Small group number 2023 of order 128

G = SD16*SD16 is Central product SD_16 * SD_16

G has 4 minimal generators, rank 3 and exponent 8. The centre has rank 1.

The 15 maximal subgroups are: 64gp124 (2x), Syl2(M12) (2x), 64gp137 (2x), 64gp152 (4x), 64gp173, 64gp258 (4x).

There are 4 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 3, 3, 3, 3.

This cohomology ring calculation is complete.

Ring structure | Completion information | Koszul information | Restriction information | Poincaré series

Ring structure

The cohomology ring has 9 generators:

There are 18 minimal relations:

A minimal Gröbner basis for the relations ideal consists of this minimal generating set, together with the following redundant relations:

Essential ideal: Zero ideal

Nilradical: There are 3 minimal generators:

Completion information

This cohomology ring was obtained from a calculation out to degree 14. The cohomology ring approximation is stable from degree 14 onwards, and Carlson's tests detect stability from degree 14 onwards.

This cohomology ring has dimension 3 and depth 2. Here is a homogeneous system of parameters:

The first 2 terms h1, h2 form a regular sequence of maximum length. The remaining term h3 is annihilated by the class y42 + y3.y4.

The first term h1 forms a complete Duflot regular sequence. That is, its restriction to the greatest central elementary abelian subgroup forms a regular sequence of maximal length.

The ideal of essential classes is the zero ideal. The essential ideal squares to zero.

Koszul information

A basis for R/(h1, h2, h3) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 11.

A basis for AnnR/(h1, h2)(h3) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 10.

Restriction information

Restrictions to maximal subgroups

Restriction to maximal subgroup number 1, which is 64gp258

Restriction to maximal subgroup number 2, which is 64gp258

Restriction to maximal subgroup number 3, which is 64gp134

Restriction to maximal subgroup number 4, which is 64gp258

Restriction to maximal subgroup number 5, which is 64gp124

Restriction to maximal subgroup number 6, which is 64gp137

Restriction to maximal subgroup number 7, which is 64gp152

Restriction to maximal subgroup number 8, which is 64gp137

Restriction to maximal subgroup number 9, which is 64gp258

Restriction to maximal subgroup number 10, which is 64gp152

Restriction to maximal subgroup number 11, which is 64gp124

Restriction to maximal subgroup number 12, which is 64gp152

Restriction to maximal subgroup number 13, which is 64gp134

Restriction to maximal subgroup number 14, which is 64gp173

Restriction to maximal subgroup number 15, which is 64gp152

Restrictions to maximal elementary abelian subgroups

Restriction to maximal elementary abelian number 1, which is V8

Restriction to maximal elementary abelian number 2, which is V8

Restriction to maximal elementary abelian number 3, which is V8

Restriction to maximal elementary abelian number 4, which is V8

Restriction to the greatest central elementary abelian subgroup

Restriction to the greatest central elementary abelian, which is C2

Poincaré series

(1 + 3t + 4t2 + 3t3 + t4 - t5 - t6 + t7 + t8) / (1 - t) (1 - t2) (1 - t8)

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