GROUP OF ORDER 32 # 9

GROUP # 9

Abelian(2,2) x Quaternion(8)

The MAGMA library number for G is 47

GrpPC : G of order 32 = 2^5 PC-Relations: G.1^2 = G.5, G.2^2 = G.5, G.3^2 = G.5, G.4^2 = G.5, G.2^G.1 = G.2 * G.5, G.3^G.1 = G.3 * G.5, G.4^G.2 = G.4 * G.5, G.4^G.3 = G.4 * G.5

The center of G is abelian of type [ 2, 2, 2 ]
The orders of the terms of the lower central series are [ 32, 2, 1 ]
The orders of the terms of the upper central series are [ 1, 8, 32 ]
The order of the Frattini subgroup is 2
The exponent of G is 4
G has a unique maximal elementary abelian subgroup which is central in G and has order 8

The cohomology ring of G is the quotient of a polynomial ring in the variables [ z, y, x, w, v ] in degrees [ 1, 1, 1, 1, 4 ] by the ideal generated by the relations
[ z^2 + z*y + y^2 + z*x + x^2 + y*w + x*w + w^2, y^3 + y^2*x + y*x^2 + x^3 ]

The Hilbert series for the cohomology ring is (-t^2 - t - 1)/(t^5 - 3*t^4 + 4*t^3 - 4*t^2 + 3*t - 1)
Its numerator factors as ( t^2 + t + 1 )
Its denominator factors as ( t - 1 )^3 ( t^2 + 1 )^1

The Krull dimension of the cohomology ring is 3
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators [ z^2, y^2, v ]

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.2 * G.3 * G.5, G.2 * G.3, G.1 * G.2 * G.3 * G.4 * G.5 ]

The cohomology ring of E is a polynomial ring in the variables [ z, y, x ]

The images of the generators of the cohomology of G restricted to E are
[ x, z + y + x, z + y + x, x, y^4 + x^4 ] in the cohomology of E

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by
[ z + w, y + x ]

This ideal is also the nilradical of the cohomology of G.

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

The Group H is Isomorphic to the Group of Order 16 Number 7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.3 * G.4, G.3, G.1 * G.2 * G.3 * G.4 ]

of type Cyclic(2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ x, x, z + y + x, z + x, w ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ z + y ]

Maximal Subgroup H # 2

The group H is abelian of type [ 4, 2, 2 ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables [ z, y, x, w ] in degrees [ 1, 1, 1, 2 ] by the ideal of relations
[ z^2 ]

The images of the generators of the cohomology of G restricted to H are
[ z + y, x, x, y, x^4 + w^2 ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ y + x ]

Maximal Subgroup H # 3

The Group H is Isomorphic to the Group of Order 16 Number 7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.2 * G.3 * G.5, G.1 * G.2 * G.5, G.4 ]

of type Cyclic(2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z, z + x, x, y, w ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ z + y + x ]

Maximal Subgroup H # 4

The Group H is Isomorphic to the Group of Order 16 Number 7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.4, G.2, G.1 * G.3 * G.5 ]

of type Cyclic(2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z + x, y, z, x, x^4 + w ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ z + x + w ]

Maximal Subgroup H # 5

The Group H is Isomorphic to the Group of Order 16 Number 7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.2 * G.5, G.1 * G.4, G.3 ]

of type Cyclic(2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ y + x, y, z, x, x^4 + w ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ z + y + w ]

Maximal Subgroup H # 6

The Group H is Isomorphic to the Group of Order 16 Number 7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.2 * G.3 * G.4 * G.5, G.1 * G.3 * G.4 * G.5, G.1 * G.3 * G.5 ]

of type Cyclic(2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z + y + x, x, z + y + x, z + x, x^4 + w ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ z + x ]

Maximal Subgroup H # 7

The Group H is Isomorphic to the Group of Order 16 Number 7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.2 * G.3 * G.5, G.1, G.2 ]

of type Cyclic(2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ y, z + x, x, 0, w ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ w ]

Maximal Subgroup H # 8

The Group H is Isomorphic to the Group of Order 16 Number 7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.3 * G.4, G.1 * G.2 * G.3 * G.4 * G.5, G.1 * G.3 * G.4 * G.5 ]

of type Cyclic(2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z + x, x, z + y + x, z + y + x, x^4 + w ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ x + w ]

Maximal Subgroup H # 9

The Group H is Isomorphic to the Group of Order 16 Number 7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.2 * G.3 * G.5, G.4, G.2 ]

of type Cyclic(2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ 0, z + x, x, y, w ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ z ]

Maximal Subgroup H # 10

The Group H is Isomorphic to the Group of Order 16 Number 7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.4 * G.5, G.4, G.2 ]

of type Cyclic(2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ x, y, 0, z + x, w ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ x ]

Maximal Subgroup H # 11

The group H is abelian of type [ 4, 2, 2 ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables [ z, y, x, w ] in degrees [ 1, 1, 1, 2 ] by the ideal of relations
[ z^2 ]

The images of the generators of the cohomology of G restricted to H are
[ x, z + y, y, x, x^4 + w^2 ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ z + w ]

Maximal Subgroup H # 12

The Group H is Isomorphic to the Group of Order 16 Number 7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.4 * G.5, G.3, G.4 ]

of type Cyclic(2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ x, 0, y, z + x, w ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ y ]

Maximal Subgroup H # 13

The group H is abelian of type [ 4, 2, 2 ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables [ z, y, x, w ] in degrees [ 1, 1, 1, 2 ] by the ideal of relations
[ z^2 ]

The images of the generators of the cohomology of G restricted to H are
[ z + x, z + y, y, x, x^4 + w^2 ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ z + y + x + w ]

Maximal Subgroup H # 14

The Group H is Isomorphic to the Group of Order 16 Number 7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.3, G.1 * G.2 * G.3 * G.4, G.1 * G.3 * G.5 ]

of type Cyclic(2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ z + x, x, z + y + x, x, w ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ y + w ]

Maximal Subgroup H # 15

The Group H is Isomorphic to the Group of Order 16 Number 7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.2 * G.3 * G.5, G.2 * G.4, G.1 ]

of type Cyclic(2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ y, z + x, x, z, w ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ y + x + w ]

The essential cohomology of G is generated as an ideal by the elements
[ z*y^2*x^2*w + z*x^4*w + z*y^2*x*w^2 + y^2*x^2*w^2 + z*x^3*w^2 + x^4*w^2 + y^2*x*w^3 + x^3*w^3, z*y*x^4*w^2 + z*x^5*w^2 + y*x^4*w^3 + x^5*w^3 + z*y*x^2*w^4 + z*x^3*w^4 + y*x^2*w^5 + x^3*w^5, y^2*x^4*w^2 + x^6*w^2 + y^2*x^2*w^4 + x^4*w^4 ]

The annihilator of the Essential Cohomology has dimension 3 no end to the new generators was found.

CHECKS

paramflag = true
qregflagflag = true
cmflag = true
essflag = true
centflag = true

THE COMPUTATION IS COMPLETE