Small group number 12 of order 625

G = E125xC5 is Direct product E125 x C_5

G has 3 minimal generators, rank 3 and exponent 5. The centre has rank 2.

The 31 maximal subgroups are: E125 (25x), V125 (6x).

There are 6 conjugacy classes of maximal elementary abelian subgroups. Their ranks are: 3, 3, 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 14 generators:

There are 51 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 9 minimal generators:

Completion information

This cohomology ring was obtained from a calculation out to degree 18. The cohomology ring approximation is stable from degree 18 onwards, and Carlson's tests detect stability from degree 18 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 y1.x2.

The first 2 terms h1, h2 form a complete Duflot regular sequence. That is, their restrictions to the greatest central elementary abelian subgroup form 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, h2. These free generators are:

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 18.

A basis for AnnR/(h1, h2)(h3) is as follows. Carlson's Koszul condition stipulates that this must be confined to degrees less than 12.

Restriction information

Restrictions to maximal subgroups

Restriction to maximal subgroup number 1, which is V125

Restriction to maximal subgroup number 2, which is V125

Restriction to maximal subgroup number 3, which is V125

Restriction to maximal subgroup number 4, which is V125

Restriction to maximal subgroup number 5, which is V125

Restriction to maximal subgroup number 6, which is V125

Restriction to maximal subgroup number 7, which is 125gp3

Restriction to maximal subgroup number 8, which is 125gp3

Restriction to maximal subgroup number 9, which is 125gp3

Restriction to maximal subgroup number 10, which is 125gp3

Restriction to maximal subgroup number 11, which is 125gp3

Restriction to maximal subgroup number 12, which is 125gp3

Restriction to maximal subgroup number 13, which is 125gp3

Restriction to maximal subgroup number 14, which is 125gp3

Restriction to maximal subgroup number 15, which is 125gp3

Restriction to maximal subgroup number 16, which is 125gp3

Restriction to maximal subgroup number 17, which is 125gp3

Restriction to maximal subgroup number 18, which is 125gp3

Restriction to maximal subgroup number 19, which is 125gp3

Restriction to maximal subgroup number 20, which is 125gp3

Restriction to maximal subgroup number 21, which is 125gp3

Restriction to maximal subgroup number 22, which is 125gp3

Restriction to maximal subgroup number 23, which is 125gp3

Restriction to maximal subgroup number 24, which is 125gp3

Restriction to maximal subgroup number 25, which is 125gp3

Restriction to maximal subgroup number 26, which is 125gp3

Restriction to maximal subgroup number 27, which is 125gp3

Restriction to maximal subgroup number 28, which is 125gp3

Restriction to maximal subgroup number 29, which is 125gp3

Restriction to maximal subgroup number 30, which is 125gp3

Restriction to maximal subgroup number 31, which is 125gp3

Restrictions to maximal elementary abelian subgroups

Restriction to maximal elementary abelian number 1, which is V125

Restriction to maximal elementary abelian number 2, which is V125

Restriction to maximal elementary abelian number 3, which is V125

Restriction to maximal elementary abelian number 4, which is V125

Restriction to maximal elementary abelian number 5, which is V125

Restriction to maximal elementary abelian number 6, which is V125

Restriction to the greatest central elementary abelian subgroup

Restriction to the greatest central elementary abelian, which is V25

Poincaré series

(1 + 3t + 6t2 + 10t3 + 13t4 + 15t5 + 16t6 + 16t7 + 15t8 + 13t9 + 11t10 + 9t11 + 7t12 + 5t13 + 3t14 + t15) / (1 - t2) (1 - t6) (1 - t10)

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