Simon King
David J. Green
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Cohomology of group number 857 of order 128
General information on the group
- The group has 3 minimal generators and exponent 4.
- It is non-abelian.
- It has p-Rank 4.
- Its center has rank 2.
- It has 2 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 4.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 4 and depth 2.
- The depth coincides with the Duflot bound.
- The Poincaré series is
( − 1) · (t7 − t6 − t5 + t3 − t2 − 1) |
| (t + 1) · (t − 1)4 · (t2 + 1) · (t4 + 1) |
- The a-invariants are -∞,-∞,-6,-4,-4. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 13 minimal generators of maximal degree 8:
- a_1_0, a nilpotent element of degree 1
- a_1_1, a nilpotent element of degree 1
- c_1_2, a Duflot regular element of degree 1
- 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_3_11, an element of degree 3
- a_5_21, a nilpotent element of degree 5
- b_5_25, an element of degree 5
- b_6_34, an element of degree 6
- b_6_36, an element of degree 6
- b_7_49, an element of degree 7
- c_8_65, a Duflot regular element of degree 8
Ring relations
There are 44 minimal relations of maximal degree 14:
- a_1_02
- a_1_0·a_1_1
- b_2_5·a_1_1 + b_2_4·a_1_1
- b_2_5·a_1_0 + b_2_4·a_1_1
- b_2_6·a_1_0 + a_1_13
- b_2_52 + b_2_4·b_2_5
- a_1_1·b_3_11
- a_1_0·b_3_11
- b_2_6·a_1_13
- b_3_112 + b_2_4·b_2_62
- a_1_1·a_5_21 + b_2_62·a_1_12
- a_1_0·a_5_21
- a_1_0·b_5_25
- b_2_6·a_5_21 + b_2_63·a_1_1 + a_1_12·b_5_25
- b_2_5·b_5_25 + b_2_4·b_2_5·b_3_11 + b_2_5·a_5_21 + b_2_43·a_1_1
- b_2_5·b_2_6·b_3_11 + b_2_4·b_5_25 + b_2_4·b_2_6·b_3_11 + b_2_4·b_2_5·b_3_11
+ b_2_5·a_5_21 + b_2_43·a_1_1
- b_6_34·a_1_1 + b_2_6·a_5_21 + b_2_63·a_1_1 + b_2_5·a_5_21 + b_2_43·a_1_1
- b_6_34·a_1_0 + b_2_4·a_5_21 + b_2_43·a_1_1
- b_6_36·a_1_1 + b_2_63·a_1_1 + b_2_5·a_5_21 + b_2_43·a_1_1
- b_6_36·a_1_0 + b_2_5·a_5_21 + b_2_43·a_1_1
- b_3_11·a_5_21
- b_3_11·b_5_25 + b_2_5·b_2_63 + b_2_4·b_2_63 + b_2_4·b_2_5·b_2_62
- b_2_5·b_6_36 + b_2_5·b_6_34 + b_2_5·b_2_63
- b_2_5·b_6_34 + b_2_4·b_6_36 + b_2_4·b_2_63
- a_1_1·b_7_49 + b_2_6·a_1_1·b_5_25 + b_2_63·a_1_12
- a_1_0·b_7_49
- b_6_36·b_3_11 + b_2_63·b_3_11 + b_2_5·b_7_49 + b_2_4·b_2_6·b_5_25
+ b_2_4·b_2_62·b_3_11 + b_2_42·b_5_25 + b_2_42·b_2_6·b_3_11 + b_2_4·b_2_5·a_5_21 + b_2_44·a_1_1
- b_6_34·b_3_11 + b_2_4·b_7_49 + b_2_4·b_2_6·b_5_25 + b_2_4·b_2_62·b_3_11
+ b_2_42·b_2_6·b_3_11 + b_2_43·b_3_11 + b_2_4·b_2_5·a_5_21 + b_2_42·a_5_21
- a_5_212 + b_2_64·a_1_12
- a_5_21·b_5_25 + b_2_62·a_1_1·b_5_25
- b_3_11·b_7_49 + b_2_62·b_6_34 + b_2_5·b_2_64 + b_2_4·b_2_5·b_2_63
+ b_2_42·b_2_63 + b_2_43·b_2_62 + b_2_62·a_1_1·b_5_25 + b_2_64·a_1_12
