Simon King
David J. Green
Cohomology
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Singular
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Cohomology of group number 1258 of order 128
General information on the group
- The group has 4 minimal generators and exponent 4.
- It is non-abelian.
- It has p-Rank 4.
- Its center has rank 3.
- 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 3.
- The depth coincides with the Duflot bound.
- The Poincaré series is
( − 1) · (t3 + t + 1) · (t4 − 3·t3 − t − 1) |
| (t + 1)2 · (t − 1)4 · (t2 + 1)3 |
- The a-invariants are -∞,-∞,-∞,-4,-4. They were obtained using the filter regular HSOP of the Benson test.
Ring generators
The cohomology ring has 16 minimal generators of maximal degree 5:
- a_1_0, a nilpotent element of degree 1
- a_1_3, a nilpotent element of degree 1
- b_1_1, an element of degree 1
- b_1_2, an element of degree 1
- a_3_2, a nilpotent element of degree 3
- b_3_6, an element of degree 3
- b_3_7, an element of degree 3
- b_3_8, an element of degree 3
- b_3_9, an element of degree 3
- b_3_10, an element of degree 3
- b_3_11, an element of degree 3
- c_4_19, a Duflot regular element of degree 4
- c_4_20, a Duflot regular element of degree 4
- c_4_21, a Duflot regular element of degree 4
- b_5_31, an element of degree 5
- b_5_32, an element of degree 5
Ring relations
There are 62 minimal relations of maximal degree 10:
- a_1_3·b_1_1 + a_1_0·b_1_1 + a_1_0·a_1_3
- a_1_0·b_1_2 + a_1_32 + a_1_02
- b_1_1·b_1_2 + b_1_12 + a_1_02
- a_1_0·a_1_32 + a_1_03
- a_1_33 + a_1_03
- a_1_32·b_1_2
- b_1_1·a_3_2 + a_1_3·b_3_7 + a_1_0·b_3_8 + a_1_0·a_3_2
- b_1_2·b_3_7 + b_1_1·b_3_7 + b_1_1·a_3_2 + a_1_3·b_3_8 + a_1_3·b_3_7 + a_1_0·b_3_6
- a_1_3·b_3_7 + a_1_3·b_3_6 + a_1_0·b_3_9 + a_1_0·b_3_6 + a_1_0·a_3_2
- b_1_2·b_3_7 + b_1_2·b_3_6 + b_1_1·b_3_7 + b_1_1·b_3_6 + a_1_3·b_3_9 + a_1_0·b_3_6
+ a_1_3·a_3_2
- b_1_1·b_3_9 + b_1_1·b_3_7 + b_1_1·b_3_6 + b_1_14 + b_1_1·a_3_2 + a_1_0·b_3_6
- b_1_1·a_3_2 + a_1_0·b_3_10 + a_1_0·b_3_7 + a_1_0·b_3_6 + a_1_3·a_3_2
- b_1_2·a_3_2 + b_1_1·a_3_2 + a_1_3·b_3_10 + a_1_3·b_3_7 + a_1_3·b_3_6 + a_1_3·a_3_2
+ a_1_0·a_3_2
- b_1_1·b_3_10 + b_1_1·b_3_8 + a_1_3·b_3_6 + a_1_0·b_3_6 + a_1_3·a_3_2
- b_1_2·b_3_11 + b_1_2·b_3_10 + b_1_2·b_3_9 + b_1_2·b_3_7 + b_1_2·b_3_6 + b_1_1·b_3_8
+ b_1_1·b_3_6 + b_1_14 + b_1_2·a_3_2 + b_1_1·a_3_2 + a_1_3·b_3_7 + a_1_3·b_3_6 + a_1_0·b_3_6 + a_1_0·a_3_2
- a_1_3·b_3_7 + a_1_3·b_3_6 + a_1_0·b_3_11 + a_1_3·a_3_2
- b_1_2·b_3_7 + b_1_2·b_3_6 + b_1_1·b_3_7 + b_1_1·b_3_6 + b_1_2·a_3_2 + a_1_3·b_3_11
+ a_1_3·b_3_6 + a_1_0·b_3_6 + a_1_0·a_3_2
- b_1_1·b_3_11 + b_1_1·b_3_6 + b_1_1·a_3_2 + a_1_3·b_3_6 + a_1_0·b_3_7 + a_1_0·b_3_6
+ a_1_3·a_3_2
- a_1_32·b_3_8 + a_1_02·b_3_8 + a_1_32·a_3_2 + a_1_0·a_1_3·a_3_2
