GROUP OF ORDER 32 # 45

GROUP # 45

The MAGMA library number for G is 44

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

The center of G is abelian of type [ 2 ]
The orders of the terms of the lower central series are [ 32, 4, 2, 1 ]
The orders of the terms of the upper central series are [ 1, 2, 8, 32 ]
The order of the Frattini subgroup is 4
The exponent of G is 8
G has 2 conjugacy classes of maximal elementary abelian p-subgroups. The orders of the maximal elementary abelian subgroups are [ 4, 4 ]
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, 16 ]

The cohomology ring of G is the quotient of a polynomial ring in the variables [ z, y, x, w, v, u ] in degrees [ 1, 1, 1, 5, 5, 8 ] by the ideal generated by the relations
[ zy, y^3 + zx^2, zx^4, yw, zv, x^2w + y^2v, y^2x^8 + wv, yx^9 + x^10 + yx^4v + y^2u + v^2, z^2u + w^2 ]

The Hilbert series for the cohomology ring is (t^6 + t^5 + t^2 + t + 1)/(t^8 - 2*t^7 + 2*t^6 - 2*t^5 + 2*t^4 - 2*t^3 + 2*t^2 - 2*t + 1)
Its numerator factors as ( t^6 + t^5 + t^2 + t + 1 )
Its denominator factors as ( t - 1 )^2 ( t^2 + 1 )^1 ( t^4 + 1 )^1

The Krull dimension of the cohomology ring is 2
The longest regular sequence consists of the generators [ u ]
A homogeneous set of parameters is the set [ u, z^2 + y^2 + x^2 ] of degrees [ 8, 2 ]

The hypercohomolgy spectral sequence has E2 term:

ROW (0) [ x, z, y ] [ y*x, x^2, y^2, z*x ] [ x^3, y*x^2, y^2*x ] [ y*x^3 ] [ v, w ] [ z*w, x*w, x*v, y*v ] [ x^2*v, y*x*v, z*x*w ] [ x^3*v ]
ROW (1) [] [] [ z*x^2 ] [ z*x^3 ]
The spectral sequence satisfies Poincaré duality

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E # 1
Generated by [ G.5, G.3 * G.4 ]

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, z, 0, z^5, z^8 + z^4*y^4 + y^8 ] 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^5 + v, w ]

Subgroup E # 2
Generated by [ G.5, G.1 * 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
[ z, 0, 0, z^3y^2 + zy^4, 0, z^4*y^4 + y^8 ] in the cohomology of E

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

The nilradical of the cohomology of G is generated by
[ y, zx, x^5 + v, xw ]

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

The Group H is Isomorphic to the Group of Order 16 Number 11 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.3^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.2, G.1 * G.2 * G.3 ]

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

The Group H is Isomorphic to the Group of Order 16 Number 14 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.3, G.1 * G.2 * G.3 * G.4 ]

of type Quaternion(16)

The images of the generators of the cohomology of G restricted to H are
[ z + y, z, z + y, zx + yx, 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 # 3

The Group H is Isomorphic to the Group of Order 16 Number 14 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.2, G.2 * G.3 * G.5 ]

of type Quaternion(16)

The images of the generators of the cohomology of G restricted to H are
[ z, z + y, y, zx, zx + 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 + 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.3 * G.4 * G.5, G.2, G.4 ]

of type Cyclic(2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ 0, y, x, z^2yx^2 + y^2x^3, y^2x^3 + zx^4 + yx^4 + x^5 + yw, zy^2x^5 + zy*x^6 + yx^7 + x^8 + y^2x^2w + x^4w + 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 # 5

The Group H is Isomorphic to the Group of Order 16 Number 13 GrpPC of order 16 = 2^4 PC-Relations: $.2^2 = $.3, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.1 = $.3 * $.4 Generated by [ G.1, G.1 * G.2 * G.4 ]

of type Semidihedral(16)

The images of the generators of the cohomology of G restricted to H are
[ z + y, y, 0, z^2x + zw + yw, yw, z^5x + z^4w + 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 13 GrpPC of order 16 = 2^4 PC-Relations: $.2^2 = $.3, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.1 = $.3 * $.4 Generated by [ G.1 * G.2 * G.3 * G.4 * G.5, G.1 ]

of type Semidihedral(16)

The images of the generators of the cohomology of G restricted to H are
[ z + y, y, y, z^2x + zw + yw, yw, z^5x + z^4w + 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 8 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.3 * G.4 * G.5, G.1 * G.3, G.1 ]

of type AlmostExtraSpecial(16)

The images of the generators of the cohomology of G restricted to H are
[ z + x, 0, y + x, zw + xw, y^5 + yx^4, y^8 + y^2x^6 + w^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 generated as an ideal by the elements
[ z*x^3 ]

The annihilator of the Essential Cohomology has dimension 1 The essential cohomology is a free module over the polynomial subring Q of the cohomology ring of G generated by [ u ] .
The essential cohomology is generated as a module over Q by the elements [] [] [] [ z*x^3 ]

