GROUP OF ORDER 32 # 44

GROUP # 44

The MAGMA library number for G is 43

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

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, 3, 4 ] by the ideal generated by the relations
[ zy + y^2 + zx + yx, z^3 + y^3 + z^2x + y^2x, zw + yw, y^6 + y^5x + y^4x^2 + y^3x^3 + y^2xw + yx^2w + y^2v + x^2v + w^2 ]

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

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

The hypercohomolgy spectral sequence has E2 term:

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

Restrictions to Maximal Elementary Abelian Subgroups

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

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

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

The nilradical of the cohomology of G is generated by
[ z^2 + y^2 + zx + yx ]

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

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.2 * G.4, G.1 * G.2 * G.4 * G.5 ]

of type Semidihedral(16)

The images of the generators of the cohomology of G restricted to H are
[ z, z + y, 0, z^3 + x, 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 # 2

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.1 * G.4, G.2 * G.3 * G.5, G.2 * G.3 * G.4 * G.5 ]

of type AlmostExtraSpecial(16)

The images of the generators of the cohomology of G restricted to H are
[ x, z + y, z + y, y*x^2 + x^3, w ] 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 12 GrpPC of order 16 = 2^4 PC-Relations: $.2^2 = $.3 * $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.1 = $.3 * $.4 Generated by [ G.1 * G.3 * G.4, G.1 * G.2 * G.4 ]

of type Dihedral(16)

The images of the generators of the cohomology of G restricted to H are
[ z + y, z, y, z^3 + y^3 + zx + yx, 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 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.3, G.4, G.1 * G.2 * G.5 ]

of type Cyclic(2) x Dihedral(8)

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

The images of the generators of the cohomology of G restricted to H are
[ y, z, y, zy^2 + x, zy^3 + 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 # 6

The Group H is Isomorphic to the Group of Order 16 Number 12 GrpPC of order 16 = 2^4 PC-Relations: $.2^2 = $.3 * $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.1 = $.3 * $.4 Generated by [ G.3, G.2 ]

of type Dihedral(16)

The images of the generators of the cohomology of G restricted to H are
[ 0, y, z, zx + yx, x^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 # 7

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.3, G.3 ]

of type Semidihedral(16)

The images of the generators of the cohomology of G restricted to H are
[ y, 0, 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
[ y ]

The essential cohomology of G is zero

CHECKS

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

THE COMPUTATION IS COMPLETE

GROUP # 44

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

The hypercohomolgy spectral sequence has E2 term:

Restrictions to Maximal Elementary Abelian Subgroups

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

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

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

The nilradical of the cohomology of G is generated by [ z^2 + y^2 + z*x + y*x ]

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

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.2 * G.4, G.1 * G.2 * G.4 * G.5 ]

of type Semidihedral(16)

The images of the generators of the cohomology of G restricted to H are [ z, z + y, 0, z^3 + x, 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 # 2

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.1 * G.4, G.2 * G.3 * G.5, G.2 * G.3 * G.4 * G.5 ]

of type AlmostExtraSpecial(16)

The images of the generators of the cohomology of G restricted to H are [ x, z + y, z + y, y*x^2 + x^3, w ] 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 12 GrpPC of order 16 = 2^4 PC-Relations: $.2^2 = $.3 * $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.1 = $.3 * $.4 Generated by [ G.1 * G.3 * G.4, G.1 * G.2 * G.4 ]

of type Dihedral(16)

The images of the generators of the cohomology of G restricted to H are [ z + y, z, y, z^3 + y^3 + 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 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.3, G.4, G.1 * G.2 * G.5 ]

of type Cyclic(2) x Dihedral(8)

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

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.3 * G.4, G.2 * G.4 * G.5 ]

The images of the generators of the cohomology of G restricted to H are [ y, z, y, z*y^2 + x, z*y^3 + 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 # 6

The Group H is Isomorphic to the Group of Order 16 Number 12 GrpPC of order 16 = 2^4 PC-Relations: $.2^2 = $.3 * $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.1 = $.3 * $.4 Generated by [ G.3, G.2 ]

of type Dihedral(16)

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

Maximal Subgroup H # 7

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.3, G.3 ]

of type Semidihedral(16)

The images of the generators of the cohomology of G restricted to H are [ y, 0, 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 [ y ]

The essential cohomology of G is zero

CHECKS

THE COMPUTATION IS COMPLETE

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

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

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

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

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

The nilradical of the cohomology of G is generated by
[ z^2 + y^2 + zx + yx ]

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