Simon King
David J. Green
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Cohomology of group number 779 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
( − 1) · (t2 + t + 1) |
| (t + 1) · (t − 1)3 · (t2 + 1) |
- The a-invariants are -∞,-∞,-3,-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_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
- a_2_3, a nilpotent element of degree 2
- b_2_4, an element of degree 2
- c_2_5, a Duflot regular element of degree 2
- a_4_9, a nilpotent element of degree 4
- a_5_15, a nilpotent element of degree 5
- b_5_16, an element of degree 5
- b_5_17, an element of degree 5
- b_6_21, an element of degree 6
- c_8_36, a Duflot regular element of degree 8
Ring relations
There are 38 minimal relations of maximal degree 12:
- a_1_1·a_1_2
- a_1_0·a_1_2
- a_1_22 + a_1_0·a_1_1
- a_2_3·a_1_2
- b_2_4·a_1_2 + a_1_13 + a_1_03
- b_2_4·a_1_12 + a_2_32 + c_2_5·a_1_12 + c_2_5·a_1_02
- b_2_4·a_1_03 + a_2_32·a_1_1 + c_2_5·a_1_13
- a_4_9·a_1_1 + a_2_3·a_1_03 + a_2_3·c_2_5·a_1_1 + c_2_5·a_1_13
- a_4_9·a_1_0 + a_2_3·b_2_4·a_1_0 + a_2_3·c_2_5·a_1_0
- a_4_9·a_1_2 + a_2_3·a_1_03
- a_2_3·a_4_9 + b_2_4·c_2_5·a_1_02 + a_2_32·c_2_5 + a_2_3·c_2_5·a_1_12
- a_1_1·a_5_15 + a_2_3·b_2_4·a_1_02 + a_2_33 + a_2_32·c_2_5 + a_2_3·c_2_5·a_1_12
+ a_2_3·c_2_5·a_1_02 + c_2_52·a_1_0·a_1_1 + c_2_52·a_1_02
- a_1_0·a_5_15 + b_2_42·a_1_02 + a_2_33 + a_2_3·c_2_5·a_1_12 + a_2_3·c_2_5·a_1_02
+ c_2_52·a_1_0·a_1_1 + c_2_52·a_1_02
- a_1_1·b_5_16 + b_2_4·a_4_9 + a_2_3·b_2_42 + a_1_2·a_5_15 + a_2_33 + a_2_3·b_2_4·c_2_5
+ a_2_32·c_2_5 + a_2_3·c_2_5·a_1_02 + c_2_52·a_1_0·a_1_1 + c_2_52·a_1_02
- a_1_2·b_5_16 + c_2_52·a_1_0·a_1_1
- a_1_0·b_5_17 + b_2_4·a_4_9 + a_1_2·a_5_15 + b_2_42·a_1_02 + a_2_3·b_2_4·c_2_5
+ a_2_32·c_2_5 + c_2_52·a_1_12 + c_2_52·a_1_0·a_1_1 + c_2_52·a_1_02
- a_1_2·b_5_17 + a_1_2·a_5_15 + c_2_52·a_1_0·a_1_1
- a_2_3·a_5_15 + a_2_3·b_2_42·a_1_0 + a_2_32·b_2_4·a_1_1 + a_2_3·b_2_4·c_2_5·a_1_1
+ a_2_3·c_2_5·a_1_03 + a_2_3·c_2_52·a_1_1 + a_2_3·c_2_52·a_1_0
- b_2_4·a_5_15 + b_2_43·a_1_0 + a_1_12·b_5_17 + a_1_02·b_5_16 + a_2_3·b_2_42·a_1_1
+ a_2_32·b_2_4·a_1_1 + b_2_42·c_2_5·a_1_1 + a_2_32·c_2_5·a_1_1 + a_2_3·c_2_5·a_1_03 + b_2_4·c_2_52·a_1_1 + b_2_4·c_2_52·a_1_0 + c_2_52·a_1_03
