Simon King
David J. Green
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Cohomology of group number 474 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 4.
- Its center has rank 2.
- It has a unique conjugacy class of maximal elementary abelian subgroups, which is 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) · (t6 − t5 − t4 + 2·t3 − 2·t2 + 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 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
- b_1_2, an element of degree 1
- a_2_3, a nilpotent element of degree 2
- a_2_4, a nilpotent element of degree 2
- b_2_5, an element of degree 2
- c_2_6, a Duflot regular element of degree 2
- b_3_11, an element of degree 3
- b_3_12, an element of degree 3
- a_5_5, a nilpotent element of degree 5
- b_5_25, an element of degree 5
- a_6_5, a nilpotent element of degree 6
- b_6_37, an element of degree 6
- c_8_75, a Duflot regular element of degree 8
Ring relations
There are 53 minimal relations of maximal degree 12:
- a_1_02
- a_1_12
- a_1_0·a_1_1
- a_2_3·a_1_0
- a_2_4·a_1_1
- a_2_4·a_1_0 + a_2_3·a_1_1
- a_1_1·b_1_22 + b_2_5·a_1_0
- a_2_32
- a_2_3·a_2_4
- a_2_42
- a_1_1·b_3_11 + a_2_3·b_2_5
- a_1_0·b_3_11 + a_2_3·b_1_22
- a_1_1·b_3_12 + a_2_4·b_2_5
- a_1_0·b_3_12 + a_2_4·b_1_22
- b_2_5·a_1_0·b_1_22 + b_2_52·a_1_0
- a_2_3·b_3_11
- a_2_4·b_3_11 + a_2_3·b_3_12
- b_2_52·a_1_1 + a_2_4·b_3_12 + c_2_6·a_1_0·b_1_22
- b_3_112 + b_2_5·b_1_24 + b_2_52·b_1_22 + a_2_3·b_1_24 + a_2_3·b_2_5·b_1_22
+ a_2_3·b_2_52
- b_3_122 + b_2_53 + a_2_4·b_2_5·b_1_22 + a_2_4·b_2_52 + c_2_6·b_1_24
- b_2_52·a_1_0·b_1_2 + a_2_4·b_1_2·b_3_12 + a_2_3·b_2_5·b_1_22 + a_2_3·b_2_52
+ a_1_1·a_5_5 + c_2_6·a_1_0·b_1_23
- a_1_0·a_5_5
- a_1_1·b_5_25
- a_1_0·b_5_25 + a_2_4·b_1_24 + a_2_4·b_2_5·b_1_22
- a_1_1·b_1_2·a_5_5 + a_2_3·a_5_5
- b_1_22·b_5_25 + b_1_24·b_3_12 + b_2_5·b_1_22·b_3_12 + b_2_5·b_1_22·b_3_11
+ b_2_52·b_3_11 + b_2_52·b_1_23 + b_2_53·b_1_2 + b_2_5·a_5_5 + a_2_4·b_1_22·b_3_12 + a_2_3·b_2_52·b_1_2 + a_1_1·b_1_2·a_5_5 + c_2_6·a_1_0·b_1_24 + b_2_52·c_2_6·a_1_0
- a_2_4·b_2_5·b_1_23 + a_2_4·b_2_52·b_1_2 + a_2_3·b_5_25 + a_2_3·b_1_22·b_3_12
+ a_2_3·b_2_5·b_3_12 + a_2_4·a_5_5
- a_2_4·b_5_25 + c_2_6·a_1_0·b_1_24 + b_2_52·c_2_6·a_1_0
