GROUP OF ORDER 64 # 85

GROUP # 85

The MAGMA library number for G is 71

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.3, G.2^G.1 = G.2 * G.6, G.4^G.1 = G.4 * G.6, G.4^G.2 = G.4 * G.5

The center of G is abelian of type [ 2, 2, 2 ]
The orders of the terms of the lower central series are [ 64, 4, 1 ]
The orders of the terms of the upper central series are [ 1, 8, 64 ]
The order of the Frattini subgroup is 8
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 [ 16, 16 ]
The orders of the centralizers of the maximal elementary abelian subgroups are [ 16, 16 ]
The orders of the normalizers of the maximal elementary abelian subgroups are [ 64, 64 ]

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, 2, 2, 2 ] by the ideal generated by the relations
[ z^2, z*y + z*x, y*x, z*v, y^2*w + x^2*w + v^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 [ w, u, t ]
A homogeneous set of parameters is the set [ w, u, t, y^4 + x^4 ] of degrees [ 2, 2, 2, 4 ]

The hypercohomolgy spectral sequence has E2 term:

ROW (0) [ y, z, x ] [ x^2, z*x, v, y^2 ] [ x*v, x^3, y^3, y*v ] [ y^2*v, x^4, x^2*v ] [ y^3*v, x^3*v ] [ x^4*v ]
ROW (1) [ z ] [ z*x ]
The spectral sequence satisfies Poincaré duality

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E # 1
Generated by [ G.3 * G.4 * G.5 * G.6, G.3 * G.5, G.5 * G.6, 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
[ 0, 0, z, z^2 + y^2 + w^2, z^2 + z*y + z*w, z*x + x^2, z*x + x^2 + z*w + w^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 ]

Subgroup E # 2
Generated by [ G.3 * G.5, G.5 * G.6, 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
[ 0, z, 0, y^2 + w^2, z*y + z*w, z*x + x^2, z*x + x^2 + z*w + w^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 generated by
[ z ]

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

The group H is abelian of type [ 4, 4, 2 ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables [ z, y, x, w, v ] in degrees [ 1, 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, z, v, y*x, y*x + x^2, x^2 + 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 # 2

The Group H is Isomorphic to the Group of Order 32 Number 11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.4^$.1 = $.4 * $.5 Generated by [ G.1 * G.2 * G.6, G.4, G.5 ]

of type Cyclic(2) x Group(16)#9

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

The Group H is Isomorphic to the Group of Order 32 Number 12 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.2 * G.3 * G.5, G.1 * G.4 * G.6, G.6 ]

of type Cyclic(2) x Group(16)#10

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

The Group H is Isomorphic to the Group of Order 32 Number 11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.4^$.1 = $.4 * $.5 Generated by [ G.1 * G.4, G.5, G.2 ]

of type Cyclic(2) x Group(16)#9

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

The Group H is Isomorphic to the Group of Order 32 Number 11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.4^$.1 = $.4 * $.5 Generated by [ G.5, G.1, G.2 ]

of type Cyclic(2) x Group(16)#9

The images of the generators of the cohomology of G restricted to H are
[ z, x, 0, w, v, u, y^2 + y*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 # 6

The Group H is Isomorphic to the Group of Order 32 Number 8 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.2 * G.4 * G.5, G.2 * G.4 * G.5 * G.6, G.3 * G.4 * G.6, G.3 ]

of type Abelian(2,2) x Dihedral(8)

The images of the generators of the cohomology of G restricted to H are
[ 0, z + y, z + y + x, x^2 + w^2, x^2 + x*w, y^2 + y*x, y*x + v ] 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 32 Number 11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.4^$.1 = $.4 * $.5 Generated by [ G.4, G.5, G.1 ]

of type Cyclic(2) x Group(16)#9

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

THE COMPUTATION IS COMPLETE