- b_6_21·a_1_1 + a_2_3·b_5_17 + a_2_3·b_2_42·a_1_0 + a_2_33·a_1_1 + b_2_42·c_2_5·a_1_1
+ b_2_42·c_2_5·a_1_0 + c_2_52·a_1_13
- b_6_21·a_1_0 + b_2_43·a_1_1 + a_2_3·b_5_16 + a_2_33·a_1_1 + b_2_42·c_2_5·a_1_1
+ b_2_42·c_2_5·a_1_0 + a_2_32·c_2_5·a_1_1 + a_2_3·c_2_52·a_1_1 + c_2_52·a_1_03
- b_6_21·a_1_2 + a_2_33·a_1_1
- a_4_92 + b_2_42·c_2_5·a_1_02 + a_2_32·c_2_52
- b_2_4·a_1_1·b_5_17 + a_2_3·b_6_21 + a_2_3·b_2_42·a_1_02 + c_2_5·a_1_1·b_5_17
+ c_2_5·a_1_0·b_5_16 + a_2_3·b_2_42·c_2_5 + a_2_3·b_2_4·c_2_5·a_1_02 + a_2_3·c_2_52·a_1_12 + a_2_3·c_2_52·a_1_02 + c_2_53·a_1_0·a_1_1
- a_4_9·a_5_15 + a_2_3·b_2_43·a_1_0 + a_2_3·a_1_02·b_5_16 + a_2_3·b_2_42·c_2_5·a_1_0
+ a_2_32·b_2_4·c_2_5·a_1_1 + a_2_3·b_2_4·c_2_52·a_1_1 + a_2_3·b_2_4·c_2_52·a_1_0 + a_2_32·c_2_52·a_1_1 + a_2_3·c_2_52·a_1_03 + a_2_3·c_2_53·a_1_1 + a_2_3·c_2_53·a_1_0
- a_4_9·b_5_16 + b_2_44·a_1_1 + a_2_3·b_2_4·b_5_16 + a_2_32·b_2_42·a_1_1
+ b_2_43·c_2_5·a_1_1 + a_2_3·c_2_5·b_5_16 + a_2_3·b_2_42·c_2_5·a_1_1 + a_2_32·b_2_4·c_2_5·a_1_1 + a_2_3·b_2_4·c_2_52·a_1_1 + a_2_3·c_2_52·a_1_03 + c_2_53·a_1_13
- a_4_9·b_5_17 + a_2_3·b_2_43·a_1_0 + b_2_43·c_2_5·a_1_0 + a_2_3·c_2_5·b_5_17
+ c_2_5·a_1_12·b_5_17
- a_5_15·b_5_17 + a_5_15·b_5_16 + a_4_9·b_6_21 + b_2_42·a_1_0·b_5_16 + b_2_43·a_4_9
+ a_5_152 + b_2_44·a_1_02 + a_2_3·b_2_4·a_1_0·b_5_16 + a_2_32·b_2_43 + a_2_32·a_1_1·b_5_17 + a_2_33·b_2_42 + b_2_4·c_2_5·a_1_0·b_5_16 + a_2_3·c_2_5·a_1_1·b_5_17 + b_2_42·c_2_52·a_1_02 + a_2_3·b_2_4·c_2_52·a_1_02 + a_2_33·c_2_52 + a_2_32·c_2_53 + a_2_3·c_2_53·a_1_12 + c_2_54·a_1_12 + c_2_54·a_1_0·a_1_1
- b_5_16·b_5_17 + b_2_42·b_6_21 + a_5_15·b_5_16 + a_5_152 + b_2_44·a_1_02
+ a_2_32·b_2_43 + a_2_32·a_1_1·b_5_17 + a_2_33·b_2_42 + b_2_44·c_2_5 + b_2_42·c_2_5·a_4_9 + a_2_3·b_2_43·c_2_5 + b_2_43·c_2_5·a_1_02 + a_2_3·c_2_5·a_1_1·b_5_17 + a_2_3·b_2_42·c_2_5·a_1_02 + c_2_52·a_1_1·b_5_17 + c_2_52·a_1_0·b_5_16 + b_2_4·c_2_52·a_4_9 + c_2_52·a_1_2·a_5_15 + b_2_42·c_2_52·a_1_02 + a_2_32·b_2_4·c_2_52 + a_2_3·b_2_4·c_2_52·a_1_02 + a_2_33·c_2_52 + a_2_3·b_2_4·c_2_53 + b_2_4·c_2_53·a_1_02 + a_2_32·c_2_53 + a_2_3·c_2_53·a_1_02 + c_2_54·a_1_12 + c_2_54·a_1_0·a_1_1
- a_5_15·b_5_16 + b_2_42·a_1_0·b_5_16 + b_2_44·a_1_02 + a_2_32·b_6_21