- a_1_02·b_3_9 + a_1_02·a_3_2
- a_1_32·b_3_9 + a_1_02·b_3_10 + a_1_02·b_3_6 + a_1_32·a_3_2
- b_3_10·b_3_11 + b_3_102 + b_3_9·b_3_10 + b_3_8·b_3_11 + b_3_8·b_3_10 + b_3_8·b_3_9
+ b_3_7·b_3_11 + b_3_7·b_3_10 + b_3_7·b_3_9 + b_3_7·b_3_8 + b_3_72 + b_3_6·b_3_11 + b_3_62 + b_1_13·b_3_7 + a_1_3·b_1_22·b_3_9 + a_3_22
- b_3_9·b_3_11 + b_3_9·b_3_10 + b_3_92 + b_3_7·b_3_10 + b_3_7·b_3_9 + b_3_6·b_3_10
+ b_3_6·b_3_9 + b_3_6·b_3_7 + b_3_62 + b_1_13·b_3_8 + b_1_13·b_3_7 + b_1_16 + a_3_2·b_3_11 + a_3_2·b_3_10 + a_3_2·b_3_9 + a_3_2·b_3_7 + a_3_2·b_3_6
- b_3_10·b_3_11 + b_3_102 + b_3_9·b_3_10 + b_3_82 + b_3_7·b_3_10 + b_3_7·b_3_9
+ b_3_6·b_3_11 + b_3_6·b_3_7 + b_3_62 + b_1_13·b_3_7 + b_1_13·b_3_6 + b_1_16 + a_3_2·b_3_7 + a_3_2·b_3_6 + a_1_3·b_1_22·b_3_8 + a_3_22 + c_4_19·b_1_22 + c_4_19·a_1_3·b_1_2 + c_4_19·a_1_32
- b_3_7·b_3_11 + b_3_6·b_3_10 + b_3_6·b_3_8 + b_3_6·b_3_7 + a_3_2·b_3_9 + a_3_22
+ c_4_19·a_1_3·b_1_2 + c_4_19·a_1_0·b_1_1 + c_4_19·a_1_02
- b_3_10·b_3_11 + b_3_102 + b_3_9·b_3_10 + b_3_82 + b_3_7·b_3_11 + b_3_7·b_3_9
+ b_3_7·b_3_8 + b_3_72 + b_3_6·b_3_10 + b_3_6·b_3_9 + b_3_6·b_3_8 + b_3_6·b_3_7 + b_3_62 + b_1_13·b_3_8 + b_1_13·b_3_7 + b_1_13·b_3_6 + a_3_2·b_3_6 + a_1_3·b_1_22·b_3_8 + a_3_22 + c_4_19·b_1_22 + c_4_19·b_1_12 + c_4_19·a_1_3·b_1_2 + c_4_19·a_1_0·a_1_3 + c_4_19·a_1_02
- b_3_92 + b_3_7·b_3_11 + b_3_7·b_3_10 + b_3_7·b_3_8 + b_3_72 + b_3_6·b_3_11
+ b_3_6·b_3_10 + b_3_6·b_3_9 + b_3_6·b_3_8 + b_1_23·b_3_9 + b_1_13·b_3_8 + b_1_13·b_3_6 + a_3_2·b_3_7 + a_1_3·b_1_22·b_3_8 + a_3_22 + c_4_20·b_1_22 + c_4_19·a_1_0·a_1_3 + c_4_19·a_1_02
- b_3_112 + b_3_10·b_3_11 + b_3_9·b_3_10 + b_3_92 + b_3_7·b_3_10 + b_3_72 + b_3_6·b_3_11
+ b_3_6·b_3_9 + b_3_6·b_3_7 + b_1_13·b_3_8 + b_1_13·b_3_6 + b_1_16 + a_3_22 + c_4_20·a_1_3·b_1_2 + c_4_19·a_1_3·b_1_2 + c_4_19·a_1_02
- b_3_7·b_3_11 + b_3_7·b_3_10 + b_3_7·b_3_8 + b_3_6·b_3_11 + b_3_6·b_3_10 + b_3_6·b_3_9
+ b_3_6·b_3_8 + b_1_13·b_3_6 + a_3_2·b_3_7 + c_4_20·a_1_02 + c_4_19·a_1_0·a_1_3
- b_3_10·b_3_11 + b_3_102 + b_3_9·b_3_10 + b_3_82 + b_3_7·b_3_8 + b_3_72 + b_3_6·b_3_11
+ b_3_6·b_3_10 + b_3_6·b_3_8 + b_3_62 + b_1_13·b_3_6 + b_1_16 + a_3_2·b_3_9 + a_3_2·b_3_7 + a_3_2·b_3_6 + a_1_3·b_1_22·b_3_8 + c_4_19·b_1_22 + c_4_19·a_1_3·b_1_2 + c_4_20·a_1_0·a_1_3
- b_3_112 + b_3_10·b_3_11 + b_3_9·b_3_10 + b_3_92 + b_3_7·b_3_11 + b_3_7·b_3_9
+ b_3_7·b_3_8 + b_3_6·b_3_10 + b_3_6·b_3_9 + b_3_6·b_3_8 + b_3_6·b_3_7 + b_3_62 + b_1_13·b_3_8 + b_1_13·b_3_7 + b_1_13·b_3_6 + b_1_16 + a_3_2·b_3_6 + a_1_3·b_1_22·b_3_9 + a_3_22 + c_4_19·a_1_3·b_1_2 + c_4_20·a_1_32 + c_4_19·a_1_0·a_1_3
- b_3_7·b_3_10 + b_3_7·b_3_8 + a_3_2·b_3_11 + a_3_2·b_3_10 + a_3_2·b_3_9 + a_3_2·b_3_8
+ c_4_20·a_1_0·b_1_1 + c_4_19·a_1_3·b_1_2 + c_4_19·a_1_0·a_1_3 + c_4_19·a_1_02
