Simon King
David J. Green
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Singular
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Cohomology of group number 342 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 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 3.
- The depth exceeds the Duflot bound, which is 2.
- The Poincaré series is
( − 1) · (t2 + t + 1) · (t6 + t5 + t3 + t + 1) |
| (t + 1)2 · (t − 1)3 · (t2 + 1) · (t4 + 1) |
- The a-invariants are -∞,-∞,-∞,-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:
- a_1_0, a nilpotent element of degree 1
- b_1_1, an element of degree 1
- b_1_2, an element of degree 1
- b_2_4, an element of degree 2
- c_2_5, a Duflot regular element of degree 2
- a_3_2, a nilpotent element of degree 3
- a_5_14, a nilpotent element of degree 5
- b_5_16, an element of degree 5
- b_5_18, an element of degree 5
- a_7_26, a nilpotent element of degree 7
- c_8_41, a Duflot regular element of degree 8
Ring relations
There are 35 minimal relations of maximal degree 14:
- a_1_0·b_1_1
- a_1_0·b_1_2
- a_1_03
- b_1_23 + b_1_1·b_1_22 + b_2_4·b_1_1
- b_2_4·b_1_1·b_1_2 + b_2_4·b_1_12
- b_2_4·b_1_22 + b_2_4·b_1_1·b_1_2 + b_1_2·a_3_2 + b_2_4·a_1_02
- b_1_1·a_3_2
- b_1_22·a_3_2
- a_1_02·a_3_2
- b_2_4·b_1_14 + b_1_2·a_5_14 + b_2_4·b_1_2·a_3_2 + b_2_42·a_1_02
+ b_2_4·c_2_5·a_1_02
- a_3_22 + a_1_0·a_5_14 + b_2_4·a_1_0·a_3_2 + b_2_42·a_1_02 + c_2_5·a_1_0·a_3_2
+ b_2_4·c_2_5·a_1_02 + c_2_52·a_1_02
- b_2_4·b_1_14 + b_1_1·a_5_14 + c_2_5·b_1_12·b_1_22 + c_2_5·b_1_13·b_1_2
+ b_2_4·c_2_5·b_1_12
- a_1_0·b_5_16 + a_3_22 + b_2_4·a_1_0·a_3_2 + b_2_4·c_2_5·a_1_02
- a_1_0·b_5_18 + a_3_22 + b_2_4·c_2_5·a_1_02 + c_2_52·a_1_02
- b_1_2·b_5_16 + b_1_1·b_5_18 + b_2_4·b_1_2·a_3_2 + b_2_42·a_1_02 + c_2_5·b_1_2·a_3_2
+ b_2_4·c_2_5·a_1_02 + c_2_52·b_1_1·b_1_2
- a_1_02·a_5_14
- b_1_22·b_5_18 + b_1_1·b_1_2·b_5_18 + b_2_4·b_5_16 + b_2_4·a_5_14 + b_2_43·a_1_0
+ b_2_4·c_2_5·a_3_2 + b_2_4·c_2_52·b_1_1 + b_2_4·c_2_52·a_1_0
- a_3_2·b_5_16 + a_3_2·a_5_14 + b_2_42·a_1_0·a_3_2 + c_2_5·a_1_0·a_5_14
+ b_2_4·c_2_5·a_1_0·a_3_2 + b_2_42·c_2_5·a_1_02 + b_2_4·c_2_52·a_1_02 + c_2_53·a_1_02
- b_2_4·b_1_2·b_5_18 + b_2_4·b_1_1·b_5_16 + a_3_2·b_5_18 + a_3_2·a_5_14 + b_2_43·a_1_02
+ b_2_4·c_2_5·b_1_2·a_3_2 + c_2_5·a_1_0·a_5_14 + b_2_4·c_2_5·a_1_0·a_3_2 + b_2_4·c_2_52·b_1_12 + c_2_52·a_1_0·a_3_2 + c_2_53·a_1_02
- b_2_4·b_1_1·b_5_18 + b_2_4·b_1_1·b_5_16 + b_2_4·c_2_52·b_1_12
- a_3_2·b_5_18 + b_1_2·a_7_26 + b_1_13·a_5_14 + a_3_2·a_5_14 + b_2_4·a_1_0·a_5_14
