Simon King
David J. Green
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Cohomology of group number 613 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 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) · (t5 − t3 + t2 + t + 1) |
| (t + 1) · (t − 1)3 · (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 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
- a_2_3, a nilpotent element of degree 2
- b_2_4, an element of degree 2
- b_2_5, an element of degree 2
- b_2_6, an element of degree 2
- b_5_19, an element of degree 5
- b_5_20, an element of degree 5
- b_8_36, an element of degree 8
- c_8_38, a Duflot regular element of degree 8
Ring relations
There are 29 minimal relations of maximal degree 16:
- a_1_02
- a_1_0·b_1_1
- b_1_1·b_1_2 + a_1_0·b_1_2
- a_2_3·a_1_0
- b_2_4·a_1_0 + a_2_3·b_1_2
- b_2_4·b_1_1 + a_2_3·b_1_2
- b_2_6·b_1_2 + b_2_6·b_1_1 + b_2_5·b_1_2 + b_2_6·a_1_0 + b_2_5·a_1_0
- a_2_32
- b_2_5·b_1_22 + b_2_42 + a_2_3·b_1_22
- b_2_5·a_1_0·b_1_2 + a_2_3·b_2_4
- b_2_5·b_2_6·b_1_1 + a_2_3·b_2_4·b_1_2
- a_1_0·b_5_19 + a_2_3·b_2_5·b_2_6
- b_1_1·b_5_19 + b_2_5·b_1_14 + b_2_52·b_1_12 + a_2_3·b_2_5·b_1_12 + a_2_3·b_2_42
- b_1_2·b_5_20 + b_2_4·b_2_5·b_2_6 + b_2_4·b_2_52 + b_2_42·b_1_22 + b_2_43
+ a_2_3·b_1_24 + a_2_3·b_2_4·b_2_5 + a_2_3·b_2_42
- b_2_4·b_2_5·b_2_6 + b_2_4·b_2_52 + a_1_0·b_5_20 + a_2_3·b_2_4·b_2_5
- b_2_5·b_2_62·a_1_0 + b_2_52·b_2_6·a_1_0 + a_2_3·b_5_19 + a_2_3·b_2_5·b_1_13
+ a_2_3·b_2_52·b_1_1 + a_2_3·b_2_4·b_2_5·b_1_2
- b_2_4·b_5_20 + b_2_42·b_2_5·b_1_2 + b_2_43·b_1_2 + b_2_5·b_2_62·a_1_0
+ b_2_52·b_2_6·a_1_0 + a_2_3·b_2_4·b_2_5·b_1_2
- b_2_4·b_2_6·b_5_19 + b_2_4·b_2_5·b_5_19 + a_2_3·b_2_6·b_5_20
+ a_2_3·b_2_42·b_2_5·b_1_2
- b_8_36·b_1_2 + b_2_4·b_2_6·b_5_19 + b_2_4·b_2_5·b_5_19 + b_2_42·b_1_25
+ b_2_44·b_1_2 + b_2_53·b_2_6·a_1_0 + b_2_54·a_1_0 + a_2_3·b_1_27 + a_2_3·b_2_63·b_1_1 + a_2_3·b_2_42·b_2_5·b_1_2 + a_2_3·b_2_43·b_1_2
- b_2_4·b_2_6·b_5_19 + b_2_4·b_2_5·b_5_19 + b_8_36·a_1_0 + b_2_53·b_2_6·a_1_0
+ b_2_54·a_1_0 + a_2_3·b_2_63·b_1_1 + a_2_3·b_2_43·b_1_2
- b_1_19 + b_8_36·b_1_1 + b_2_53·b_1_13 + b_2_54·b_1_1 + a_2_3·b_1_17
+ a_2_3·b_2_5·b_1_15 + a_2_3·b_2_53·b_1_1 + a_2_3·b_2_43·b_1_2
- b_5_192 + b_2_52·b_1_16 + b_2_52·b_2_63 + b_2_53·b_2_62 + b_2_54·b_1_12
+ b_2_42·b_2_53 + b_2_44·b_1_22 + b_2_44·b_2_5 + a_2_3·b_2_43·b_2_5
- b_5_202 + b_1_15·b_5_20 + b_8_36·b_1_12 + b_2_5·b_1_13·b_5_20 + b_2_5·b_1_18