- b_3_112 + b_3_102 + b_3_92 + b_3_82 + b_3_62 + b_1_13·b_3_7 + a_1_3·b_1_22·b_3_9
+ a_1_3·b_1_22·b_3_8 + c_4_20·b_1_12 + c_4_19·b_1_22
- b_3_112 + b_3_102 + b_3_92 + b_3_82 + b_1_13·b_3_8 + b_1_13·b_3_6
+ a_1_3·b_1_22·b_3_9 + a_1_3·b_1_22·b_3_8 + a_3_22 + c_4_19·b_1_22 + c_4_21·a_1_02 + c_4_19·a_1_02
- b_3_112 + b_3_10·b_3_11 + b_3_9·b_3_10 + b_3_92 + b_3_7·b_3_10 + b_3_72 + b_3_6·b_3_11
+ b_3_6·b_3_9 + b_3_6·b_3_7 + b_1_13·b_3_8 + b_1_13·b_3_6 + b_1_16 + a_3_2·b_3_11 + a_3_2·b_3_9 + a_3_2·b_3_6 + c_4_21·a_1_0·a_1_3 + c_4_19·a_1_02
- b_3_102 + b_3_92 + b_3_82 + b_3_7·b_3_11 + b_3_7·b_3_10 + b_3_7·b_3_8 + b_3_72
+ b_3_6·b_3_11 + b_3_6·b_3_10 + b_3_6·b_3_9 + b_3_6·b_3_8 + b_3_62 + b_1_23·b_3_9 + b_1_13·b_3_7 + a_3_2·b_3_7 + a_1_3·b_1_22·b_3_9 + c_4_21·a_1_32 + c_4_19·a_1_0·a_1_3
- b_3_112 + b_3_10·b_3_11 + b_3_9·b_3_10 + b_3_92 + b_3_7·b_3_11 + b_3_7·b_3_10 + b_3_72
+ b_3_6·b_3_9 + b_3_62 + b_1_13·b_3_8 + b_1_13·b_3_6 + b_1_16 + a_3_2·b_3_11 + a_3_2·b_3_10 + a_3_2·b_3_7 + a_3_2·b_3_6 + a_1_3·b_1_22·b_3_9 + a_3_22 + c_4_21·a_1_0·b_1_1 + c_4_19·a_1_3·b_1_2
- b_3_82 + b_3_7·b_3_11 + b_3_7·b_3_10 + b_3_7·b_3_8 + b_3_72 + b_3_6·b_3_11
+ b_3_6·b_3_10 + b_3_6·b_3_9 + b_3_6·b_3_8 + b_1_13·b_3_8 + b_1_13·b_3_7 + b_1_13·b_3_6 + a_3_2·b_3_7 + a_1_3·b_1_22·b_3_9 + a_1_3·b_1_22·b_3_8 + a_3_22 + c_4_21·b_1_12 + c_4_19·b_1_22 + c_4_19·a_1_0·a_1_3
- b_3_102 + b_3_9·b_3_10 + b_3_92 + b_3_8·b_3_10 + b_3_7·b_3_10 + b_3_72 + b_3_6·b_3_11
+ b_3_6·b_3_9 + b_3_6·b_3_7 + b_1_2·b_5_31 + b_1_23·b_3_10 + b_1_23·b_3_9 + b_1_23·b_3_8 + b_1_13·b_3_8 + b_1_13·b_3_7 + b_1_13·b_3_6 + a_3_2·b_3_10 + a_3_2·b_3_7 + a_1_3·b_1_22·b_3_10 + a_1_3·b_1_22·b_3_8 + c_4_19·b_1_22 + c_4_21·a_1_3·b_1_2 + c_4_19·a_1_3·b_1_2 + c_4_19·a_1_0·a_1_3 + c_4_19·a_1_02
- b_3_10·b_3_11 + b_3_102 + b_3_9·b_3_10 + b_3_82 + b_3_7·b_3_10 + b_3_72 + b_3_6·b_3_11
+ b_3_6·b_3_9 + b_3_6·b_3_7 + b_1_16 + a_3_2·b_3_11 + a_3_2·b_3_9 + a_1_3·b_1_22·b_3_9 + a_1_3·b_1_22·b_3_8 + a_1_0·b_5_31 + a_3_22 + c_4_19·b_1_22 + c_4_19·a_1_0·a_1_3
- b_3_112 + b_3_7·b_3_11 + b_3_7·b_3_10 + b_3_7·b_3_9 + b_3_7·b_3_8 + b_3_6·b_3_11
+ b_3_6·b_3_10 + b_3_6·b_3_8 + b_1_23·b_3_9 + b_1_13·b_3_8 + a_3_2·b_3_10 + a_3_2·b_3_9 + a_3_2·b_3_8 + a_3_2·b_3_7 + a_3_2·b_3_6 + a_1_3·b_5_31 + a_1_3·b_1_22·b_3_10 + a_1_3·b_1_22·b_3_9 + c_4_19·b_1_22 + c_4_19·a_1_3·b_1_2 + c_4_19·a_1_0·a_1_3 + c_4_19·a_1_02
- b_3_112 + b_3_10·b_3_11 + b_3_9·b_3_10 + b_3_92 + b_3_7·b_3_10 + b_3_7·b_3_9 + b_3_72
+ b_3_6·b_3_11 + b_3_6·b_3_9 + b_3_6·b_3_8 + b_3_62 + b_1_1·b_5_31 + b_1_13·b_3_7 + b_1_13·b_3_6 + a_3_2·b_3_11 + a_3_2·b_3_9 + a_3_2·b_3_7 + c_4_19·a_1_3·b_1_2 + c_4_19·a_1_0·a_1_3
- b_3_112 + b_3_102 + b_3_8·b_3_10 + b_3_82 + b_3_7·b_3_11 + b_3_7·b_3_9 + b_3_7·b_3_8
