GROUP OF ORDER 32 # 40

GROUP # 40

The MAGMA library number for G is 32

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

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

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

The Hilbert series for the cohomology ring is (t^4 + t^3 + t^2 + t + 1)/(t^6 - 2*t^5 + 3*t^4 - 4*t^3 + 3*t^2 - 2*t + 1)
Its numerator factors as ( t^4 + t^3 + t^2 + t + 1 )
Its denominator factors as ( t - 1 )^2 ( t^2 + 1 )^2

The Krull dimension of the cohomology ring is 2
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators [ u, t ]

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.4, G.4 * G.5 ]

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

The images of the generators of the cohomology of G restricted to E are
[ 0, 0, 0, 0, 0, z^4, z^4 + y^4 ] in the cohomology of E

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

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 10 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.3, G.2 * G.3 ]

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

Maximal Subgroup H # 2

The Group H is Isomorphic to the Group of Order 16 Number 10 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.2 * G.3 * G.4, G.2 * G.3 ]

The images of the generators of the cohomology of G restricted to H are
[ y, z + y, z + y, z*x + z*w, z*w, y^2*w + w^2, y^2*w + x^2 + 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 10 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.2 * G.5, G.1 * G.2 * G.3 * G.4 * G.5 ]

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

The Group H is Isomorphic to the Group of Order 16 Number 10 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.1, G.3 ]

The images of the generators of the cohomology of G restricted to H are
[ y, 0, z, z*x + y*w, z*x + y*x + z*w, x^2, y^2*x + w^2 ] in the cohomology of H

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

Maximal Subgroup H # 5

The Group H is Isomorphic to the Group of Order 16 Number 10 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.2, G.1 * G.2 * G.4 ]

The images of the generators of the cohomology of G restricted to H are
[ y, z + y, 0, z*x + z*w + y*w, z*x + y*x + y*w, w^2, y^2*x + x^2 ] in the cohomology of H

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

Maximal Subgroup H # 6

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

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

The images of the generators of the cohomology of G restricted to H are
[ y, z, y, z*x + y*x + z*w + y*w, z*x + y*x, z*y*x + w^2, z*y*x + x^2 ] 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 10 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.3, G.2 ]

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

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

The essential cohomology of G is generated as an ideal by the elements
[ x^3, y*x^2, y^2*v, x^2*v, y*x*v ]

The annihilator of the Essential Cohomology has dimension 2 The essential cohomology is a free module over the polynomial subring Q of the cohomology ring of G generated by [ u, t ] .
The essential cohomology is generated as a module over Q by the elements [] [] [ y*x^2, x^3 ] [] [ y*x*v, y^2*v, x^2*v ] [ w^2 ]

CHECKS

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

THE COMPUTATION IS COMPLETE