+ b_2_52·b_1_1·b_5_20 + b_2_52·b_2_63 + b_2_53·b_1_14 + b_2_54·b_1_12 + b_2_54·b_2_6 + b_2_44·b_1_22 + b_2_44·b_2_5 + a_2_3·b_1_18 + a_2_3·b_2_43·b_2_5 + c_8_38·b_1_12
- b_5_19·b_5_20 + b_2_5·b_1_13·b_5_20 + b_2_5·b_1_18 + b_2_5·b_8_36
+ b_2_52·b_1_1·b_5_20 + b_2_54·b_1_12 + b_2_54·b_2_6 + b_2_55 + b_2_4·b_2_5·b_1_2·b_5_19 + b_2_42·b_1_2·b_5_19 + b_2_44·b_1_22 + b_2_44·b_2_5 + b_2_62·a_1_0·b_5_20 + b_2_52·a_1_0·b_5_20 + a_2_3·b_2_5·b_1_1·b_5_20 + a_2_3·b_2_5·b_1_16 + a_2_3·b_2_5·b_2_63 + a_2_3·b_2_52·b_1_14 + a_2_3·b_2_52·b_2_62 + a_2_3·b_2_53·b_2_6 + a_2_3·b_2_54 + a_2_3·b_2_43·b_2_5
- b_2_4·b_8_36 + b_2_43·b_1_24 + b_2_45 + b_2_52·a_1_0·b_5_20 + a_2_3·b_2_5·b_2_63
+ a_2_3·b_2_52·b_2_62 + a_2_3·b_2_43·b_2_5 + a_2_3·b_2_44
- b_2_62·a_1_0·b_5_20 + b_2_5·b_2_6·a_1_0·b_5_20 + a_2_3·b_1_18 + a_2_3·b_8_36
+ a_2_3·b_2_53·b_1_12 + a_2_3·b_2_53·b_2_6 + a_2_3·b_2_54
- b_1_18·b_5_20 + b_8_36·b_5_20 + b_2_5·b_2_63·b_5_19 + b_2_53·b_1_12·b_5_20
+ b_2_53·b_2_6·b_5_20 + b_2_53·b_2_6·b_5_19 + b_2_54·b_5_20 + b_2_44·b_1_25 + b_2_45·b_1_23 + b_2_45·b_2_5·b_1_2 + b_2_46·b_1_2 + a_2_3·b_1_16·b_5_20 + a_2_3·b_2_63·b_5_20 + a_2_3·b_2_63·b_5_19 + a_2_3·b_2_5·b_1_14·b_5_20 + a_2_3·b_2_5·b_2_62·b_5_20 + a_2_3·b_2_5·b_2_62·b_5_19 + a_2_3·b_2_52·b_2_6·b_5_20 + a_2_3·b_2_52·b_2_6·b_5_19 + a_2_3·b_2_53·b_5_20 + a_2_3·b_2_53·b_5_19 + a_2_3·b_2_54·b_1_13 + a_2_3·b_2_55·b_1_1 + a_2_3·b_2_44·b_2_5·b_1_2 + a_2_3·b_2_6·c_8_38·b_1_1
- b_8_36·b_5_19 + b_2_5·b_8_36·b_1_13 + b_2_5·b_2_63·b_5_20 + b_2_52·b_8_36·b_1_1
+ b_2_52·b_2_62·b_5_20 + b_2_53·b_2_6·b_5_19 + b_2_54·b_5_19 + b_2_55·b_1_13 + b_2_56·b_1_1 + b_2_42·b_1_24·b_5_19 + b_2_44·b_5_19 + a_2_3·b_2_63·b_5_20 + a_2_3·b_2_63·b_5_19 + a_2_3·b_2_5·b_8_36·b_1_1 + a_2_3·b_2_5·b_2_62·b_5_19 + a_2_3·b_2_52·b_2_6·b_5_20 + a_2_3·b_2_52·b_2_6·b_5_19 + a_2_3·b_2_53·b_5_19 + a_2_3·b_2_54·b_1_13 + a_2_3·b_2_6·c_8_38·b_1_1
- b_8_36·b_1_18 + b_8_362 + b_2_52·b_2_66 + b_2_53·b_8_36·b_1_12
+ b_2_53·b_2_65 + b_2_54·b_1_18 + b_2_54·b_2_64 + b_2_55·b_2_63 + b_2_56·b_2_62 + b_2_57·b_1_12 + b_2_58 + b_2_44·b_1_28 + b_2_48 + a_2_3·b_8_36·b_1_16 + a_2_3·b_2_5·b_8_36·b_1_14 + a_2_3·b_2_53·b_1_18 + a_2_3·b_2_54·b_1_16 + a_2_3·b_2_55·b_1_14 + a_2_3·b_2_56·b_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_8_38, a Duflot regular element of degree 8
- b_1_24 + b_1_14 + b_2_62 + b_2_5·b_2_6 + b_2_52 + b_2_4·b_2_6 + b_2_4·b_2_5 + b_2_42, an element of degree 4