+ b_3_72 + b_3_6·b_3_10 + b_3_6·b_3_8 + b_3_6·b_3_7 + b_1_2·b_5_32 + b_1_23·b_3_8 + b_1_13·b_3_7 + b_1_13·b_3_6 + a_3_2·b_3_7 + a_1_3·b_1_22·b_3_9 + c_4_19·b_1_22 + c_4_19·a_1_3·b_1_2
- b_3_10·b_3_11 + b_3_102 + b_3_9·b_3_10 + b_3_82 + b_3_7·b_3_10 + b_3_7·b_3_9
+ b_3_6·b_3_11 + b_3_6·b_3_7 + b_3_62 + b_1_13·b_3_7 + b_1_13·b_3_6 + b_1_16 + a_3_2·b_3_11 + a_3_2·b_3_10 + a_3_2·b_3_9 + a_3_2·b_3_7 + a_3_2·b_3_6 + a_1_3·b_1_22·b_3_8 + a_1_0·b_5_32 + c_4_19·b_1_22 + c_4_19·a_1_3·b_1_2 + c_4_19·a_1_0·a_1_3 + c_4_19·a_1_02
- b_3_10·b_3_11 + b_3_102 + b_3_9·b_3_10 + b_3_82 + b_3_7·b_3_10 + b_3_7·b_3_9
+ b_3_6·b_3_11 + b_3_6·b_3_7 + b_3_62 + b_1_13·b_3_7 + b_1_13·b_3_6 + b_1_16 + a_3_2·b_3_11 + a_3_2·b_3_9 + a_3_2·b_3_8 + a_3_2·b_3_7 + a_1_3·b_5_32 + a_1_3·b_1_22·b_3_9 + a_3_22 + c_4_19·b_1_22
- b_3_112 + b_3_10·b_3_11 + b_3_9·b_3_10 + b_3_92 + b_3_82 + b_3_7·b_3_11 + b_3_7·b_3_10
+ b_3_7·b_3_8 + b_3_72 + b_3_6·b_3_11 + b_1_1·b_5_32 + b_1_13·b_3_6 + b_1_16 + a_3_2·b_3_11 + a_3_2·b_3_10 + a_3_2·b_3_9 + a_3_2·b_3_8 + a_3_2·b_3_6 + a_1_3·b_1_22·b_3_8 + c_4_19·b_1_22 + c_4_19·a_1_02
- a_1_32·b_5_32 + a_1_02·b_5_32 + c_4_21·a_1_02·a_1_3 + c_4_21·a_1_03
+ c_4_19·a_1_02·a_1_3 + c_4_19·a_1_03
- b_3_11·b_5_31 + b_3_8·b_5_31 + b_3_6·b_5_31 + b_1_22·b_3_8·b_3_9 + b_1_12·b_3_6·b_3_7
+ b_1_13·b_5_31 + b_1_15·b_3_8 + b_1_15·b_3_7 + b_1_15·b_3_6 + b_1_18 + a_1_3·b_1_22·b_5_31 + a_1_3·b_1_24·b_3_10 + a_1_3·b_1_24·b_3_9 + a_1_03·b_5_31 + c_4_21·b_1_1·b_3_6 + c_4_21·b_1_14 + c_4_20·b_1_2·b_3_10 + c_4_20·b_1_2·b_3_9 + c_4_20·b_1_2·b_3_8 + c_4_20·b_1_24 + c_4_20·b_1_1·b_3_8 + c_4_20·b_1_1·b_3_7 + c_4_20·b_1_1·b_3_6 + c_4_19·b_1_2·b_3_10 + c_4_19·b_1_2·b_3_9 + c_4_19·b_1_2·b_3_8 + c_4_19·b_1_1·b_3_8 + c_4_19·b_1_1·b_3_7 + c_4_19·b_1_1·b_3_6 + c_4_21·a_1_3·b_3_10 + c_4_21·a_1_3·b_3_9 + c_4_21·a_1_3·b_3_8 + c_4_21·a_1_0·b_3_9 + c_4_21·a_1_0·b_3_6 + c_4_20·a_1_3·b_3_9 + c_4_20·a_1_0·b_3_10 + c_4_20·a_1_0·b_3_9 + c_4_20·a_1_0·b_3_8 + c_4_20·a_1_0·b_3_7 + c_4_19·a_1_3·b_3_10 + c_4_19·a_1_3·b_3_8 + c_4_19·a_1_3·b_1_23 + c_4_19·a_1_0·b_3_10 + c_4_19·a_1_0·b_3_8 + c_4_19·a_1_0·b_3_7 + c_4_19·a_1_0·b_3_6 + c_4_21·a_1_3·a_3_2
- b_3_11·b_5_31 + b_3_10·b_5_31 + b_3_9·b_5_31 + b_3_7·b_5_31 + b_3_6·b_5_31
+ b_1_13·b_5_31 + b_1_15·b_3_8 + b_1_15·b_3_7 + a_3_2·b_5_31 + a_1_03·b_5_31 + c_4_21·b_1_1·b_3_6 + c_4_20·b_1_1·b_3_6 + c_4_20·b_1_14 + c_4_19·b_1_1·b_3_8 + c_4_19·b_1_1·b_3_6 + c_4_21·a_1_0·b_3_10 + c_4_21·a_1_0·b_3_9 + c_4_21·a_1_0·b_3_8 + c_4_21·a_1_0·b_3_7 + c_4_21·a_1_0·b_3_6 + c_4_20·a_1_0·b_3_10 + c_4_20·a_1_0·b_3_8 + c_4_19·a_1_0·b_3_9 + c_4_19·a_1_0·b_3_6 + c_4_20·a_1_0·a_3_2
- b_3_10·b_5_31 + b_1_23·b_5_32 + b_1_25·b_3_9 + b_1_25·b_3_8 + b_1_15·b_3_7