+ c_2_5·b_1_14·b_1_22 + c_2_5·b_1_15·b_1_2 + c_2_5·b_1_1·a_5_14 + b_2_4·c_2_5·b_1_2·a_3_2 + c_2_5·a_1_0·a_5_14 + b_2_42·c_2_5·a_1_02 + c_2_52·b_1_12·b_1_22 + c_2_52·b_1_13·b_1_2 + b_2_4·c_2_52·b_1_12 + c_2_52·a_1_0·a_3_2 + b_2_4·c_2_52·a_1_02 + c_2_53·a_1_02
- b_2_42·b_1_2·a_3_2 + a_3_2·a_5_14 + a_1_0·a_7_26 + b_2_4·a_1_0·a_5_14
+ b_2_43·a_1_02 + b_2_4·c_2_5·b_1_2·a_3_2 + b_2_4·c_2_5·a_1_0·a_3_2 + b_2_42·c_2_5·a_1_02 + c_2_52·a_1_0·a_3_2 + b_2_4·c_2_52·a_1_02 + c_2_53·a_1_02
- b_1_12·b_1_2·b_5_18 + b_1_13·b_5_18 + b_2_4·b_1_1·b_5_16 + b_1_1·a_7_26
+ b_1_13·a_5_14 + b_2_4·c_2_52·b_1_12
- b_1_22·a_7_26 + b_1_14·a_5_14 + a_1_02·a_7_26 + c_2_5·b_1_15·b_1_22
+ c_2_5·b_1_16·b_1_2 + c_2_5·b_1_12·a_5_14 + c_2_52·b_1_13·b_1_22 + c_2_52·b_1_14·b_1_2 + b_2_4·c_2_52·b_1_13
- a_5_142 + a_3_2·a_7_26 + b_2_42·a_1_0·a_5_14 + b_2_43·a_1_0·a_3_2 + b_2_44·a_1_02
+ c_2_5·a_3_2·a_5_14 + b_2_43·c_2_5·a_1_02 + b_2_4·c_2_52·b_1_2·a_3_2 + b_2_4·c_2_52·a_1_0·a_3_2 + b_2_42·c_2_52·a_1_02 + c_2_54·a_1_02
- b_2_4·b_1_13·b_5_16 + a_5_14·b_5_18 + b_2_4·b_1_2·a_7_26 + a_5_142
+ b_2_4·a_3_2·a_5_14 + c_2_5·a_1_0·a_7_26 + b_2_42·c_2_5·a_1_0·a_3_2 + c_2_52·b_1_1·a_5_14 + b_2_4·c_2_52·b_1_2·a_3_2 + b_2_42·c_2_52·a_1_02 + c_2_53·b_1_12·b_1_22 + c_2_53·b_1_13·b_1_2 + b_2_4·c_2_53·b_1_12 + c_2_53·a_1_0·a_3_2 + b_2_4·c_2_53·a_1_02 + c_2_54·a_1_02
- b_2_4·b_1_13·b_5_16 + a_5_14·b_5_16 + a_5_142 + b_2_42·a_1_0·a_5_14
+ c_2_5·b_1_1·a_7_26 + c_2_5·b_1_13·a_5_14 + c_2_5·a_3_2·a_5_14 + c_2_52·a_1_0·a_5_14 + c_2_53·b_1_12·b_1_22 + c_2_53·b_1_13·b_1_2 + b_2_4·c_2_53·b_1_12
- b_5_182 + b_1_18·b_1_22 + b_2_4·b_1_13·b_5_16 + a_5_14·b_5_16
+ b_2_43·a_1_0·a_3_2 + c_8_41·b_1_22 + c_2_5·b_1_13·b_5_18 + c_2_5·b_1_13·a_5_14 + c_2_5·a_3_2·a_5_14 + b_2_42·c_2_5·a_1_0·a_3_2 + b_2_43·c_2_5·a_1_02 + c_2_52·b_1_15·b_1_2 + b_2_4·c_2_52·b_1_2·a_3_2 + b_2_4·c_2_52·a_1_0·a_3_2 + b_2_42·c_2_52·a_1_02 + c_2_53·b_1_12·b_1_22 + c_2_53·b_1_13·b_1_2 + c_2_53·b_1_2·a_3_2 + c_2_53·a_1_0·a_3_2 + c_2_54·a_1_02
- b_5_16·b_5_18 + b_1_18·b_1_22 + a_5_14·b_5_18 + a_5_14·b_5_16 + b_1_15·a_5_14
+ a_5_142 + b_2_43·a_1_0·a_3_2 + b_2_44·a_1_02 + c_8_41·b_1_1·b_1_2 + c_2_5·b_1_13·b_5_18 + c_2_5·b_1_16·b_1_22 + c_2_5·b_1_17·b_1_2 + b_2_4·c_2_5·b_1_1·b_5_16 + c_2_5·b_1_2·a_7_26 + c_2_5·b_1_13·a_5_14 + b_2_4·c_2_5·a_1_0·a_5_14 + b_2_42·c_2_5·a_1_0·a_3_2 + c_2_52·b_1_1·b_5_18 + c_2_52·b_1_15·b_1_2 + b_2_4·c_2_52·b_1_2·a_3_2 + c_2_52·a_1_0·a_5_14 + b_2_4·c_2_52·a_1_0·a_3_2 + b_2_4·c_2_53·a_1_02 + c_2_54·a_1_02
- b_2_4·b_1_13·b_5_16 + a_5_14·b_5_18 + b_2_4·b_1_2·a_7_26 + b_2_4·a_1_0·a_7_26
