Simon King
David J. Green
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Cohomology of group number 471 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 5.
- Its center has rank 3.
- It has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 5.
Structure of the cohomology ring
General information
- The cohomology ring is of dimension 5 and depth 4.
- The depth exceeds the Duflot bound, which is 3.
- The Poincaré series is
t3 − 2·t2 + t − 1 |
| (t + 1) · (t − 1)5 · (t2 + 1) |
- The a-invariants are -∞,-∞,-∞,-∞,-5,-5. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 14 minimal generators of maximal degree 4:
- 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
- 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
- c_2_7, a Duflot regular element of degree 2
- b_3_13, an element of degree 3
- b_3_14, an element of degree 3
- b_3_15, an element of degree 3
- b_3_16, an element of degree 3
- b_4_26, an element of degree 4
- c_4_31, a Duflot regular element of degree 4
Ring relations
There are 44 minimal relations of maximal degree 8:
- a_1_02
- a_1_12
- a_1_0·a_1_1
- a_2_3·a_1_1
- a_2_3·a_1_0
- 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 + b_2_4·a_1_1
- a_2_32
- b_2_52 + b_2_4·b_2_6
- a_1_1·b_3_13 + a_2_3·b_2_5
- a_1_0·b_3_13 + a_2_3·b_2_4
- a_1_1·b_3_14
- a_1_0·b_3_14
- a_1_1·b_3_15
- a_1_0·b_3_15 + a_2_3·b_2_5 + a_2_3·b_2_4
- a_1_1·b_3_16 + a_2_3·b_2_6
- a_1_0·b_3_16 + a_2_3·b_2_5
- b_2_42·a_1_1 + a_2_3·b_3_13 + b_2_4·c_2_7·a_1_0
- a_2_3·b_3_14
- b_2_6·b_3_14 + b_2_6·b_3_13 + b_2_5·b_3_15 + b_2_5·b_3_13 + b_2_42·a_1_1
- b_2_5·b_3_14 + b_2_5·b_3_13 + b_2_4·b_3_15 + b_2_4·b_3_13 + b_2_42·a_1_1
- a_2_3·b_3_15 + b_2_4·c_2_7·a_1_1 + b_2_4·c_2_7·a_1_0
- b_2_6·b_3_13 + b_2_5·b_3_16
- b_2_5·b_3_13 + b_2_4·b_3_16
- b_2_62·a_1_1 + a_2_3·b_3_16 + b_2_4·c_2_7·a_1_1
- b_4_26·a_1_1 + b_2_62·a_1_1 + b_2_42·a_1_1
- b_4_26·a_1_0
- b_3_132 + b_2_4·b_2_62 + b_2_42·c_2_7
- b_3_142 + b_2_4·b_2_62 + b_2_42·b_2_6
- b_3_14·b_3_15 + b_3_13·b_3_15 + b_3_13·b_3_14 + b_2_4·b_2_62 + b_2_4·b_2_5·b_2_6
+ a_2_3·b_2_4·b_2_5 + b_2_4·b_2_5·c_2_7 + b_2_42·c_2_7
- b_3_152 + b_2_4·b_2_6·c_2_7 + b_2_42·c_2_7
- b_3_162 + b_2_63 + a_2_3·b_2_62 + a_2_3·b_2_4·b_2_5 + b_2_4·b_2_6·c_2_7
- b_3_13·b_3_16 + b_2_5·b_2_62 + b_2_4·b_2_5·c_2_7
- b_3_14·b_3_16 + b_3_13·b_3_15 + b_2_5·b_2_62 + b_2_4·b_2_62 + a_2_3·b_2_4·b_2_5
+ b_2_4·b_2_5·c_2_7 + b_2_42·c_2_7
- b_3_13·b_3_15 + b_2_5·b_4_26 + b_2_5·b_2_62 + b_2_4·b_2_5·b_2_6 + b_2_4·b_2_5·c_2_7
+ b_2_42·c_2_7
- b_3_13·b_3_14 + b_2_4·b_4_26 + b_2_4·b_2_5·b_2_6 + b_2_42·b_2_6 + a_2_3·b_2_4·b_2_5