+ b_1_15·b_3_6 + b_1_18 + a_3_2·b_5_31 + a_1_3·b_1_2·b_3_8·b_3_9 + a_1_3·b_1_22·b_5_31 + a_1_3·b_1_24·b_3_10 + a_1_3·b_1_24·b_3_8 + a_1_03·b_5_31 + c_4_21·b_1_1·b_3_6 + c_4_20·b_1_2·b_3_9 + c_4_20·b_1_2·b_3_8 + c_4_20·b_1_1·b_3_7 + c_4_20·b_1_1·b_3_6 + c_4_20·b_1_14 + c_4_19·b_1_2·b_3_9 + c_4_19·b_1_2·b_3_8 + c_4_19·b_1_24 + c_4_19·b_1_1·b_3_7 + c_4_19·b_1_1·b_3_6 + c_4_19·b_1_14 + c_4_21·a_1_3·b_3_10 + c_4_21·a_1_0·b_3_9 + c_4_21·a_1_0·b_3_8 + c_4_21·a_1_0·b_3_7 + c_4_21·a_1_0·b_3_6 + c_4_20·a_1_3·b_3_10 + c_4_20·a_1_3·b_3_8 + c_4_20·a_1_0·b_3_10 + c_4_19·a_1_3·b_3_8 + c_4_19·a_1_3·b_1_23 + c_4_19·a_1_0·b_3_10 + c_4_21·a_1_0·a_3_2 + c_4_20·a_1_3·a_3_2 + c_4_20·a_1_0·a_3_2
- a_3_2·b_5_31 + a_1_3·b_1_22·b_5_32 + a_1_3·b_1_24·b_3_9 + a_1_3·b_1_24·b_3_8
+ c_4_21·a_1_0·b_3_6 + c_4_20·a_1_3·b_3_9 + c_4_20·a_1_3·b_3_8 + c_4_20·a_1_0·b_3_9 + c_4_20·a_1_0·b_3_8 + c_4_19·a_1_3·b_3_9 + c_4_19·a_1_3·b_3_8 + c_4_19·a_1_3·b_1_23 + c_4_19·a_1_0·b_3_9 + c_4_19·a_1_0·b_3_8 + c_4_21·a_1_3·a_3_2 + c_4_21·a_1_0·a_3_2 + c_4_20·a_1_3·a_3_2
- b_3_6·b_5_31 + b_1_13·b_5_32 + b_1_15·b_3_7 + a_3_2·b_5_31 + a_1_3·b_1_2·b_3_8·b_3_9
+ a_1_03·b_5_31 + c_4_20·b_1_1·b_3_8 + c_4_20·b_1_1·b_3_6 + c_4_20·b_1_14 + c_4_19·b_1_1·b_3_6 + c_4_19·b_1_14 + c_4_21·a_1_0·b_3_10 + c_4_21·a_1_0·b_3_9 + c_4_21·a_1_0·b_3_8 + c_4_21·a_1_0·b_3_7 + c_4_20·a_1_3·b_3_10 + c_4_20·a_1_3·b_3_9 + c_4_20·a_1_3·b_3_8 + c_4_20·a_1_3·b_1_23 + c_4_20·a_1_0·b_3_9 + c_4_20·a_1_0·b_3_8 + c_4_20·a_1_0·b_3_7 + c_4_19·a_1_3·b_3_10 + c_4_19·a_1_3·b_3_9 + c_4_19·a_1_3·b_3_8 + c_4_21·a_1_0·a_3_2 + c_4_20·a_1_0·a_3_2 + c_4_19·a_1_3·a_3_2
- a_3_2·b_5_32 + a_1_3·b_1_22·b_5_31 + a_1_3·b_1_24·b_3_10 + a_1_3·b_1_24·b_3_8
+ a_1_03·b_5_31 + c_4_21·a_1_0·b_3_10 + c_4_21·a_1_0·b_3_6 + c_4_20·a_1_3·b_3_10 + c_4_20·a_1_3·b_3_8 + c_4_20·a_1_0·b_3_9 + c_4_20·a_1_0·b_3_7 + c_4_20·a_1_0·b_3_6 + c_4_19·a_1_3·b_3_8 + c_4_19·a_1_0·b_3_10 + c_4_19·a_1_0·b_3_6 + c_4_21·a_1_3·a_3_2 + c_4_20·a_1_0·a_3_2 + c_4_19·a_1_3·a_3_2 + c_4_19·a_1_0·a_3_2
- b_3_10·b_5_32 + b_3_7·b_5_31 + b_3_6·b_5_31 + b_1_23·b_5_31 + b_1_25·b_3_10
+ b_1_25·b_3_8 + b_1_15·b_3_7 + b_1_15·b_3_6 + b_1_18 + a_3_2·b_5_31 + a_1_3·b_1_2·b_3_8·b_3_9 + a_1_3·b_1_22·b_5_31 + a_1_3·b_1_24·b_3_8 + a_1_03·b_5_31 + c_4_21·b_1_1·b_3_8 + c_4_21·b_1_1·b_3_7 + c_4_21·b_1_14 + c_4_20·b_1_2·b_3_10 + c_4_20·b_1_2·b_3_8 + c_4_20·b_1_1·b_3_7 + c_4_20·b_1_1·b_3_6 + c_4_20·b_1_14 + c_4_19·b_1_2·b_3_8 + c_4_19·b_1_1·b_3_7 + c_4_19·b_1_1·b_3_6 + c_4_19·b_1_14 + c_4_21·a_1_3·b_1_23 + c_4_21·a_1_0·b_3_7 + c_4_21·a_1_0·b_3_6 + c_4_20·a_1_3·b_3_10 + c_4_20·a_1_3·b_3_8 + c_4_20·a_1_0·b_3_9 + c_4_20·a_1_0·b_3_7 + c_4_20·a_1_0·b_3_6 + c_4_19·a_1_3·b_3_10 + c_4_19·a_1_3·b_3_8 + c_4_19·a_1_3·b_1_23 + c_4_19·a_1_0·b_3_10 + c_4_19·a_1_0·b_3_9 + c_4_21·a_1_0·a_3_2 + c_4_20·a_1_3·a_3_2 + c_4_20·a_1_0·a_3_2