- b_5_252 + b_2_5·b_2_64 + b_2_4·b_2_64 + b_2_42·b_2_5·b_2_62
+ b_2_62·a_1_1·b_5_25 + c_8_65·a_1_12
- b_6_36·b_5_25 + b_2_63·b_5_25 + b_2_4·b_2_5·b_7_49 + b_2_42·b_2_6·b_5_25
+ b_2_42·b_2_62·b_3_11 + b_2_43·b_5_25 + b_2_43·b_2_6·b_3_11 + b_6_36·a_5_21 + b_2_65·a_1_1
- b_6_36·b_5_25 + b_6_34·b_5_25 + b_2_63·b_5_25 + b_2_5·b_2_6·b_7_49 + b_2_4·b_2_6·b_7_49
+ b_2_42·b_2_6·b_5_25 + b_2_43·b_2_6·b_3_11 + b_2_62·a_1_12·b_5_25 + c_8_65·a_1_13
- b_6_36·a_5_21 + b_2_65·a_1_1 + b_2_42·b_2_5·a_5_21 + b_2_62·a_1_12·b_5_25
+ b_2_4·c_8_65·a_1_1
- b_6_34·a_5_21 + b_2_43·a_5_21 + b_2_62·a_1_12·b_5_25 + b_2_4·c_8_65·a_1_0
- b_6_362 + b_6_34·b_6_36 + b_2_63·b_6_34 + b_2_66
- a_5_21·b_7_49 + b_2_63·a_1_1·b_5_25 + b_2_65·a_1_12
- b_6_362 + b_2_66 + b_2_42·b_2_6·b_6_36 + b_2_42·b_2_64 + b_2_42·b_2_5·b_2_63
+ b_2_43·b_2_5·b_2_62 + b_2_45·b_2_5 + b_2_4·b_2_5·c_8_65
- b_6_342 + b_2_42·b_2_6·b_6_34 + b_2_42·b_2_5·b_2_63 + b_2_43·b_6_36
+ b_2_43·b_6_34 + b_2_43·b_2_63 + b_2_43·b_2_5·b_2_62 + b_2_44·b_2_5·b_2_6 + b_2_45·b_2_6 + b_2_45·b_2_5 + b_2_42·c_8_65
- b_5_25·b_7_49 + b_2_63·b_6_36 + b_2_63·b_6_34 + b_2_66 + b_2_4·b_2_62·b_6_36
+ b_2_4·b_2_65 + b_2_42·b_2_64 + b_2_42·b_2_5·b_2_63 + b_2_43·b_2_63 + b_2_43·b_2_5·b_2_62 + b_2_63·a_1_1·b_5_25 + b_2_6·c_8_65·a_1_12
- b_6_36·b_7_49 + b_2_63·b_7_49 + b_2_5·b_2_62·b_7_49 + b_2_4·b_2_63·b_5_25
+ b_2_4·b_2_64·b_3_11 + b_2_4·b_2_5·b_2_6·b_7_49 + b_2_42·b_2_62·b_5_25 + b_2_42·b_2_63·b_3_11 + b_2_42·b_2_5·b_7_49 + b_2_44·b_5_25 + b_2_44·b_2_6·b_3_11 + b_2_44·b_2_5·b_3_11 + b_2_43·b_2_5·a_5_21 + b_2_46·a_1_1 + b_2_63·a_1_12·b_5_25 + b_2_5·c_8_65·b_3_11
- b_6_34·b_7_49 + b_2_5·b_2_62·b_7_49 + b_2_4·b_2_63·b_5_25 + b_2_4·b_2_64·b_3_11
+ b_2_4·b_2_5·b_2_6·b_7_49 + b_2_42·b_2_62·b_5_25 + b_2_42·b_2_63·b_3_11 + b_2_42·b_2_5·b_7_49 + b_2_44·b_2_6·b_3_11 + b_2_43·b_2_5·a_5_21 + b_2_44·a_5_21 + b_2_63·a_1_12·b_5_25 + b_2_4·c_8_65·b_3_11 + b_2_42·c_8_65·a_1_1 + b_2_42·c_8_65·a_1_0
- b_7_492 + b_2_5·b_2_66 + b_2_4·b_2_63·b_6_34 + b_2_4·b_2_5·b_2_65
+ b_2_42·b_2_62·b_6_36 + b_2_42·b_2_62·b_6_34 + b_2_42·b_2_65 + b_2_43·b_2_64 + b_2_43·b_2_5·b_2_63 + b_2_44·b_2_63 + b_2_44·b_2_5·b_2_62 + b_2_45·b_2_62 + b_2_64·a_1_1·b_5_25 + b_2_66·a_1_12 + b_2_4·b_2_62·c_8_65 + b_2_62·c_8_65·a_1_12
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_1_2, a Duflot regular element of degree 1
- c_8_65, a Duflot regular element of degree 8
- b_2_6 + b_2_4, an element of degree 2
- b_3_11, an element of degree 3
- The Raw Filter Degree Type of that HSOP is [-1, -1, 3, 7, 10].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
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
- c_1_2 → c_1_0, an element of degree 1