CHECKS

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

THE COMPUTATION IS COMPLETE

GROUP # 45

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

The cohomology ring of G is the quotient of a polynomial ring in the variables [ z, y, x, w, v, u ] in degrees [ 1, 1, 1, 5, 5, 8 ] by the ideal generated by the relations [ z*y, y^3 + z*x^2, z*x^4, y*w, z*v, x^2*w + y^2*v, y^2*x^8 + w*v, y*x^9 + x^10 + y*x^4*v + y^2*u + v^2, z^2*u + w^2 ]

The hypercohomolgy spectral sequence has E2 term:

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E # 1 Generated by [ G.5, G.3 * G.4 ]

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, z, 0, z^5, z^8 + z^4*y^4 + y^8 ] 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^5 + v, w ]

Subgroup E # 2 Generated by [ G.5, G.1 * 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 [ z, 0, 0, z^3*y^2 + z*y^4, 0, z^4*y^4 + y^8 ] in the cohomology of E

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

The nilradical of the cohomology of G is generated by [ y, z*x, x^5 + v, x*w ]

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

The Group H is Isomorphic to the Group of Order 16 Number 11 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.3^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.2, G.1 * G.2 * G.3 ]

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

The Group H is Isomorphic to the Group of Order 16 Number 14 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.3, G.1 * G.2 * G.3 * G.4 ]

of type Quaternion(16)

The images of the generators of the cohomology of G restricted to H are [ z + y, z, z + y, z*x + 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 # 3

The Group H is Isomorphic to the Group of Order 16 Number 14 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.2, G.2 * G.3 * G.5 ]

of type Quaternion(16)

The images of the generators of the cohomology of G restricted to H are [ z, z + y, y, z*x, z*x + 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 + 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.3 * G.4 * G.5, G.2, G.4 ]

of type Cyclic(2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are [ 0, y, x, z^2*y*x^2 + y^2*x^3, y^2*x^3 + z*x^4 + y*x^4 + x^5 + y*w, z*y^2*x^5 + z*y*x^6 + y*x^7 + x^8 + y^2*x^2*w + x^4*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 # 5

The Group H is Isomorphic to the Group of Order 16 Number 13 GrpPC of order 16 = 2^4 PC-Relations: $.2^2 = $.3, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.1 = $.3 * $.4 Generated by [ G.1, G.1 * G.2 * G.4 ]

of type Semidihedral(16)

The images of the generators of the cohomology of G restricted to H are [ z + y, y, 0, z^2*x + z*w + y*w, y*w, z^5*x + z^4*w + 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 13 GrpPC of order 16 = 2^4 PC-Relations: $.2^2 = $.3, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.1 = $.3 * $.4 Generated by [ G.1 * G.2 * G.3 * G.4 * G.5, G.1 ]

of type Semidihedral(16)

The images of the generators of the cohomology of G restricted to H are [ z + y, y, y, z^2*x + z*w + y*w, y*w, z^5*x + z^4*w + 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 8 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.3 * G.4 * G.5, G.1 * G.3, G.1 ]

of type AlmostExtraSpecial(16)

The images of the generators of the cohomology of G restricted to H are [ z + x, 0, y + x, z*w + x*w, y^5 + y*x^4, y^8 + y^2*x^6 + w^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 generated as an ideal by the elements [ z*x^3 ]

CHECKS

THE COMPUTATION IS COMPLETE

The cohomology ring of G is the quotient of a polynomial ring in the variables [ z, y, x, w, v, u ] in degrees [ 1, 1, 1, 5, 5, 8 ] by the ideal generated by the relations
[ zy, y^3 + zx^2, zx^4, yw, zv, x^2w + y^2v, y^2x^8 + wv, yx^9 + x^10 + yx^4v + y^2u + v^2, z^2u + w^2 ]

Subgroup E # 1
Generated by [ G.5, G.3 * G.4 ]

The images of the generators of the cohomology of G restricted to E are
[ 0, 0, z, 0, z^5, z^8 + z^4*y^4 + y^8 ] 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^5 + v, w ]

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

The images of the generators of the cohomology of G restricted to E are
[ z, 0, 0, z^3y^2 + zy^4, 0, z^4*y^4 + y^8 ] in the cohomology of E

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

The nilradical of the cohomology of G is generated by
[ y, zx, x^5 + v, xw ]

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

The images of the generators of the cohomology of G restricted to H are
[ z + y, z, z + y, zx + yx, 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 ]

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

The images of the generators of the cohomology of G restricted to H are
[ 0, y, x, z^2yx^2 + y^2x^3, y^2x^3 + zx^4 + yx^4 + x^5 + yw, zy^2x^5 + zy*x^6 + yx^7 + x^8 + y^2x^2w + x^4w + w^2 ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ z + y, y, 0, z^2x + zw + yw, yw, z^5x + z^4w + w^2 ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ z + y, y, y, z^2x + zw + yw, yw, z^5x + z^4w + w^2 ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ z + x, 0, y + x, zw + xw, y^5 + yx^4, y^8 + y^2x^6 + w^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 generated as an ideal by the elements
[ z*x^3 ]