- b_1_2·b_5_19 + b_2_5·b_2_62 + b_2_52·b_1_12 + b_2_52·b_2_6 + b_2_4·b_2_62
+ b_2_4·b_2_52 + b_2_43, an element of degree 6
- The Raw Filter Degree Type of that HSOP is [-1, -1, 8, 15].
- 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
- 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
- a_2_3 → 0, an element of degree 2
- b_2_4 → 0, an element of degree 2
- b_2_5 → 0, an element of degree 2
- b_2_6 → 0, an element of degree 2
- b_5_19 → 0, an element of degree 5
- b_5_20 → 0, an element of degree 5
- b_8_36 → 0, an element of degree 8
- c_8_38 → 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 → c_1_1, an element of degree 1
- a_2_3 → 0, an element of degree 2
- b_2_4 → c_1_1·c_1_2, an element of degree 2
- b_2_5 → c_1_22, an element of degree 2
- b_2_6 → c_1_22, an element of degree 2
- b_5_19 → c_1_1·c_1_24 + c_1_12·c_1_23 + c_1_13·c_1_22, an element of degree 5
- b_5_20 → c_1_12·c_1_23 + c_1_13·c_1_22, an element of degree 5
- b_8_36 → c_1_14·c_1_24 + c_1_16·c_1_22, an element of degree 8
- c_8_38 → c_1_12·c_1_26 + c_1_15·c_1_23 + 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
- 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
- a_2_3 → 0, an element of degree 2
- b_2_4 → 0, an element of degree 2
- b_2_5 → c_1_22 + c_1_12, an element of degree 2
- b_2_6 → c_1_12, an element of degree 2
- b_5_19 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
- b_5_20 → c_1_1·c_1_24 + c_1_13·c_1_22, an element of degree 5
- b_8_36 → c_1_28 + 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, an element of degree 8
- c_8_38 → c_1_12·c_1_26 + c_1_14·c_1_24 + 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
- a_1_0 → 0, an element of degree 1
- b_1_1 → c_1_1, an element of degree 1
- b_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
- b_2_5 → c_1_22 + c_1_1·c_1_2, an element of degree 2
- b_2_6 → 0, an element of degree 2
- b_5_19 → c_1_1·c_1_24 + c_1_14·c_1_2, an element of degree 5
- b_5_20 → c_1_1·c_1_24 + c_1_13·c_1_22 + c_1_02·c_1_13 + c_1_04·c_1_1, an element of degree 5
- b_8_36 → c_1_28 + c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_15·c_1_23 + c_1_18, an element of degree 8
- c_8_38 → c_1_12·c_1_26 + c_1_13·c_1_25 + c_1_15·c_1_23 + c_1_17·c_1_2 + c_1_18
+ c_1_02·c_1_12·c_1_24 + c_1_02·c_1_15·c_1_2 + c_1_02·c_1_16 + c_1_04·c_1_24 + c_1_04·c_1_13·c_1_2 + c_1_08, an element of degree 8
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