Mod-2-Cohomology of Normalizer(G2(5),Centre(SylowSubgroup(G2(5),2))) (Group of Order 14400)
Mod-2-Cohomology of Normalizer(G2(5),Centre(SylowSubgroup(G2(5),2))), a group of order 14400
General information on the group
Normalizer(G2(5),Centre(SylowSubgroup(G2(5),2))) is a group of order 14400.
The group order factors as 26 & 32 & 52.
The group is defined by 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55)(00,,41)(52,,,58)(74,,49)(60,,39)(49,,85)(28,,01)()(37,,16)()(74,,20)(21,,13)(74,,50)(66,,69)(50,,79)(31,,66)(01,,15)(66,,87)(27,,83)(64,,,84)(19,,56)(39,,48)(74,,74)(84,,50)(99,,18)(30,,80)(22,,69)(26,,3823)]).
It is non-abelian.
It has 2-Rank 3.
The centre of a Sylow 2-subgroup has rank 1.
Its Sylow 2-subgroup has 5 conjugacy classes of maximal elementary abelian subgroups, which are all of rank 3.
Structure of the cohomology ring
The computation was based on 7 stability conditions for .
General information
The cohomology ring is of dimension 3 and depth 3.
The depth exceeds the Duflot bound, which is 1.
The Poincar& series is
(&−&1)&(1 &−& t &+& t2)
(&−&1 &+& t)3 & (1 &+& t &+& t2)
The a-invariants are -&,-&,-&,-3.
They were obtained using the filter regular HSOP of the .
The filter degree type of any filter regular HSOP is [-1, -2, -3, -3].
Ring generators
The cohomology ring has 5 minimal generators of maximal degree 4:
b_1_0, an element of degree 1
b_2_0, an element of degree 2
b_3_1, an element of degree 3
b_3_0, an element of degree 3
c_4_4, a Duflot element of degree 4
Ring relations
There are 2 minimal relations of maximal degree 6:
b_1_0&b_3_1
b_3_0&b_3_1&+&b_3_02&+&b_1_03&b_3_0&+&b_2_0&b_1_0&b_3_0&+&c_4_4&b_1_02
Data used for the Hilbert-Poincar& test
We proved completion in degree 6 using the Hilbert-Poincar& criterion.
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_4, an element of degree 4
b_1_02&+&b_2_0, an element of degree 2
b_3_1&+&b_2_0&b_1_0, an element of degree 3
A Duflot regular sequence is given by c_4_4.
The Raw Filter Degree Type of the filter regular HSOP is [-1, -1, -1, 6].
Restriction maps
Expressing the generators as elements of
b_1_0 & b_1_0
b_2_0 & b_1_22&+&b_1_1&b_1_2&+&b_1_12&+&b_2_5
b_3_1 & b_1_1&b_1_22&+&b_1_12&b_1_2&+&b_2_5&b_1_2
b_3_0 & b_3_9&+&b_2_5&b_1_2&+&b_2_4&b_1_1
c_4_4 & b_2_4&b_1_1&b_1_2&+&b_2_42&+&c_4_14
Restriction map to the greatest el. ab. subgp. in the centre of a Sylow subgroup, which is of rank 1
b_1_0 & 0, an element of degree 1
b_2_0 & 0, an element of degree 2
b_3_1 & 0, an element of degree 3
b_3_0 & 0, an element of degree 3
c_4_4 & c_1_04, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup
b_1_0 & c_1_1, an element of degree 1
b_2_0 & c_1_22&+&c_1_1&c_1_2, an element of degree 2
b_3_1 & 0, an element of degree 3
b_3_0 & c_1_0&c_1_12&+&c_1_02&c_1_1, an element of degree 3
c_4_4 & c_1_0&c_1_1&c_1_22&+&c_1_0&c_1_12&c_1_2&+&c_1_0&c_1_13&+&c_1_02&c_1_22&&&+&c_1_02&c_1_1&c_1_2&+&c_1_04, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup
b_1_0 & 0, an element of degree 1
b_2_0 & c_1_22&+&c_1_1&c_1_2&+&c_1_12, an element of degree 2
b_3_1 & c_1_1&c_1_22&+&c_1_12&c_1_2, an element of degree 3
b_3_0 & 0, an element of degree 3
c_4_4 & c_1_0&c_1_1&c_1_22&+&c_1_0&c_1_12&c_1_2&+&c_1_02&c_1_22&+&c_1_02&c_1_1&c_1_2&&&+&c_1_02&c_1_12&+&c_1_04, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup
b_1_0 & 0, an element of degree 1
b_2_0 & c_1_22&+&c_1_1&c_1_2&+&c_1_12, an element of degree 2
b_3_1 & c_1_1&c_1_22&+&c_1_12&c_1_2, an element of degree 3
b_3_0 & c_1_1&c_1_22&+&c_1_12&c_1_2, an element of degree 3
c_4_4 & c_1_0&c_1_1&c_1_22&+&c_1_0&c_1_12&c_1_2&+&c_1_02&c_1_22&+&c_1_02&c_1_1&c_1_2&&&+&c_1_02&c_1_12&+&c_1_04, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup
b_1_0 & 0, an element of degree 1
b_2_0 & c_1_22&+&c_1_1&c_1_2&+&c_1_12, an element of degree 2
b_3_1 & c_1_1&c_1_22&+&c_1_12&c_1_2, an element of degree 3
b_3_0 & c_1_1&c_1_22&+&c_1_12&c_1_2, an element of degree 3
c_4_4 & c_1_0&c_1_1&c_1_22&+&c_1_0&c_1_12&c_1_2&+&c_1_02&c_1_22&+&c_1_02&c_1_1&c_1_2&&&+&c_1_02&c_1_12&+&c_1_04, an element of degree 4
Restriction map to a maximal el. ab. subgp. of rank 3 in a Sylow subgroup
b_1_0 & 0, an element of degree 1
b_2_0 & c_1_22&+&c_1_1&c_1_2&+&c_1_12, an element of degree 2
b_3_1 & c_1_1&c_1_22&+&c_1_12&c_1_2, an element of degree 3
b_3_0 & 0, an element of degree 3
c_4_4 & c_1_0&c_1_1&c_1_22&+&c_1_0&c_1_12&c_1_2&+&c_1_02&c_1_22&+&c_1_02&c_1_1&c_1_2&&&+&c_1_02&c_1_12&+&c_1_04, an element of degree 4
Simon King
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Last change: 31.03.2010}