_1_02 + c_2_44·a_1_02
- a_5_11·b_5_13 + b_2_3·a_8_20 + b_2_33·a_4_6 + a_5_11·a_5_12 + b_2_32·c_2_4·a_4_6
+ b_2_33·c_2_4·a_1_02 + b_2_3·c_2_4·a_4_6·a_1_02 + c_2_42·a_1_0·b_5_13 + b_2_3·c_2_42·a_4_6 + b_2_3·c_2_42·a_4_5 + b_2_32·c_2_42·a_1_02 + c_2_43·a_1_0·a_3_6 + b_2_3·c_2_43·a_1_02
- a_4_6·a_7_19 + b_2_3·a_1_02·a_7_19 + b_2_3·c_2_4·a_1_02·b_5_13
+ b_2_3·c_2_42·a_4_6·a_1_0
- a_4_5·a_7_19 + b_2_3·a_1_02·a_7_19 + c_2_42·a_4_6·a_1_03
- a_8_20·a_3_6 + b_2_3·a_1_02·a_7_19 + c_2_4·a_1_02·a_7_19 + c_2_42·a_4_6·a_1_03
+ c_2_43·a_1_02·a_3_6
- a_5_11·a_7_19 + b_2_32·a_1_0·a_7_19 + b_2_3·c_2_4·a_1_0·a_7_19
+ b_2_32·c_2_4·a_4_6·a_1_02 + c_2_4·c_8_26·a_1_0·a_1_1 + c_2_4·c_8_26·a_1_02 + c_2_42·a_1_0·a_7_19
- a_5_12·a_7_19 + b_2_33·a_4_6·a_1_02 + c_2_4·c_8_26·a_1_1·a_1_2
+ c_2_43·a_4_6·a_1_02
- b_5_13·a_7_19 + b_2_34·a_4_6 + a_5_12·a_7_19 + b_2_35·a_1_02 + b_2_3·c_2_4·a_8_20
+ b_2_32·c_2_4·a_1_0·b_5_13 + b_2_33·c_2_4·a_4_6 + b_2_33·c_2_4·a_4_5 + c_8_26·a_1_0·a_3_6 + b_2_3·c_2_4·a_1_0·a_7_19 + b_2_34·c_2_4·a_1_02 + b_2_32·c_2_4·a_4_6·a_1_02 + b_2_32·c_2_42·a_4_6 + b_2_32·c_2_42·a_4_5 + c_2_42·a_1_0·a_7_19 + b_2_33·c_2_42·a_1_02 + b_2_3·c_2_42·a_4_6·a_1_02 + b_2_3·c_2_43·a_4_6 + b_2_3·c_2_43·a_4_5 + b_2_32·c_2_43·a_1_02
- a_4_6·a_8_20 + b_2_32·a_1_0·a_7_19 + b_2_33·a_4_6·a_1_02
+ b_2_3·c_2_4·a_1_0·a_7_19 + b_2_34·c_2_4·a_1_02 + b_2_3·c_2_42·a_4_6·a_1_02 + c_2_43·a_4_6·a_1_02
- a_4_5·a_8_20 + c_2_43·a_4_6·a_1_02
- a_8_20·b_5_13 + b_2_33·a_7_19 + b_2_34·a_5_11 + b_2_34·a_4_6·a_1_0
+ b_2_32·a_1_02·a_7_19 + b_2_32·c_2_4·a_7_19 + b_2_33·c_2_4·a_5_11 + b_2_35·c_2_4·a_1_1 + b_2_32·c_2_4·a_1_02·b_5_13 + c_8_26·a_1_02·a_3_6 + b_2_3·c_2_42·a_7_19 + b_2_32·c_2_42·a_5_11 + b_2_34·c_2_42·a_1_1 + b_2_34·c_2_42·a_1_0 + b_2_3·c_2_42·a_1_02·b_5_13 + b_2_32·c_2_42·a_4_6·a_1_0 + c_2_4·c_8_26·a_1_03 + c_2_42·a_1_02·a_7_19 + b_2_3·c_2_43·a_5_11 + b_2_32·c_2_44·a_1_1 + c_2_44·a_4_6·a_1_0 + c_2_44·a_1_02·a_3_6 + b_2_3·c_2_45·a_1_0 + c_2_45·a_1_03
- a_8_20·a_5_11 + b_2_33·a_1_02·b_5_13 + b_2_34·a_4_6·a_1_0 + c_8_26·a_1_02·a_3_6
+ b_2_32·c_2_42·a_4_6·a_1_0 + c_2_43·a_1_02·b_5_13 + c_2_43·a_4_6·a_1_03 + c_2_44·a_4_6·a_1_0 + c_2_44·a_1_02·a_3_6
- a_8_20·a_5_12 + c_8_26·a_1_02·a_3_6 + b_2_3·c_2_4·a_1_02·a_7_19
+ c_2_42·a_1_02·a_7_19 + c_2_44·a_1_02·a_3_6
- a_7_192 + b_2_35·c_2_4·a_1_02 + c_2_42·c_8_26·a_1_0·a_1_1
+ c_2_42·c_8_26·a_1_02 + b_2_32·c_2_44·a_1_02
- a_8_20·a_7_19 + b_2_35·a_4_6·a_1_0 + b_2_33·a_1_02·a_7_19
+ b_2_33·c_2_4·a_1_02·b_5_13 + b_2_34·c_2_4·a_4_6·a_1_0 + b_2_32·c_2_4·a_1_02·a_7_19 + a_4_6·c_8_26·a_1_03 + b_2_33·c_2_42·a_4_6·a_1_0 + b_2_32·c_2_43·a_4_6·a_1_0 + b_2_3·c_2_44·a_4_6·a_1_0 + c_2_44·a_4_6·a_1_03
- a_8_202 + b_2_37·a_1_02 + b_2_36·c_2_4·a_1_02 + b_2_35·c_2_42·a_1_02
+ b_2_34·c_2_43·a_1_02 + b_2_32·c_2_45·a_1_02
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_26, a Duflot regular element of degree 8
- b_2_3, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 7, 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_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- b_2_3 → 0, an element of degree 2
- c_2_4 → c_1_02, an element of degree 2
- a_3_6 → 0, an element of degree 3
- a_4_5 → 0, an element of degree 4
- a_4_6 → 0, an element of degree 4
- a_5_11 → 0, an element of degree 5
- a_5_12 → 0, an element of degree 5
- b_5_13 → 0, an element of degree 5
- a_7_19 → 0, an element of degree 7
- a_8_20 → 0, an element of degree 8
- c_8_26 → c_1_18, 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
- a_1_1 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- b_2_3 → c_1_22, an element of degree 2
- c_2_4 → c_1_02, an element of degree 2
- a_3_6 → 0, an element of degree 3
- a_4_5 → 0, an element of degree 4
- a_4_6 → 0, an element of degree 4
- a_5_11 → 0, an element of degree 5
- a_5_12 → 0, an element of degree 5
- b_5_13 → c_1_25, an element of degree 5
- a_7_19 → 0, an element of degree 7
- a_8_20 → 0, an element of degree 8
- c_8_26 → c_1_14·c_1_24 + c_1_18 + c_1_04·c_1_24 + c_1_06·c_1_22, an element of degree 8
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