- b_3_11·b_5_32 + b_3_11·b_5_31 + b_3_10·b_5_31 + b_3_7·b_5_31 + b_1_23·b_5_31
+ b_1_25·b_3_10 + b_1_25·b_3_9 + b_1_25·b_3_8 + b_1_12·b_3_6·b_3_7 + b_1_13·b_5_31 + b_1_15·b_3_7 + b_1_15·b_3_6 + b_1_18 + a_3_2·b_5_31 + a_1_3·b_1_2·b_3_8·b_3_9 + a_1_3·b_1_22·b_5_31 + a_1_3·b_1_24·b_3_9 + a_1_03·b_5_31 + c_4_20·b_1_2·b_3_9 + c_4_20·b_1_2·b_3_8 + c_4_20·b_1_24 + c_4_20·b_1_1·b_3_8 + c_4_20·b_1_1·b_3_7 + c_4_20·b_1_1·b_3_6 + c_4_19·b_1_2·b_3_8 + c_4_19·b_1_1·b_3_7 + c_4_21·a_1_3·b_3_9 + c_4_21·a_1_3·b_1_23 + c_4_21·a_1_0·b_3_10 + c_4_20·a_1_3·b_3_9 + c_4_20·a_1_0·b_3_10 + c_4_20·a_1_0·b_3_8 + c_4_19·a_1_3·b_3_8 + c_4_19·a_1_3·b_1_23 + c_4_19·a_1_0·b_3_9 + c_4_19·a_1_0·b_3_8 + c_4_19·a_1_0·b_3_6 + c_4_21·a_1_3·a_3_2 + c_4_21·a_1_0·a_3_2 + c_4_20·a_1_3·a_3_2 + c_4_20·a_1_0·a_3_2 + c_4_19·a_1_3·a_3_2
- b_3_7·b_5_31 + b_3_6·b_5_32 + b_1_12·b_3_6·b_3_7 + b_1_15·b_3_8 + b_1_15·b_3_7
+ a_3_2·b_5_31 + a_1_3·b_1_24·b_3_9 + c_4_21·b_1_1·b_3_6 + c_4_20·b_1_1·b_3_6 + c_4_20·b_1_14 + c_4_19·b_1_1·b_3_7 + c_4_19·b_1_1·b_3_6 + c_4_21·a_1_0·b_3_10 + c_4_21·a_1_0·b_3_8 + c_4_20·a_1_0·b_3_10 + c_4_20·a_1_0·b_3_8 + c_4_20·a_1_0·b_3_7 + c_4_20·a_1_0·b_3_6 + c_4_19·a_1_3·b_3_9 + c_4_19·a_1_3·b_3_8 + c_4_19·a_1_3·b_1_23 + c_4_19·a_1_0·b_3_8 + c_4_19·a_1_0·b_3_7 + c_4_20·a_1_3·a_3_2 + c_4_20·a_1_0·a_3_2 + c_4_19·a_1_0·a_3_2
- b_3_7·b_5_32 + b_1_12·b_3_6·b_3_7 + b_1_15·b_3_7 + b_1_15·b_3_6 + b_1_18
+ a_1_3·b_1_2·b_3_8·b_3_9 + a_1_03·b_5_31 + c_4_21·b_1_1·b_3_7 + c_4_21·b_1_14 + c_4_20·b_1_1·b_3_7 + c_4_20·b_1_14 + c_4_19·b_1_1·b_3_8 + c_4_19·b_1_1·b_3_7 + c_4_19·b_1_1·b_3_6 + c_4_20·a_1_3·b_3_8 + c_4_20·a_1_0·b_3_10 + c_4_19·a_1_3·b_3_10 + c_4_19·a_1_3·b_1_23 + c_4_19·a_1_0·b_3_10 + c_4_19·a_1_0·b_3_8 + c_4_20·a_1_3·a_3_2 + c_4_20·a_1_0·a_3_2
- b_3_8·b_5_32 + b_3_7·b_5_31 + b_1_22·b_3_8·b_3_9 + b_1_12·b_3_6·b_3_7 + b_1_15·b_3_7
+ a_3_2·b_5_31 + a_1_3·b_1_22·b_5_31 + a_1_3·b_1_24·b_3_10 + a_1_3·b_1_24·b_3_9 + a_1_03·b_5_31 + c_4_21·b_1_1·b_3_8 + c_4_21·b_1_1·b_3_7 + c_4_20·b_1_2·b_3_8 + c_4_20·b_1_1·b_3_7 + c_4_19·b_1_2·b_3_10 + c_4_19·b_1_24 + c_4_19·b_1_1·b_3_7 + c_4_19·b_1_14 + c_4_21·a_1_0·b_3_10 + c_4_21·a_1_0·b_3_8 + c_4_21·a_1_0·b_3_7 + c_4_20·a_1_3·b_3_9 + c_4_20·a_1_3·b_3_8 + c_4_20·a_1_3·b_1_23 + c_4_20·a_1_0·b_3_7 + c_4_20·a_1_0·b_3_6 + c_4_19·a_1_3·b_3_10 + c_4_20·a_1_3·a_3_2 + c_4_19·a_1_3·a_3_2
- b_3_11·b_5_31 + b_3_10·b_5_31 + b_3_9·b_5_32 + b_3_7·b_5_31 + b_3_6·b_5_31 + b_1_25·b_3_9
+ b_1_15·b_3_7 + a_3_2·b_5_31 + a_1_3·b_1_2·b_3_8·b_3_9 + a_1_3·b_1_22·b_5_31 + a_1_3·b_1_24·b_3_10 + a_1_03·b_5_31 + c_4_21·b_1_1·b_3_7 + c_4_21·b_1_14 + c_4_20·b_1_2·b_3_10 + c_4_20·b_1_2·b_3_9 + c_4_20·b_1_24 + c_4_19·b_1_14 + c_4_21·a_1_3·b_3_9 + c_4_21·a_1_0·b_3_8 + c_4_21·a_1_0·b_3_7 + c_4_20·a_1_3·b_3_9 + c_4_20·a_1_3·b_3_8 + c_4_20·a_1_0·b_3_9 + c_4_20·a_1_0·b_3_8 + c_4_20·a_1_0·b_3_6 + c_4_19·a_1_3·b_3_9 + c_4_19·a_1_0·b_3_10 + c_4_19·a_1_0·b_3_7 + c_4_21·a_1_3·a_3_2 + c_4_20·a_1_3·a_3_2 + c_4_20·a_1_0·a_3_2 + c_4_19·a_1_3·a_3_2