+ a_2_32·b_2_43 + b_2_42·c_2_5·a_4_9 + a_2_3·b_2_43·c_2_5 + a_2_3·c_2_5·a_1_1·b_5_17 + a_2_3·c_2_5·a_1_0·b_5_16 + a_2_32·b_2_42·c_2_5 + a_2_33·b_2_4·c_2_5 + c_2_52·a_1_0·b_5_16 + b_2_4·c_2_52·a_4_9 + a_2_32·b_2_4·c_2_52 + a_2_3·b_2_4·c_2_53 + b_2_4·c_2_53·a_1_02 + a_2_32·c_2_53 + a_2_3·c_2_53·a_1_12 + c_2_54·a_1_0·a_1_1 + c_2_54·a_1_02
- a_5_15·b_5_17 + a_5_15·b_5_16 + b_2_42·a_1_0·b_5_16 + b_2_43·a_4_9 + a_5_152
+ b_2_44·a_1_02 + a_2_3·b_2_4·a_1_0·b_5_16 + a_2_32·b_2_43 + a_2_33·b_2_42 + b_2_42·c_2_5·a_4_9 + a_2_3·c_2_5·b_6_21 + a_2_3·b_2_42·c_2_52 + b_2_42·c_2_52·a_1_02 + a_2_33·c_2_52 + a_2_32·c_2_53 + a_2_3·c_2_53·a_1_12 + c_2_54·a_1_12 + c_2_54·a_1_0·a_1_1
- b_5_172 + a_5_15·b_5_17 + b_2_43·a_4_9 + a_2_3·b_2_4·b_6_21
+ a_2_3·b_2_4·a_1_0·b_5_16 + a_2_33·b_2_42 + b_2_44·c_2_5 + b_2_4·c_2_5·a_1_0·b_5_16 + c_8_36·a_1_12 + a_2_3·c_2_5·a_1_1·b_5_17 + c_2_52·a_1_1·b_5_17 + b_2_4·c_2_52·a_4_9 + b_2_42·c_2_52·a_1_02 + a_2_32·b_2_4·c_2_52 + a_2_3·b_2_4·c_2_52·a_1_02 + a_2_3·b_2_4·c_2_53 + b_2_4·c_2_53·a_1_02 + a_2_32·c_2_53 + c_2_54·a_1_02
- a_5_152 + b_2_44·a_1_02 + c_8_36·a_1_0·a_1_1 + a_2_32·b_2_4·c_2_52
+ b_2_4·c_2_53·a_1_02 + a_2_32·c_2_53
- b_5_162 + b_2_45 + b_2_42·a_1_0·b_5_16 + a_2_3·b_2_4·a_1_0·b_5_16
+ a_2_33·b_2_42 + b_2_44·c_2_5 + c_8_36·a_1_02 + b_2_43·c_2_5·a_1_02 + a_2_3·c_2_5·a_1_0·b_5_16 + a_2_33·b_2_4·c_2_5 + a_2_3·b_2_4·c_2_52·a_1_02 + a_2_33·c_2_52 + b_2_4·c_2_53·a_1_02 + a_2_3·c_2_53·a_1_12 + a_2_3·c_2_53·a_1_02 + c_2_54·a_1_12 + c_2_54·a_1_0·a_1_1 + c_2_54·a_1_02
- b_6_21·a_5_15 + b_2_45·a_1_1 + a_2_3·b_2_42·b_5_16 + a_2_3·b_2_44·a_1_0
+ a_2_32·b_2_4·b_5_17 + a_2_33·b_5_17 + b_2_44·c_2_5·a_1_1 + b_2_44·c_2_5·a_1_0 + a_2_3·b_2_4·c_2_5·b_5_17 + b_2_4·c_2_5·a_1_02·b_5_16 + a_2_3·b_2_43·c_2_5·a_1_0 + a_2_32·c_2_5·b_5_17 + a_2_3·c_2_5·a_1_02·b_5_16 + a_2_32·b_2_42·c_2_5·a_1_1 + b_2_43·c_2_52·a_1_0 + a_2_3·c_2_52·b_5_17 + a_2_3·c_2_52·b_5_16 + c_2_52·a_1_12·b_5_17 + a_2_3·b_2_42·c_2_52·a_1_1 + a_2_32·b_2_4·c_2_52·a_1_1 + a_2_33·c_2_52·a_1_1 + a_2_32·c_2_53·a_1_1 + a_2_3·c_2_54·a_1_1 + c_2_54·a_1_13 + c_2_54·a_1_03
- b_6_21·b_5_17 + b_2_45·a_1_1 + a_2_3·b_2_42·b_5_17 + a_2_3·b_2_42·b_5_16
