GROUP OF ORDER 32 # 38

GROUP # 38

The MAGMA library number for G is 30

GrpPC : G of order 32 = 2^5 PC-Relations: G.1^2 = G.4 * G.5, G.2^2 = G.5, G.2^G.1 = G.2 * G.4, G.3^G.1 = 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 2 conjugacy classes of maximal elementary abelian p-subgroups. The orders of the maximal elementary abelian subgroups are [ 8, 8 ]
The orders of the centralizers of the maximal elementary abelian subgroups are [ 16, 8 ]
The orders of the normalizers of the maximal elementary abelian subgroups are [ 32, 32 ]

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, 2, 3, 3, 4 ] by the ideal generated by the relations
[ z^2 + y^2 + z*x, z*y + y^2 + z*x, z*x^2, z*v + z*u, y*v + z*u, x*v + z*u + y*u, y*x^3*w + v^2 + v*u, x^4*w + y*x^2*u + z*x*t + v^2 + u^2, y^3*u + y^2*t + z*x*t + v^2 ]

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

The Krull dimension of the cohomology ring is 3
The longest regular sequence consists of the generators [ w, t ]
A homogeneous set of parameters is the set [ w, t, y^2 + x^2 ] of degrees [ 2, 4, 2 ]

The hypercohomolgy spectral sequence has E2 term:

ROW (0) [ z, y, x ] [ z*x, x^2, y*x ] [ u, v ] [ z*u, x*u, y*u ] [ x^2*u ]
ROW (1) [] [ z*x ]
The spectral sequence satisfies Poincaré duality

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E # 1
Generated by [ G.5, G.4 * G.5, G.3 * 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
[ 0, 0, z, y^2, 0, z^2*y, z^2*y^2 + y^4 + z^2*x^2 + 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, y, v ]

Subgroup E # 2
Generated by [ G.5, G.4 * G.5, G.1 * G.2 * G.4 ]

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
[ z, z, 0, z^2 + z*y + y^2, z^2*y + z*y^2 + z^2*x + z*x^2, z^2*y + z*y^2 + z^2*x + z*x^2, z^3*y + y^4 + z^3*x + 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 + y, x, v + u ]

The nilradical of the cohomology of G is generated by
[ z + y ]

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.2, G.1 * G.3 ]

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

Maximal Subgroup H # 3

The Group H is Isomorphic to the Group of Order 16 Number 9 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.2 * G.5, G.1 * G.3 ]

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

The Group H is Isomorphic to the Group of Order 16 Number 6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.2 * G.3 * G.4 * G.5, G.3 * G.4, G.1 * G.2 * G.4 * G.5 ]

of type Cyclic(2) x Dihedral(8)

The images of the generators of the cohomology of G restricted to H are
[ z + x, z + x, y + x, z^2 + y^2 + z*x, z^2*x + z*x^2 + z*w + x*w, y^3 + z^2*x + z*x^2 + y*x^2 + z*w + x*w, z^3*y + z^2*y*x + z*y^2*x + z^2*w + x^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 ]

Maximal Subgroup H # 5

The Group H is Isomorphic to the Group of Order 16 Number 9 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.2 * G.5, G.1 ]

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

Maximal Subgroup H # 6

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.3, G.1 ]

The images of the generators of the cohomology of G restricted to H are
[ y, z, z, y^2 + w, y*x + y*w, z*x + z*w, y^2*x + 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 # 7

The Group H is Isomorphic to the Group of Order 16 Number 9 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^$.1 = $.2 * $.4 Generated by [ G.3, G.1 ]

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

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

The essential cohomology of G is zero

CHECKS

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

THE COMPUTATION IS COMPLETE