Simon King
David J. Green
Cohomology
→Theory
→Implementation
Jena:
Faculty
External links:
Singular
Gap
|
Cohomology of group number 1715 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 a unique conjugacy class of maximal elementary abelian subgroups, which is 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) · (t4 + t + 1) |
| (t − 1)3 · (t2 + 1) · (t4 + 1) |
- The a-invariants are -∞,-∞,-3,-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
- a_1_1, a nilpotent element of degree 1
- a_1_2, a nilpotent element of degree 1
- b_1_3, an element of degree 1
- a_3_10, a nilpotent element of degree 3
- a_3_11, a nilpotent element of degree 3
- c_4_16, a Duflot regular element of degree 4
- a_5_22, a nilpotent element of degree 5
- b_6_28, an element of degree 6
- a_7_34, a nilpotent element of degree 7
- c_8_41, a Duflot regular element of degree 8
Ring relations
There are 28 minimal relations of maximal degree 14:
- a_1_02
- a_1_0·b_1_3 + a_1_22 + a_1_1·a_1_2 + a_1_12 + a_1_0·a_1_1
- a_1_0·a_1_1·a_1_2 + a_1_0·a_1_12
- a_1_22·b_1_3 + a_1_1·a_1_2·b_1_3 + a_1_12·b_1_3 + a_1_13
- a_1_0·a_3_10
- a_1_2·a_3_10 + a_1_0·a_3_11
- a_1_13·b_1_32 + a_1_14·b_1_3 + a_1_15
- b_1_32·a_3_10 + a_1_1·a_1_2·a_3_11 + a_1_12·a_3_11 + a_1_0·a_1_1·a_3_11
- a_3_102
- a_3_10·a_3_11 + c_4_16·a_1_0·a_1_2
- a_3_112 + a_1_1·a_1_22·a_3_11 + a_1_12·a_1_2·a_3_11 + a_1_0·a_1_12·a_3_11
+ c_4_16·a_1_22
- a_1_0·a_5_22 + a_1_23·a_3_11 + a_1_1·a_1_22·a_3_11 + a_1_12·a_1_2·a_3_11
+ a_1_13·a_3_11 + a_1_0·a_1_12·a_3_11
- a_1_1·a_1_2·a_5_22 + a_1_12·a_5_22 + a_1_12·a_1_2·b_1_3·a_3_11
+ a_1_13·a_1_2·a_3_11
- b_6_28·a_1_0 + a_1_13·b_1_3·a_3_11 + a_1_13·a_1_2·a_3_11 + a_1_14·a_3_11
+ c_4_16·a_1_1·a_1_22 + c_4_16·a_1_12·a_1_2
- b_1_32·a_5_22 + b_6_28·a_1_2 + a_1_1·b_1_33·a_3_11 + a_1_1·a_1_2·b_1_32·a_3_11
+ a_1_12·a_5_22 + a_1_12·a_1_2·b_1_3·a_3_11 + a_1_13·b_1_3·a_3_11 + a_1_14·a_3_11 + c_4_16·a_1_2·b_1_32 + c_4_16·a_1_23 + c_4_16·a_1_1·a_1_22 + c_4_16·a_1_0·a_1_12
- a_3_10·a_5_22 + c_4_16·a_1_13·a_1_2
- a_1_0·a_7_34 + a_1_13·a_1_2·b_1_3·a_3_11 + a_1_14·b_1_3·a_3_11 + c_4_16·a_1_0·a_3_11
+ c_4_16·a_1_12·a_1_22 + c_4_16·a_1_13·a_1_2