- 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_3_11 → 0, an element of degree 3
- a_5_21 → 0, an element of degree 5
- b_5_25 → 0, an element of degree 5
- b_6_34 → 0, an element of degree 6
- b_6_36 → 0, an element of degree 6
- b_7_49 → 0, an element of degree 7
- c_8_65 → c_1_18, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- c_1_2 → c_1_0, an element of degree 1
- b_2_4 → c_1_22, an element of degree 2
- b_2_5 → 0, an element of degree 2
- b_2_6 → c_1_32 + c_1_2·c_1_3, an element of degree 2
- b_3_11 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
- a_5_21 → 0, an element of degree 5
- b_5_25 → c_1_2·c_1_34 + c_1_23·c_1_32, an element of degree 5
- b_6_34 → c_1_22·c_1_34 + c_1_25·c_1_3 + c_1_1·c_1_23·c_1_32 + c_1_1·c_1_24·c_1_3
+ c_1_12·c_1_22·c_1_32 + c_1_12·c_1_23·c_1_3 + c_1_12·c_1_24 + c_1_14·c_1_22, an element of degree 6
- b_6_36 → c_1_36 + c_1_2·c_1_35 + c_1_22·c_1_34 + c_1_23·c_1_33, an element of degree 6
- b_7_49 → c_1_2·c_1_36 + c_1_22·c_1_35 + c_1_23·c_1_34 + c_1_24·c_1_33
+ c_1_25·c_1_32 + c_1_26·c_1_3 + c_1_1·c_1_22·c_1_34 + c_1_1·c_1_24·c_1_32 + c_1_12·c_1_2·c_1_34 + c_1_12·c_1_24·c_1_3 + c_1_14·c_1_2·c_1_32 + c_1_14·c_1_22·c_1_3, an element of degree 7
- c_8_65 → c_1_38 + c_1_22·c_1_36 + c_1_23·c_1_35 + c_1_24·c_1_34 + c_1_25·c_1_33
+ c_1_26·c_1_32 + c_1_1·c_1_23·c_1_34 + c_1_1·c_1_26·c_1_3 + c_1_12·c_1_26 + c_1_14·c_1_34 + c_1_14·c_1_23·c_1_3 + c_1_18, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- c_1_2 → c_1_0, an element of degree 1
- b_2_4 → c_1_32, an element of degree 2
- b_2_5 → c_1_32, an element of degree 2
- b_2_6 → c_1_2·c_1_3 + c_1_22, an element of degree 2
- b_3_11 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
- a_5_21 → 0, an element of degree 5
- b_5_25 → c_1_2·c_1_34 + c_1_22·c_1_33, an element of degree 5
- b_6_34 → c_1_2·c_1_35 + c_1_24·c_1_32 + c_1_1·c_1_2·c_1_34 + c_1_1·c_1_22·c_1_33
+ c_1_12·c_1_34 + c_1_12·c_1_2·c_1_33 + c_1_12·c_1_22·c_1_32 + c_1_14·c_1_32, an element of degree 6
- b_6_36 → c_1_2·c_1_35 + c_1_23·c_1_33 + c_1_25·c_1_3 + c_1_26 + c_1_1·c_1_2·c_1_34
+ c_1_1·c_1_22·c_1_33 + c_1_12·c_1_34 + c_1_12·c_1_2·c_1_33 + c_1_12·c_1_22·c_1_32 + c_1_14·c_1_32, an element of degree 6
- b_7_49 → c_1_2·c_1_36 + c_1_24·c_1_33 + c_1_1·c_1_22·c_1_34 + c_1_1·c_1_24·c_1_32
+ c_1_12·c_1_2·c_1_34 + c_1_12·c_1_24·c_1_3 + c_1_14·c_1_2·c_1_32 + c_1_14·c_1_22·c_1_3, an element of degree 7
- c_8_65 → c_1_38 + c_1_22·c_1_36 + c_1_28 + c_1_1·c_1_22·c_1_35 + c_1_1·c_1_24·c_1_33
+ c_1_12·c_1_2·c_1_35 + c_1_12·c_1_22·c_1_34 + c_1_14·c_1_34 + c_1_14·c_1_2·c_1_33 + c_1_14·c_1_24 + c_1_18, an element of degree 8
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