- a_6_5·a_1_1 + a_2_4·a_5_5 + a_2_3·c_2_62·a_1_1
- a_6_5·a_1_0
- b_6_37·a_1_1
- b_6_37·a_1_0 + a_2_4·b_2_5·b_1_23 + a_2_4·b_2_52·b_1_2 + a_2_3·b_1_22·b_3_12
+ a_2_3·b_2_5·b_3_12 + a_2_4·a_5_5
- b_3_11·a_5_5 + a_6_5·b_1_22 + b_2_5·b_1_2·a_5_5 + a_2_4·b_1_26
+ a_2_4·b_2_5·b_1_2·b_3_12 + a_2_4·b_2_52·b_1_22 + a_2_3·b_1_23·b_3_12 + a_2_3·b_1_26 + a_2_3·b_2_52·b_1_22 + b_2_5·a_1_1·a_5_5 + c_2_6·a_1_0·b_1_25 + a_2_4·b_2_5·c_2_6·b_1_22 + a_2_3·c_2_6·b_1_24 + a_2_3·b_2_5·c_2_6·b_1_22 + a_2_3·c_2_62·b_1_22
- b_3_11·b_5_25 + b_1_22·b_3_11·b_3_12 + b_2_5·b_3_11·b_3_12 + b_2_5·b_1_2·b_5_25
+ b_2_5·b_1_23·b_3_12 + b_2_52·b_1_2·b_3_12 + b_2_52·b_1_24 + b_2_53·b_1_22 + b_2_5·a_6_5 + a_2_3·b_2_5·b_1_2·b_3_12 + a_2_3·b_2_52·b_1_22 + a_2_3·b_2_53 + a_2_4·c_2_6·b_1_2·b_3_12 + a_2_4·b_2_52·c_2_6 + a_2_3·b_2_5·c_2_6·b_1_22 + c_2_6·a_1_1·a_5_5 + c_2_62·a_1_0·b_1_23 + a_2_3·b_2_5·c_2_62
- a_2_4·b_1_2·a_5_5 + a_2_3·a_6_5
- b_2_5·a_1_1·a_5_5 + a_2_4·a_6_5
- b_1_22·b_3_11·b_3_12 + b_6_37·b_1_22 + b_2_5·b_3_11·b_3_12 + b_2_5·b_1_23·b_3_12
+ b_2_52·b_1_2·b_3_12 + b_2_52·b_1_24 + b_2_53·b_1_22 + b_3_12·a_5_5 + a_2_4·b_2_52·b_1_22 + a_2_3·b_1_23·b_3_12 + a_2_4·c_2_6·b_1_24
- b_3_12·b_5_25 + b_2_5·b_6_37 + c_2_6·b_1_26 + b_2_5·c_2_6·b_1_24
- a_2_3·b_6_37
- a_2_4·b_6_37 + a_2_3·c_2_6·b_1_24 + a_2_3·b_2_5·c_2_6·b_1_22
- a_6_5·b_3_11 + b_2_5·b_1_22·a_5_5 + b_2_5·a_6_5·b_1_2 + a_2_4·b_2_5·b_1_22·b_3_12
+ a_2_3·b_1_24·b_3_12 + a_2_3·b_2_5·b_1_22·b_3_12 + a_2_3·b_2_52·b_3_12 + a_2_3·b_2_53·b_1_2 + a_2_4·c_2_6·b_1_22·b_3_12 + a_2_4·b_2_52·c_2_6·b_1_2 + a_2_3·c_2_6·b_1_25 + a_2_3·b_2_5·c_2_6·b_3_12 + a_2_3·b_2_52·c_2_6·b_1_2 + c_2_62·a_1_0·b_1_24 + a_2_3·b_2_5·c_2_62·b_1_2
- b_6_37·b_3_12 + b_2_52·b_5_25 + c_2_6·b_1_24·b_3_11 + b_2_5·c_2_6·b_1_22·b_3_11
+ b_2_5·c_2_6·b_1_25 + b_2_52·c_2_6·b_1_23 + c_2_6·b_1_22·a_5_5 + a_2_4·c_2_6·b_1_22·b_3_12 + a_2_3·c_2_6·b_1_25
- b_6_37·b_3_11 + b_2_5·b_1_24·b_3_12 + b_2_5·b_6_37·b_1_2 + b_2_52·b_1_22·b_3_12
+ b_2_52·b_1_22·b_3_11 + b_2_53·b_3_11 + b_2_53·b_1_23 + b_2_54·b_1_2 + a_6_5·b_3_12 + a_2_3·b_2_5·b_1_22·b_3_12 + a_2_3·b_2_52·b_3_12 + a_2_3·b_2_53·b_1_2 + a_2_4·b_2_5·a_5_5 + c_2_6·a_1_0·b_1_26 + a_2_4·c_2_6·b_1_22·b_3_12 + a_2_4·c_2_6·b_1_25 + a_2_4·b_2_5·c_2_6·b_3_12 + a_2_3·c_2_6·b_1_25 + a_2_3·b_2_5·c_2_6·b_3_12 + c_2_62·a_1_0·b_1_24 + a_2_3·c_2_62·b_3_12
