GROUP OF ORDER 32 # 33

GROUP # 33

The MAGMA library number for G is 27

GrpPC : G of order 32 = 2^5 PC-Relations: G.1^2 = G.5, G.3^2 = G.4 * 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 2 conjugacy classes of maximal elementary abelian p-subgroups. The orders of the maximal elementary abelian subgroups are [ 8, 16 ]
The orders of the centralizers of the maximal elementary abelian subgroups are [ 8, 16 ]
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 ] in degrees [ 1, 1, 1, 2, 2, 2 ] by the ideal generated by the relations
[ z^2 + z*y + y*x + x^2, z*x + x^2, z*y^2 + y^2*x + z*w + x*w, y*x^3 + z*y*w + x^2*w + x^2*v + z*y*u + y^2*u + y*x*u + x^2*u + w^2 ]

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

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

The hypercohomolgy spectral sequence has E2 term:

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

Restrictions to Maximal Elementary Abelian Subgroups

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

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

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

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

The nilradical of the cohomology of G is zero

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

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

The cohomology ring of H is a polynomial ring with variables [ z, y, x, w ] in degrees [ 1, 1, 1, 1 ]

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

Maximal Subgroup H # 2

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.2 ]

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

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

of type Cyclic(2) x Dihedral(8)

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

of type Cyclic(2) x Dihedral(8)

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

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

Maximal Subgroup H # 5

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.2 * G.3, G.1 * G.3, 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 + y, y + x, z + x, z^2 + y^2 + z*x + x^2, w, z*x + y*x + x^2 + 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 # 6

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

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

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

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

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

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

The essential cohomology of G is zero

CHECKS

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

THE COMPUTATION IS COMPLETE