GROUP OF ORDER 64 # 99

GROUP # 99

The MAGMA library number for G is 116

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

The center of G is abelian of type [ 2, 4 ]
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 8
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 [ 32, 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, 3, 3, 4 ] by the ideal generated by the relations
[ z^2 + y^2, z*y + y^2 + z*x, y^2*x, z*x*w + z*u + y*u, z*v + y*u, y*v + y*u + x*u, y^6 + y^2*w^2 + u^2, y*x*w^2 + z*x*t + y*x*t + v*u + u^2, x^2*w^2 + x^2*t + v^2 + u^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 + z*x + y*x + x^2 ] of degrees [ 2, 4, 2 ]

The hypercohomolgy spectral sequence has E2 term:

ROW (0) [ y, x, z ] [ y*x, z*x, x^2 ] [ v, u ] [ x*u, x*v, 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.6, G.6, 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, z*y + y^2, z*y^2 + z*x^2, 0, z^2*y^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, u ]

Subgroup E # 2
Generated by [ G.5 * G.6, G.1 * G.2 * G.4 * G.5 * G.6, G.6 ]

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*y + z*y^2, z^3*y + z^2*y^2 + 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, x, y^3 + y*w + u, v + u ]

The nilradical of the cohomology of G is generated by
[ z + y, y^3 + y*w + u ]

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

The Group H is Isomorphic to the Group of Order 32 Number 14 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.3 * G.4, G.1 * G.2 * G.5, G.1 * G.2 * G.3 ]

of type Cyclic(4) x Dihedral(8)

The images of the generators of the cohomology of G restricted to H are
[ z + y, z + y, y + x, z*y + y^2 + x^2 + w, z*x^2 + x^3 + z*w + y*w + y*v + x*v, z*x^2 + x^3 + z*w + y*w, z*y*x^2 + x^2*w + x^2*v + w^2 + v^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 32 Number 20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1, G.1 * G.2 * G.6 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 21 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.2 * G.4, G.1 * G.3 ]

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

Maximal Subgroup H # 4

The Group H is Isomorphic to the Group of Order 32 Number 19 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.5, $.2^$.1 = $.2 * $.5 Generated by [ G.1, G.1 * G.2 * G.3 * G.4 * G.5 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 13 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3 * $.5, $.2^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2 * G.3 * G.5, G.1 * G.2, G.1 * G.3 ]

of type Cyclic(2) x Group(16)#11

The images of the generators of the cohomology of G restricted to H are
[ y + x, z + x, z + y, z*y + z*x + x^2, z^2*y + x^3 + w, z*y^2 + x^3 + w, z^2*y*x + y*x^3 + v ] 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 abelian of type [ 8, 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 + x, z*y + y*x, z*y*x + y*x^2 + x^3 + y*w + x*w, z*x^2 + z*w, y^2*x^2 + x^4 + 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 # 7

The Group H is Isomorphic to the Group of Order 32 Number 20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1, G.3 * G.4 * G.6 ]

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

THE COMPUTATION IS COMPLETE