- a_2_3·b_2_53·b_1_22 + a_2_3·b_2_54 + a_5_52
- b_5_252 + c_2_6·b_1_28 + b_2_52·c_2_6·b_1_24
- a_5_5·b_5_25 + b_1_22·b_3_12·a_5_5 + b_2_5·b_3_12·a_5_5 + b_2_5·a_6_5·b_1_22
+ b_2_52·a_6_5 + a_2_3·b_2_53·b_1_22 + a_2_3·b_2_54 + a_2_3·b_2_5·a_6_5 + a_2_4·b_2_52·c_2_6·b_1_22 + a_2_4·b_2_53·c_2_6 + a_2_3·b_2_52·c_2_6·b_1_22 + a_2_3·b_2_53·c_2_6 + a_2_3·b_2_5·c_2_62·b_1_22 + a_2_3·b_2_52·c_2_62
- a_2_3·b_2_52·b_1_22·b_3_12 + a_2_3·b_2_53·b_3_12 + a_6_5·a_5_5
+ a_2_3·b_2_52·a_5_5 + a_2_4·b_2_5·c_2_6·a_5_5 + a_2_3·b_2_5·c_2_6·a_5_5 + a_2_3·c_2_62·a_5_5
- b_6_37·b_5_25 + c_2_6·b_1_26·b_3_11 + b_2_5·c_2_6·b_1_27
+ b_2_52·c_2_6·b_1_22·b_3_11 + b_2_53·c_2_6·b_1_23 + c_2_6·b_1_24·a_5_5 + b_2_5·c_2_6·b_1_22·a_5_5 + a_2_3·c_2_6·b_1_27 + a_2_3·b_2_53·c_2_6·b_1_2 + a_2_3·b_2_5·c_2_6·a_5_5 + c_2_62·a_1_0·b_1_26 + a_2_4·c_2_62·b_1_22·b_3_12 + c_2_63·a_1_0·b_1_24
- b_6_37·a_5_5 + a_6_5·b_5_25 + a_2_3·b_1_26·b_3_12 + a_2_3·b_2_52·b_1_22·b_3_12
+ c_2_6·a_1_0·b_1_28 + a_2_4·c_2_6·b_1_27 + a_2_4·b_2_5·c_2_6·b_1_22·b_3_12 + a_2_4·b_2_53·c_2_6·b_1_2 + a_2_3·c_2_6·b_1_24·b_3_12 + a_2_3·c_2_6·b_1_27 + a_2_3·b_2_52·c_2_6·b_3_12 + a_2_3·b_2_53·c_2_6·b_1_2 + a_2_4·b_2_5·c_2_6·a_5_5 + a_2_3·b_2_5·c_2_6·a_5_5 + a_2_4·c_2_62·b_1_22·b_3_12 + a_2_3·c_2_62·b_5_25 + c_2_63·a_1_0·b_1_24
- a_6_5·b_5_25 + a_6_5·b_1_22·b_3_12 + b_2_5·a_6_5·b_3_12 + b_2_52·b_1_22·a_5_5
+ b_2_53·a_5_5 + a_2_3·b_2_52·b_1_22·b_3_12 + a_2_3·b_2_53·b_3_12 + a_2_4·b_2_52·a_5_5 + a_2_3·b_2_52·a_5_5 + a_2_4·b_2_5·c_2_6·b_1_22·b_3_12 + a_2_4·b_2_52·c_2_6·b_3_12 + a_2_3·b_2_5·c_2_6·b_1_22·b_3_12 + a_2_3·b_2_52·c_2_6·b_3_12 + a_2_3·c_8_75·a_1_1 + a_2_3·c_2_62·b_5_25 + a_2_3·c_2_62·b_1_22·b_3_12 + a_2_3·b_2_5·c_2_62·b_3_12
- a_6_52 + b_2_5·a_5_52
- b_6_372 + b_2_5·c_2_6·b_1_28 + b_2_53·c_2_6·b_1_24 + a_2_3·c_2_6·b_1_28
+ a_2_3·b_2_54·c_2_6 + c_2_6·a_5_52
- a_6_5·b_6_37 + b_2_5·b_1_22·b_3_12·a_5_5 + b_2_52·b_3_12·a_5_5