+ a_2_32·b_2_4·b_5_17 + b_2_42·c_2_5·b_5_17 + b_2_42·c_2_5·b_5_16 + b_2_44·c_2_5·a_1_1 + b_2_44·c_2_5·a_1_0 + a_2_3·c_8_36·a_1_1 + a_2_32·c_2_5·b_5_17 + a_2_32·c_2_5·b_5_16 + c_2_52·a_1_12·b_5_17 + c_2_52·a_1_02·b_5_16 + a_2_33·c_2_52·a_1_1 + b_2_42·c_2_53·a_1_1 + a_2_3·b_2_4·c_2_53·a_1_1 + a_2_3·c_2_53·a_1_03 + a_2_3·c_2_54·a_1_1 + c_2_54·a_1_13
- b_6_21·b_5_16 + b_2_43·b_5_17 + b_2_45·a_1_1 + b_2_45·a_1_0 + a_2_3·b_2_42·b_5_16
+ b_2_42·a_1_02·b_5_16 + a_2_32·b_2_4·b_5_17 + b_2_42·c_2_5·b_5_17 + b_2_42·c_2_5·b_5_16 + b_2_44·c_2_5·a_1_1 + b_2_44·c_2_5·a_1_0 + a_2_3·c_8_36·a_1_0 + a_2_32·c_2_5·b_5_17 + a_2_3·c_2_5·a_1_02·b_5_16 + a_2_3·c_2_52·b_5_17 + a_2_32·b_2_4·c_2_52·a_1_1 + a_2_33·c_2_52·a_1_1 + b_2_42·c_2_53·a_1_0 + a_2_3·b_2_4·c_2_53·a_1_0 + a_2_32·c_2_53·a_1_1 + a_2_3·c_2_53·a_1_03 + a_2_3·c_2_54·a_1_0
- b_6_212 + a_2_3·b_2_42·b_6_21 + a_2_32·b_2_4·b_6_21 + b_2_45·c_2_5
+ a_2_3·b_2_44·c_2_5 + a_2_32·c_8_36 + a_2_32·c_2_5·b_6_21 + a_2_32·b_2_43·c_2_5 + b_2_43·c_2_52·a_1_02 + a_2_33·b_2_4·c_2_52 + b_2_42·c_2_53·a_1_02 + a_2_32·b_2_4·c_2_53 + a_2_3·b_2_4·c_2_53·a_1_02 + a_2_32·c_2_54
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_2_5, a Duflot regular element of degree 2
- c_8_36, a Duflot regular element of degree 8
- b_2_4, 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
- a_2_3 → 0, an element of degree 2
- b_2_4 → 0, an element of degree 2
- c_2_5 → c_1_12, an element of degree 2
- a_4_9 → 0, an element of degree 4
- a_5_15 → 0, an element of degree 5
- b_5_16 → 0, an element of degree 5
- b_5_17 → 0, an element of degree 5
- b_6_21 → 0, an element of degree 6
- c_8_36 → 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
- a_1_1 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- a_2_3 → 0, an element of degree 2
- b_2_4 → c_1_22, an element of degree 2
- c_2_5 → c_1_12, an element of degree 2
- a_4_9 → 0, an element of degree 4
- a_5_15 → 0, an element of degree 5
- b_5_16 → c_1_25 + c_1_1·c_1_24, an element of degree 5
- b_5_17 → c_1_1·c_1_24, an element of degree 5
- b_6_21 → c_1_1·c_1_25, an element of degree 6
- c_8_36 → c_1_1·c_1_27 + c_1_13·c_1_25 + c_1_16·c_1_22 + c_1_04·c_1_24 + c_1_08, an element of degree 8
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