- b_3_15·b_3_16 + b_2_6·b_4_26 + b_2_63 + b_2_4·b_2_62 + b_2_4·b_2_6·c_2_7
+ b_2_4·b_2_5·c_2_7
- a_2_3·b_4_26 + a_2_3·b_2_62 + a_2_3·b_2_4·b_2_5
- b_4_26·b_3_16 + b_2_62·b_3_16 + b_2_62·b_3_15 + b_2_4·b_2_6·b_3_16
+ b_2_4·c_2_7·b_3_16 + b_2_4·c_2_7·b_3_15 + b_2_4·c_2_7·b_3_13
- b_4_26·b_3_13 + b_2_5·b_2_6·b_3_16 + b_2_5·b_2_6·b_3_15 + b_2_4·b_2_5·b_3_16
+ b_2_4·c_2_7·b_3_14 + a_2_3·c_2_7·b_3_13 + b_2_4·c_2_72·a_1_0
- b_4_26·b_3_14 + b_2_5·b_2_6·b_3_16 + b_2_4·b_2_6·b_3_16 + b_2_4·b_2_6·b_3_15
+ b_2_4·b_2_5·b_3_16 + b_2_4·b_2_5·b_3_15 + b_2_42·b_3_16
- b_4_26·b_3_15 + b_2_62·b_3_15 + b_2_4·b_2_6·b_3_15 + b_2_4·c_2_7·b_3_16
+ b_2_4·c_2_7·b_3_15 + b_2_4·c_2_7·b_3_14 + b_2_4·c_2_7·b_3_13 + a_2_3·c_2_7·b_3_13 + b_2_4·c_2_72·a_1_0
- b_4_262 + b_2_64 + b_2_42·b_2_62 + b_2_4·b_2_62·c_2_7 + b_2_42·b_2_6·c_2_7
Data used for Benson′s test
- Benson′s completion test succeeded in degree 8.
- 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_2_7, a Duflot regular element of degree 2
- c_4_31, a Duflot regular element of degree 4
- b_2_6 + b_2_5 + b_2_4, an element of degree 2
- b_3_14, an element of degree 3
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, -1, 4, 7].
- The filter degree type of any filter regular HSOP is [-1, -2, -3, -4, -5, -5].
Restriction maps
Restriction map to the greatest central el. ab. subgp., which is of rank 3
- 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
- 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
- c_2_7 → c_1_12, an element of degree 2
- b_3_13 → 0, an element of degree 3
- b_3_14 → 0, an element of degree 3
- b_3_15 → 0, an element of degree 3
- b_3_16 → 0, an element of degree 3
- b_4_26 → 0, an element of degree 4
- c_4_31 → c_1_24, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 5
- 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
- a_2_3 → 0, an element of degree 2
- b_2_4 → c_1_42, an element of degree 2
- b_2_5 → c_1_3·c_1_4, an element of degree 2
- b_2_6 → c_1_32, an element of degree 2
- c_2_7 → c_1_12, an element of degree 2
- b_3_13 → c_1_32·c_1_4 + c_1_1·c_1_42, an element of degree 3
- b_3_14 → c_1_3·c_1_42 + c_1_32·c_1_4, an element of degree 3
- b_3_15 → c_1_1·c_1_42 + c_1_1·c_1_3·c_1_4, an element of degree 3
- b_3_16 → c_1_33 + c_1_1·c_1_3·c_1_4, an element of degree 3
- b_4_26 → c_1_32·c_1_42 + c_1_34 + c_1_1·c_1_3·c_1_42 + c_1_1·c_1_32·c_1_4, an element of degree 4
- c_4_31 → c_1_3·c_1_43 + c_1_33·c_1_4 + c_1_2·c_1_3·c_1_42 + c_1_2·c_1_32·c_1_4
+ c_1_22·c_1_42 + c_1_22·c_1_3·c_1_4 + c_1_22·c_1_32 + c_1_24, an element of degree 4
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