+ b_2_42·a_1_0·a_5_14 + b_2_44·a_1_02 + c_8_41·a_1_02 + c_2_5·a_3_2·a_5_14 + b_2_4·c_2_5·a_1_0·a_5_14 + c_2_52·b_1_1·a_5_14 + b_2_4·c_2_52·b_1_2·a_3_2 + c_2_52·a_1_0·a_5_14 + b_2_4·c_2_52·a_1_0·a_3_2 + b_2_42·c_2_52·a_1_02 + c_2_53·b_1_12·b_1_22 + c_2_53·b_1_13·b_1_2 + b_2_4·c_2_53·b_1_12 + c_2_53·a_1_0·a_3_2 + b_2_4·c_2_53·a_1_02 + c_2_54·a_1_02
- b_5_162 + b_1_18·b_1_22 + b_1_13·a_7_26 + b_1_15·a_5_14 + a_5_142
+ b_2_44·a_1_02 + c_8_41·b_1_12 + c_2_5·b_1_13·b_5_18 + c_2_5·b_1_16·b_1_22 + c_2_5·b_1_17·b_1_2 + c_2_52·b_1_14·b_1_22 + c_2_52·b_1_1·a_5_14 + c_2_52·a_1_0·a_5_14 + b_2_4·c_2_52·a_1_0·a_3_2 + b_2_42·c_2_52·a_1_02 + c_2_53·b_1_13·b_1_2 + c_2_53·b_1_14 + c_2_53·a_1_0·a_3_2 + b_2_4·c_2_53·a_1_02 + c_2_54·b_1_12
- b_2_42·b_1_2·a_7_26 + a_5_14·a_7_26 + b_2_43·a_1_0·a_5_14 + b_2_44·a_1_0·a_3_2
+ b_2_45·a_1_02 + b_2_4·c_2_5·b_1_2·a_7_26 + c_8_41·a_1_0·a_3_2 + b_2_4·c_8_41·a_1_02 + b_2_4·c_2_5·a_3_2·a_5_14 + b_2_42·c_2_5·a_1_0·a_5_14 + b_2_4·c_2_52·a_1_0·a_5_14 + b_2_42·c_2_52·a_1_0·a_3_2 + b_2_4·c_2_53·b_1_2·a_3_2 + b_2_4·c_2_53·a_1_0·a_3_2 + b_2_4·c_2_54·a_1_02 + c_2_55·a_1_02
- b_5_18·a_7_26 + b_5_16·a_7_26 + b_2_42·b_1_2·a_7_26 + a_5_14·a_7_26
+ b_2_42·a_3_2·a_5_14 + b_2_42·a_1_0·a_7_26 + c_8_41·b_1_12·b_1_22 + c_8_41·b_1_13·b_1_2 + b_2_4·c_8_41·b_1_12 + c_8_41·b_1_2·a_3_2 + c_2_5·b_1_13·a_7_26 + c_2_5·b_1_15·a_5_14 + c_8_41·a_1_0·a_3_2 + b_2_4·c_2_5·a_3_2·a_5_14 + b_2_4·c_2_5·a_1_0·a_7_26 + b_2_42·c_2_5·a_1_0·a_5_14 + c_2_52·b_1_1·a_7_26 + c_2_52·a_3_2·a_5_14 + b_2_43·c_2_52·a_1_02 + c_2_53·b_1_14·b_1_22 + c_2_53·b_1_15·b_1_2 + c_2_53·b_1_1·a_5_14 + b_2_4·c_2_53·b_1_2·a_3_2 + b_2_4·c_2_53·a_1_0·a_3_2 + b_2_42·c_2_53·a_1_02 + c_2_54·b_1_12·b_1_22 + c_2_54·b_1_13·b_1_2 + b_2_4·c_2_54·b_1_12 + c_2_54·a_1_0·a_3_2 + b_2_4·c_2_54·a_1_02
- b_5_18·a_7_26 + b_1_12·a_5_14·b_5_16 + b_2_42·b_1_2·a_7_26 + b_2_42·a_3_2·a_5_14
+ c_8_41·b_1_2·a_3_2 + c_2_5·b_1_13·a_7_26 + c_2_5·b_1_15·a_5_14 + c_8_41·a_1_0·a_3_2 + b_2_42·c_2_5·a_1_0·a_5_14 + b_2_43·c_2_5·a_1_0·a_3_2 + c_2_52·b_1_13·a_5_14 + c_2_5·c_8_41·a_1_02 + c_2_52·a_3_2·a_5_14 + b_2_4·c_2_52·a_1_0·a_5_14 + b_2_42·c_2_52·a_1_0·a_3_2 + c_2_53·a_1_0·a_5_14 + b_2_4·c_2_53·a_1_0·a_3_2 + c_2_54·a_1_0·a_3_2 + b_2_4·c_2_54·a_1_02 + c_2_55·a_1_02
- a_7_262 + b_2_43·a_3_2·a_5_14 + b_2_43·a_1_0·a_7_26 + b_2_45·a_1_0·a_3_2
+ b_2_46·a_1_02 + c_8_41·a_1_0·a_5_14 + b_2_4·c_8_41·a_1_0·a_3_2 + b_2_42·c_2_5·a_3_2·a_5_14 + b_2_42·c_2_5·a_1_0·a_7_26 + b_2_43·c_2_5·a_1_0·a_5_14 + b_2_45·c_2_5·a_1_02 + c_2_5·c_8_41·a_1_0·a_3_2 + b_2_4·c_2_5·c_8_41·a_1_02 + b_2_42·c_2_52·a_1_0·a_5_14 + b_2_43·c_2_52·a_1_0·a_3_2 + b_2_4·c_2_53·a_1_0·a_5_14 + b_2_42·c_2_53·a_1_0·a_3_2 + b_2_43·c_2_53·a_1_02 + b_2_4·c_2_54·b_1_2·a_3_2 + c_2_54·a_1_0·a_5_14 + b_2_42·c_2_54·a_1_02 + c_2_55·a_1_0·a_3_2 + b_2_4·c_2_55·a_1_02 + c_2_56·a_1_02