- b_5_322 + b_1_27·b_3_9 + b_1_15·b_5_31 + a_1_3·b_1_26·b_3_9 + c_4_21·b_1_13·b_3_8
+ c_4_21·b_1_13·b_3_6 + c_4_20·b_1_26 + c_4_20·b_1_13·b_3_8 + c_4_20·b_1_13·b_3_6 + c_4_19·b_1_26 + c_4_19·b_1_13·b_3_8 + c_4_19·b_1_16 + c_4_20·a_1_3·b_1_22·b_3_9 + c_4_20·a_1_3·b_1_22·b_3_8 + c_4_19·a_1_3·b_1_22·b_3_9 + c_4_19·a_1_3·b_1_22·b_3_8 + c_4_212·b_1_12 + c_4_202·b_1_22 + c_4_19·c_4_21·b_1_12 + c_4_19·c_4_20·b_1_22 + c_4_192·b_1_22 + c_4_20·c_4_21·a_1_02 + c_4_19·c_4_21·a_1_32 + c_4_19·c_4_21·a_1_02 + c_4_19·c_4_20·a_1_02 + c_4_192·a_1_32 + c_4_192·a_1_02
- b_5_312 + b_1_14·b_3_6·b_3_7 + b_1_15·b_5_32 + b_1_15·b_5_31 + b_1_17·b_3_8
+ a_1_3·b_1_26·b_3_9 + a_1_3·b_1_26·b_3_8 + c_4_21·b_1_13·b_3_8 + c_4_21·b_1_13·b_3_7 + c_4_21·b_1_13·b_3_6 + c_4_20·b_1_23·b_3_9 + c_4_20·b_1_26 + c_4_19·b_1_23·b_3_9 + c_4_19·b_1_13·b_3_7 + c_4_19·b_1_13·b_3_6 + c_4_19·b_1_16 + c_4_20·a_1_3·b_1_22·b_3_9 + c_4_19·a_1_3·b_1_22·b_3_9 + c_4_20·c_4_21·b_1_12 + c_4_202·b_1_22 + c_4_192·b_1_22 + c_4_212·a_1_32 + c_4_212·a_1_02 + c_4_20·c_4_21·a_1_32 + c_4_20·c_4_21·a_1_02 + c_4_202·a_1_32 + c_4_19·c_4_21·a_1_32 + c_4_19·c_4_20·a_1_02 + c_4_192·a_1_02
- b_5_31·b_5_32 + b_1_24·b_3_8·b_3_9 + b_1_25·b_5_32 + b_1_27·b_3_9 + b_1_27·b_3_8
+ b_1_17·b_3_7 + a_1_3·b_1_23·b_3_8·b_3_9 + a_1_3·b_1_24·b_5_32 + a_1_3·b_1_24·b_5_31 + a_1_3·b_1_26·b_3_10 + a_1_3·b_1_26·b_3_8 + c_4_21·b_3_6·b_3_7 + c_4_21·b_1_1·b_5_31 + c_4_20·b_3_8·b_3_9 + c_4_20·b_1_2·b_5_31 + c_4_20·b_1_23·b_3_10 + c_4_20·b_1_23·b_3_8 + c_4_20·b_1_26 + c_4_20·b_1_1·b_5_32 + c_4_20·b_1_1·b_5_31 + c_4_20·b_1_13·b_3_8 + c_4_19·b_3_8·b_3_9 + c_4_19·b_3_6·b_3_7 + c_4_19·b_1_23·b_3_8 + c_4_19·b_1_26 + c_4_19·b_1_13·b_3_8 + c_4_19·b_1_13·b_3_7 + c_4_19·b_1_13·b_3_6 + c_4_21·a_1_3·b_5_32 + c_4_21·a_1_0·b_5_32 + c_4_20·a_1_3·b_1_22·b_3_10 + c_4_20·a_1_3·b_1_22·b_3_8 + c_4_20·a_1_3·b_1_25 + c_4_20·a_1_0·b_5_31 + c_4_19·a_1_3·b_5_32 + c_4_19·a_1_3·b_1_22·b_3_10 + c_4_19·a_1_3·b_1_22·b_3_9 + c_4_19·a_1_3·b_1_25 + c_4_19·a_1_0·b_5_32 + c_4_19·a_1_0·b_5_31 + c_4_20·c_4_21·b_1_12 + c_4_19·c_4_20·b_1_22 + c_4_192·b_1_22 + c_4_20·c_4_21·a_1_3·b_1_2 + c_4_20·c_4_21·a_1_0·b_1_1 + c_4_202·a_1_3·b_1_2 + c_4_202·a_1_0·b_1_1 + c_4_19·c_4_21·a_1_0·b_1_1 + c_4_19·c_4_20·a_1_0·b_1_1 + c_4_212·a_1_0·a_1_3 + c_4_20·c_4_21·a_1_32 + c_4_20·c_4_21·a_1_02 + c_4_202·a_1_32 + c_4_202·a_1_02 + c_4_19·c_4_20·a_1_32 + c_4_192·a_1_32 + c_4_192·a_1_0·a_1_3 + c_4_192·a_1_02
Data used for Benson′s test
- Benson′s completion test succeeded in degree 10.