+ b_2_52·a_6_5·b_1_22 + b_2_53·a_6_5 + b_2_5·a_5_52 + a_2_3·b_2_52·a_6_5 + a_2_4·b_2_53·c_2_6·b_1_22 + a_2_4·b_2_54·c_2_6 + a_2_3·c_2_6·b_1_25·b_3_12 + a_2_3·c_2_6·b_1_28 + a_2_3·b_2_52·c_2_6·b_1_2·b_3_12 + a_2_3·b_2_54·c_2_6 + a_2_3·b_2_5·c_2_6·a_6_5 + a_2_3·b_2_52·c_2_62·b_1_22 + a_2_3·b_2_53·c_2_62
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_6, a Duflot regular element of degree 2
- c_8_75, a Duflot regular element of degree 8
- b_1_24 + b_2_5·b_1_22 + b_2_52, an element of degree 4
- b_3_11, an element of degree 3
- The Raw Filter Degree Type of that HSOP is [-1, -1, 4, 10, 13].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -4].
- We found that there exists some filter regular HSOP formed by the first 2 terms of the above HSOP, together with 2 elements of degree 2.
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
- b_1_2 → 0, an element of degree 1
- a_2_3 → 0, an element of degree 2
- a_2_4 → 0, an element of degree 2
- b_2_5 → 0, an element of degree 2
- c_2_6 → c_1_02, an element of degree 2
- b_3_11 → 0, an element of degree 3
- b_3_12 → 0, an element of degree 3
- a_5_5 → 0, an element of degree 5
- b_5_25 → 0, an element of degree 5
- a_6_5 → 0, an element of degree 6
- b_6_37 → 0, an element of degree 6
- c_8_75 → 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
- b_1_2 → c_1_3, an element of degree 1
- a_2_3 → 0, an element of degree 2
- a_2_4 → 0, an element of degree 2
- b_2_5 → c_1_22, an element of degree 2
- c_2_6 → c_1_02, 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
- b_3_12 → c_1_23 + c_1_0·c_1_32, an element of degree 3
- a_5_5 → 0, an element of degree 5
- b_5_25 → c_1_0·c_1_34 + c_1_0·c_1_22·c_1_32, an element of degree 5
- a_6_5 → 0, an element of degree 6
- b_6_37 → c_1_0·c_1_2·c_1_34 + c_1_0·c_1_23·c_1_32, an element of degree 6
- c_8_75 → c_1_22·c_1_36 + c_1_24·c_1_34 + c_1_25·c_1_33 + c_1_26·c_1_32
+ c_1_12·c_1_22·c_1_34 + c_1_12·c_1_24·c_1_32 + c_1_14·c_1_34 + c_1_14·c_1_22·c_1_32 + c_1_14·c_1_24 + c_1_18 + c_1_0·c_1_2·c_1_36 + c_1_0·c_1_23·c_1_34 + c_1_0·c_1_24·c_1_33 + c_1_0·c_1_25·c_1_32 + c_1_02·c_1_22·c_1_34 + c_1_02·c_1_24·c_1_32 + c_1_02·c_1_26 + c_1_03·c_1_2·c_1_34 + c_1_03·c_1_23·c_1_32 + c_1_04·c_1_34 + c_1_06·c_1_22, an element of degree 8
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