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_2_5, a Duflot regular element of degree 2
- c_8_41, a Duflot regular element of degree 8
- b_1_12 + b_2_4, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 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
- b_1_1 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_2_4 → 0, an element of degree 2
- c_2_5 → c_1_02, an element of degree 2
- a_3_2 → 0, an element of degree 3
- a_5_14 → 0, an element of degree 5
- b_5_16 → 0, an element of degree 5
- b_5_18 → 0, an element of degree 5
- a_7_26 → 0, an element of degree 7
- c_8_41 → c_1_18 + 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
- b_1_1 → c_1_2, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_2_4 → 0, an element of degree 2
- c_2_5 → c_1_0·c_1_2 + c_1_02, an element of degree 2
- a_3_2 → 0, an element of degree 3
- a_5_14 → 0, an element of degree 5
- b_5_16 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
- b_5_18 → 0, an element of degree 5
- a_7_26 → 0, an element of degree 7
- c_8_41 → c_1_14·c_1_24 + c_1_18 + c_1_03·c_1_25 + c_1_05·c_1_23 + c_1_06·c_1_22
+ 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
- b_1_1 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_2_4 → c_1_22, an element of degree 2
- c_2_5 → c_1_02, an element of degree 2
- a_3_2 → 0, an element of degree 3
- a_5_14 → 0, an element of degree 5
- b_5_16 → 0, an element of degree 5
- b_5_18 → 0, an element of degree 5
- a_7_26 → 0, an element of degree 7
- c_8_41 → c_1_14·c_1_24 + c_1_18 + c_1_04·c_1_24 + c_1_06·c_1_22 + 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
- b_1_1 → c_1_2, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- b_2_4 → 0, an element of degree 2
- c_2_5 → c_1_0·c_1_2 + c_1_02, an element of degree 2
- a_3_2 → 0, an element of degree 3
- a_5_14 → 0, an element of degree 5
- b_5_16 → c_1_25 + c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
- b_5_18 → c_1_25 + c_1_12·c_1_23 + c_1_14·c_1_2 + c_1_02·c_1_23 + c_1_04·c_1_2, an element of degree 5
- a_7_26 → 0, an element of degree 7
- c_8_41 → c_1_14·c_1_24 + c_1_18 + c_1_0·c_1_27 + c_1_0·c_1_12·c_1_25
+ c_1_0·c_1_14·c_1_23 + c_1_02·c_1_12·c_1_24 + c_1_02·c_1_14·c_1_22 + c_1_03·c_1_25 + c_1_04·c_1_24 + c_1_05·c_1_23 + c_1_06·c_1_22 + c_1_08, an element of degree 8
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