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Mod-3-Cohomology of SymmetricGroup(8), a group of order 40320
General information on the group
- SymmetricGroup(8) is a group of order 40320.
- The group order factors as 27 · 32 · 5 · 7.
- The group is defined by Group([(1,2,3,4,5,6,7,8),(1,2)]).
- It is non-abelian.
- It has 3-Rank 2.
- The centre of a Sylow 3-subgroup has rank 2.
- Its Sylow 3-subgroup has a unique conjugacy class of maximal elementary abelian subgroups, which is of rank 2.
Structure of the cohomology ring
This cohomology ring is isomorphic to the cohomology ring of a subgroup, namely H*(SmallGroup(144,186); GF(3)).
General information
- The cohomology ring is of dimension 2 and depth 2.
- The depth coincides with the Duflot bound.
- The Poincaré series is
(1 − t + t2) · (1 − t + t2 − t3 + t4 − t5 + t6) |
| ( − 1 + t)2 · (1 + t2)2 · (1 + t4) |
- The a-invariants are -∞,-∞,-2. They were obtained using the filter regular HSOP of the Hilbert-Poincaré test.
- The filter degree type of any filter regular HSOP is [-1, -2, -2].
Ring generators
The cohomology ring has 4 minimal generators of maximal degree 8:
- a_3_0, a nilpotent element of degree 3
- c_4_0, a Duflot element of degree 4
- a_7_1, a nilpotent element of degree 7
- c_8_1, a Duflot element of degree 8
Ring relations
There are 2 "obvious" relations:
a_3_02, a_7_12
Apart from that, there are no relations.
Data used for the Hilbert-Poincaré test
- We proved completion in degree 10 using the Hilbert-Poincaré criterion.
- However, the last relation was already found in degree 0 and the last generator in degree 8.
- The following is a filter regular homogeneous system of parameters:
- c_4_0, an element of degree 4
- c_8_1, an element of degree 8
- The above filter regular HSOP forms a Duflot regular sequence.
- The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, 10].
Restriction maps
- a_3_0 → a_3_0
- c_4_0 → c_4_0
- a_7_1 → a_7_1
- c_8_1 → c_8_1
Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 2
- a_3_0 → c_2_2·a_1_1 + c_2_1·a_1_0, an element of degree 3
- c_4_0 → c_2_22 + c_2_12, an element of degree 4
- a_7_1 → c_2_1·c_2_22·a_1_0 + c_2_12·c_2_2·a_1_1, an element of degree 7
- c_8_1 → c_2_12·c_2_22, an element of degree 8
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