- 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_19, a Duflot regular element of degree 4
- c_4_20, a Duflot regular element of degree 4
- c_4_21, a Duflot regular element of degree 4
- b_1_22, an element of degree 2
- The Raw Filter Degree Type of that HSOP is [-1, -1, -1, 8, 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 3
- a_1_0 → 0, an element of degree 1
- a_1_3 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → 0, an element of degree 1
- a_3_2 → 0, an element of degree 3
- b_3_6 → 0, an element of degree 3
- b_3_7 → 0, an element of degree 3
- b_3_8 → 0, an element of degree 3
- b_3_9 → 0, an element of degree 3
- b_3_10 → 0, an element of degree 3
- b_3_11 → 0, an element of degree 3
- c_4_19 → c_1_24, an element of degree 4
- c_4_20 → c_1_04, an element of degree 4
- c_4_21 → c_1_24 + c_1_14, an element of degree 4
- b_5_31 → 0, an element of degree 5
- b_5_32 → 0, an element of degree 5
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_0 → 0, an element of degree 1
- a_1_3 → 0, an element of degree 1
- b_1_1 → 0, an element of degree 1
- b_1_2 → c_1_3, an element of degree 1
- a_3_2 → 0, an element of degree 3
- b_3_6 → 0, an element of degree 3
- b_3_7 → 0, an element of degree 3
- b_3_8 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
- b_3_9 → c_1_02·c_1_3, an element of degree 3
- b_3_10 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
- b_3_11 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_0·c_1_32, an element of degree 3
- c_4_19 → c_1_22·c_1_32 + c_1_24, an element of degree 4
- c_4_20 → c_1_02·c_1_32 + c_1_04, an element of degree 4
- c_4_21 → c_1_2·c_1_33 + c_1_24 + c_1_12·c_1_32 + c_1_14 + c_1_02·c_1_32, an element of degree 4
- b_5_31 → c_1_22·c_1_33 + c_1_24·c_1_3 + c_1_0·c_1_34 + c_1_0·c_1_2·c_1_33
+ c_1_0·c_1_22·c_1_32 + c_1_02·c_1_33 + c_1_03·c_1_32 + c_1_04·c_1_3, an element of degree 5
- b_5_32 → c_1_2·c_1_34 + c_1_24·c_1_3 + c_1_0·c_1_2·c_1_33 + c_1_0·c_1_22·c_1_32
+ c_1_02·c_1_2·c_1_32 + c_1_02·c_1_22·c_1_3 + c_1_04·c_1_3, an element of degree 5
Restriction map to a maximal el. ab. subgp. of rank 4
- a_1_0 → 0, an element of degree 1
- a_1_3 → 0, an element of degree 1
- b_1_1 → c_1_3, an element of degree 1
- b_1_2 → c_1_3, an element of degree 1
- a_3_2 → 0, an element of degree 3
- b_3_6 → c_1_33 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
- b_3_7 → c_1_2·c_1_32 + c_1_22·c_1_3, an element of degree 3
- b_3_8 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_32 + c_1_12·c_1_3, an element of degree 3
- b_3_9 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
- b_3_10 → c_1_2·c_1_32 + c_1_22·c_1_3 + c_1_1·c_1_32 + c_1_12·c_1_3, an element of degree 3
- b_3_11 → c_1_33 + c_1_0·c_1_32 + c_1_02·c_1_3, an element of degree 3
- c_4_19 → c_1_2·c_1_33 + c_1_24 + c_1_1·c_1_33 + c_1_12·c_1_32 + c_1_0·c_1_33
+ c_1_02·c_1_32, an element of degree 4
- c_4_20 → c_1_1·c_1_33 + c_1_12·c_1_32 + c_1_0·c_1_33 + c_1_04, an element of degree 4
- c_4_21 → c_1_2·c_1_33 + c_1_24 + c_1_12·c_1_32 + c_1_14 + c_1_0·c_1_33
+ c_1_02·c_1_32, an element of degree 4
- b_5_31 → c_1_22·c_1_33 + c_1_24·c_1_3 + c_1_0·c_1_2·c_1_33 + c_1_0·c_1_22·c_1_32
+ c_1_0·c_1_1·c_1_33 + c_1_0·c_1_12·c_1_32 + c_1_02·c_1_33 + c_1_02·c_1_2·c_1_32 + c_1_02·c_1_22·c_1_3 + c_1_02·c_1_1·c_1_32 + c_1_02·c_1_12·c_1_3 + c_1_04·c_1_3, an element of degree 5
- b_5_32 → c_1_2·c_1_34 + c_1_24·c_1_3 + c_1_1·c_1_2·c_1_33 + c_1_1·c_1_22·c_1_32
+ c_1_12·c_1_33 + c_1_12·c_1_2·c_1_32 + c_1_12·c_1_22·c_1_3 + c_1_14·c_1_3 + c_1_0·c_1_2·c_1_33 + c_1_0·c_1_22·c_1_32 + c_1_02·c_1_33 + c_1_02·c_1_2·c_1_32 + c_1_02·c_1_22·c_1_3 + c_1_04·c_1_3, an element of degree 5
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