Small group number 3 of order 343

G = E343 is Extraspecial 7-group of order 343 and exponent 7

G has 2 minimal generators, rank 2 and exponent 7. The centre has rank 1.

The 8 maximal subgroups are: V49 (8x).

There are 8 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 2, 2, 2, 2, 2, 2, 2, 2.

This cohomology ring calculation is complete.

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

Ring structure

The cohomology ring has 16 generators:

There are 104 minimal relations:

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

Essential ideal: There are 8 minimal generators:

Nilradical: There are 12 minimal generators:

Completion information

This cohomology ring was obtained from a calculation out to degree 28. The cohomology ring approximation is stable from degree 26 onwards, and Carlson's tests detect stability from degree 26 onwards.

This cohomology ring has dimension 2 and depth 1. Here is a homogeneous system of parameters:

The first term h1 forms a regular sequence of maximum length. The remaining term h2 is annihilated by the class y1.x2.

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 free of rank 8 as a module over the polynomial algebra on h1. These free generators are:

The essential ideal squares to zero.

Koszul information

A basis for R/(h1, h2) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 18.

A basis for AnnR/(h1)(h2) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 14.

Restriction information

Restrictions to maximal subgroups

Restriction to maximal subgroup number 1, which is V49

Restriction to maximal subgroup number 2, which is V49

Restriction to maximal subgroup number 3, which is V49

Restriction to maximal subgroup number 4, which is V49

Restriction to maximal subgroup number 5, which is V49

Restriction to maximal subgroup number 6, which is V49

Restriction to maximal subgroup number 7, which is V49

Restriction to maximal subgroup number 8, which is V49

Restrictions to maximal elementary abelian subgroups

Restriction to maximal elementary abelian number 1, which is V49

Restriction to maximal elementary abelian number 2, which is V49

Restriction to maximal elementary abelian number 3, which is V49

Restriction to maximal elementary abelian number 4, which is V49

Restriction to maximal elementary abelian number 5, which is V49

Restriction to maximal elementary abelian number 6, which is V49

Restriction to maximal elementary abelian number 7, which is V49

Restriction to maximal elementary abelian number 8, which is V49

Restriction to the greatest central elementary abelian subgroup

Restriction to the greatest central elementary abelian, which is C7

Poincaré series

(1 + 2t + 4t2 + 6t3 + 6t4 + 6t5 + 5t6 + 4t7 + 4t8 + 4t9 + 4t10 + 4t11 + 4t12 + 4t13 + 3t14 + 2t15 + t16) / (1 - t4) (1 - t14)

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