- a_3_11·a_5_22 + a_1_2·a_7_34 + a_1_2·b_1_34·a_3_11 + b_6_28·a_1_12
+ a_1_12·a_1_2·b_1_32·a_3_11 + a_1_14·b_1_3·a_3_11 + c_4_16·a_1_2·a_3_11 + c_4_16·a_1_12·a_1_2·b_1_3 + c_4_16·a_1_14
- b_1_32·a_7_34 + b_1_36·a_3_11 + b_6_28·a_3_11 + b_6_28·a_3_10 + b_6_28·a_1_2·b_1_32
+ b_6_28·a_1_1·b_1_32 + a_1_12·a_7_34 + a_1_12·a_1_2·b_1_36 + b_6_28·a_1_12·a_1_2 + c_4_16·a_1_22·a_3_11 + c_4_16·a_1_12·a_1_2·b_1_32 + c_4_16·a_1_0·a_1_2·a_3_11 + c_4_16·a_1_0·a_1_1·a_3_11 + c_4_16·a_1_14·b_1_3 + c_4_16·a_1_15
- b_1_32·a_7_34 + b_1_36·a_3_11 + b_6_28·a_3_11 + b_6_28·a_1_2·b_1_32
+ b_6_28·a_1_1·b_1_32 + a_1_1·a_1_2·a_7_34 + a_1_1·a_1_2·b_1_34·a_3_11 + a_1_12·b_1_34·a_3_11 + a_1_12·a_1_2·b_1_36 + c_4_16·a_1_22·a_3_11 + c_4_16·a_1_12·a_1_2·b_1_32 + c_4_16·a_1_0·a_1_1·a_3_11
- a_5_222 + c_4_16·a_1_1·a_1_2·b_1_34 + c_4_16·a_1_12·b_1_34
+ c_4_16·a_1_15·b_1_3
- a_3_11·a_7_34 + b_6_28·a_1_2·a_3_11 + b_6_28·a_1_1·a_3_11
+ a_1_12·a_1_2·b_1_34·a_3_11 + c_4_16·a_1_2·a_5_22 + c_4_16·a_1_1·a_1_2·b_1_34 + c_4_16·a_1_12·b_1_34 + c_4_16·a_1_1·a_1_2·b_1_3·a_3_11 + c_4_16·a_1_13·a_3_11 + c_4_162·a_1_22
- a_3_10·a_7_34 + a_1_12·a_1_2·a_7_34 + a_1_12·a_1_2·b_1_34·a_3_11
+ c_4_16·a_1_23·a_3_11 + c_4_16·a_1_1·a_1_22·a_3_11 + c_4_16·a_1_12·a_1_2·a_3_11 + c_4_16·a_1_13·a_3_11 + c_4_16·a_1_0·a_1_12·a_3_11 + c_4_16·a_1_15·b_1_3 + c_4_162·a_1_0·a_1_2
- b_6_28·a_5_22 + b_6_28·a_1_1·b_1_3·a_3_11 + b_6_28·a_1_1·a_1_2·a_3_11
+ a_1_12·a_1_2·b_1_3·a_7_34 + a_1_12·a_1_2·b_1_35·a_3_11 + c_4_16·a_1_2·b_1_36 + c_4_16·b_6_28·a_1_2 + c_4_16·a_1_12·a_1_2·b_1_34 + c_4_16·a_1_13·b_1_3·a_3_11 + c_4_16·a_1_12·a_1_22·a_3_11 + c_4_16·a_1_13·a_1_2·a_3_11 + c_4_16·a_1_14·a_3_11 + c_4_162·a_1_2·b_1_32 + c_4_162·a_1_23 + c_4_162·a_1_1·a_1_22 + c_4_162·a_1_0·a_1_12
- a_5_22·a_7_34 + b_6_28·a_1_2·b_1_32·a_3_11 + b_6_28·a_1_1·a_1_2·b_1_3·a_3_11
+ b_6_28·a_1_12·b_1_3·a_3_11 + c_4_16·a_1_2·a_7_34 + c_4_16·a_1_2·b_1_34·a_3_11 + c_4_16·a_1_12·b_1_36 + c_4_16·a_1_12·b_1_3·a_5_22 + c_4_16·a_1_12·a_1_2·b_1_35 + c_4_16·a_1_13·a_1_2·b_1_3·a_3_11 + c_4_16·a_1_14·b_1_3·a_3_11 + c_4_16·a_1_15·a_3_11 + c_4_162·a_1_2·a_3_11 + c_4_162·a_1_12·b_1_32 + c_4_162·a_1_12·a_1_2·b_1_3 + c_4_162·a_1_12·a_1_22 + c_4_162·a_1_13·a_1_2 + c_4_162·a_1_14
