Simon King
David J. Green
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Cohomology of group number 1679 of order 128
General information on the group
- The group has 4 minimal generators and exponent 8.
- It is non-abelian.
- It has p-Rank 3.
- Its center has rank 2.
- It has 2 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 coincides with the Duflot bound.
- The Poincaré series is
( − 1) · (t6 + t2 + t + 1) |
| (t − 1)3 · (t2 + 1) · (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 8 minimal generators of maximal degree 8:
- a_1_1, a nilpotent element of degree 1
- a_1_0, a nilpotent element of degree 1
- b_1_2, an element of degree 1
- b_1_3, an element of degree 1
- c_2_8, a Duflot regular element of degree 2
- b_5_26, an element of degree 5
- a_6_28, a nilpotent element of degree 6
- c_8_50, a Duflot regular element of degree 8
Ring relations
There are 11 minimal relations of maximal degree 12:
- a_1_02
- a_1_1·b_1_2 + a_1_1·a_1_0
- b_1_2·b_1_32 + a_1_0·b_1_2·b_1_3 + a_1_1·a_1_0·b_1_3 + a_1_13
- a_1_13·b_1_32 + a_1_13·a_1_0·b_1_3 + a_1_14·b_1_3 + a_1_15
- a_1_1·b_5_26 + c_2_8·a_1_12·b_1_32 + c_2_8·a_1_13·b_1_3
- b_1_2·b_1_3·b_5_26 + a_1_0·b_1_2·b_5_26 + a_1_0·b_1_25·b_1_3 + a_6_28·b_1_2
+ c_2_8·a_1_0·b_1_23·b_1_3 + c_2_8·a_1_15 + c_2_82·b_1_22·b_1_3 + c_2_82·a_1_0·b_1_2·b_1_3 + c_2_82·a_1_1·a_1_0·b_1_3
- a_1_0·b_1_3·b_5_26 + a_6_28·a_1_0 + c_2_8·a_1_13·a_1_0·b_1_3 + c_2_8·a_1_15
+ c_2_82·a_1_0·b_1_2·b_1_3 + c_2_82·a_1_1·a_1_0·b_1_3
- a_6_28·a_1_0·b_1_3 + a_6_28·a_1_12 + c_2_8·a_1_15·b_1_3
+ c_2_82·a_1_12·a_1_0·b_1_3 + c_2_82·a_1_13·b_1_3 + c_2_82·a_1_13·a_1_0 + c_2_82·a_1_14
- b_5_262 + b_1_25·b_5_26 + a_1_0·b_1_24·b_5_26 + a_6_28·a_1_0·b_1_23
+ c_8_50·b_1_22 + c_2_8·a_6_28·b_1_22 + c_2_82·b_1_25·b_1_3 + c_2_82·a_1_12·b_1_34 + c_2_82·a_1_15·b_1_3 + c_2_83·b_1_23·b_1_3 + c_2_83·a_1_13·a_1_0
- a_1_0·b_1_29·b_1_3 + a_6_28·b_5_26 + a_6_28·b_1_25 + c_8_50·b_1_22·b_1_3
+ c_8_50·a_1_0·b_1_22 + c_2_8·a_1_0·b_1_27·b_1_3 + c_2_8·a_6_28·a_1_1·b_1_32 + c_2_82·b_1_26·b_1_3 + c_2_82·a_1_0·b_1_2·b_5_26 + c_2_82·a_6_28·b_1_2 + c_2_82·a_6_28·a_1_0 + c_2_83·a_1_0·b_1_23·b_1_3 + c_2_83·a_1_13·a_1_0·b_1_3 + c_2_83·a_1_14·b_1_3 + c_2_83·a_1_15 + c_2_84·b_1_22·b_1_3
- a_6_28·a_1_0·b_1_25 + a_6_282 + c_8_50·a_1_0·b_1_22·b_1_3
+ c_2_8·a_1_12·b_1_38 + c_8_50·a_1_13·a_1_0 + c_2_82·a_1_0·b_1_26·b_1_3 + c_2_82·a_1_12·b_1_36 + c_2_84·a_1_0·b_1_22·b_1_3 + c_2_84·a_1_12·b_1_32 + c_2_84·a_1_13·a_1_0
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_8, a Duflot regular element of degree 2
- c_8_50, a Duflot regular element of degree 8
- b_1_32 + b_1_22, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 6, 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_1 → 0, an element of degree 1
- a_1_0 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_1_3 → 0, an element of degree 1
- c_2_8 → c_1_02, an element of degree 2
- b_5_26 → 0, an element of degree 5
- a_6_28 → 0, an element of degree 6
- c_8_50 → c_1_18, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_1 → 0, an element of degree 1
- a_1_0 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- b_1_3 → c_1_2, an element of degree 1
- c_2_8 → c_1_02, an element of degree 2
- b_5_26 → 0, an element of degree 5
- a_6_28 → 0, an element of degree 6
- c_8_50 → c_1_14·c_1_24 + c_1_18 + c_1_06·c_1_22, an element of degree 8
Restriction map to a maximal el. ab. subgp. of rank 3
- a_1_1 → 0, an element of degree 1
- a_1_0 → 0, an element of degree 1
- b_1_2 → c_1_2, an element of degree 1
- b_1_3 → 0, an element of degree 1
- c_2_8 → c_1_22 + c_1_02, an element of degree 2
- b_5_26 → c_1_12·c_1_23 + c_1_14·c_1_2, an element of degree 5
- a_6_28 → 0, an element of degree 6
- c_8_50 → c_1_12·c_1_26 + c_1_18, an element of degree 8
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