- b_6_282 + b_6_28·a_1_1·a_1_2·b_1_34 + b_6_28·a_1_12·b_1_34
+ a_1_1·a_1_2·b_1_37·a_3_11 + a_1_12·b_1_37·a_3_11 + a_1_12·a_1_2·b_1_36·a_3_11 + c_4_16·b_1_38 + c_8_41·a_1_12·a_1_22 + c_8_41·a_1_14 + c_4_16·a_1_13·a_1_2·b_1_3·a_3_11 + c_4_16·a_1_15·a_3_11 + c_4_162·b_1_34 + c_4_162·a_1_12·a_1_22
- b_6_28·a_7_34 + b_6_28·b_1_34·a_3_11 + b_6_28·a_1_1·a_1_2·b_1_32·a_3_11
+ a_1_12·a_1_2·b_1_37·a_3_11 + c_4_16·b_1_36·a_3_11 + c_4_16·a_1_2·b_1_38 + c_4_16·a_1_1·b_1_38 + c_8_41·a_1_0·a_1_2·a_3_11 + c_4_16·a_1_1·a_1_2·a_7_34 + c_4_16·a_1_12·a_7_34 + c_4_16·a_1_12·b_1_34·a_3_11 + c_8_41·a_1_15 + c_4_162·b_1_32·a_3_11 + c_4_162·a_1_2·b_1_34 + c_4_162·a_1_1·b_1_34 + c_4_162·a_1_22·a_3_11 + c_4_162·a_1_12·a_3_11 + c_4_162·a_1_0·a_1_2·a_3_11 + c_4_162·a_1_0·a_1_1·a_3_11 + c_4_162·a_1_13·a_1_2·b_1_3 + c_4_162·a_1_14·b_1_3 + c_4_162·a_1_15
- a_7_342 + c_4_16·a_1_12·b_1_38 + c_4_162·a_1_12·b_1_34
+ c_4_162·a_1_1·a_1_22·a_3_11 + c_4_162·a_1_12·a_1_2·a_3_11 + c_4_162·a_1_0·a_1_12·a_3_11 + c_4_163·a_1_22
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_4_16, a Duflot regular element of degree 4
- c_8_41, a Duflot regular element of degree 8
- b_1_32, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, 9, 11].
- 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_0 → 0, an element of degree 1
- a_1_1 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- b_1_3 → 0, an element of degree 1
- a_3_10 → 0, an element of degree 3
- a_3_11 → 0, an element of degree 3
- c_4_16 → c_1_14, an element of degree 4
- a_5_22 → 0, an element of degree 5
- b_6_28 → 0, an element of degree 6
- a_7_34 → 0, an element of degree 7
- c_8_41 → c_1_18 + 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
- a_1_1 → 0, an element of degree 1
- a_1_2 → 0, an element of degree 1
- b_1_3 → c_1_2, an element of degree 1
- a_3_10 → 0, an element of degree 3
- a_3_11 → 0, an element of degree 3
- c_4_16 → c_1_14, an element of degree 4
- a_5_22 → 0, an element of degree 5
- b_6_28 → c_1_12·c_1_24 + c_1_14·c_1_22, an element of degree 6
- a_7_34 → 0, an element of degree 7
- c_8_41 → c_1_14·c_1_24 + c_1_18 + c_1_04·c_1_24 + c_1_08, an element of degree 8
|