GROUP OF ORDER 64 # 98

GROUP # 98

The MAGMA library number for G is 113

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

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 a unique maximal elementary abelian subgroup which is normal and has order 8. Its centralizer has order 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, 3, 4 ] by the ideal generated by the relations
[ z^2 + z*x + y*x + x^2, y^2 + z*x + y*x, z*x^2 + y*x^2 + x^3, z*v + y*v + x*v, x^6 + v^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 [ w, u ]
A homogeneous set of parameters is the set [ w, u, y^2 ] of degrees [ 2, 4, 2 ]

The hypercohomolgy spectral sequence has E2 term:

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

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.5 * G.6, G.5, G.2 * G.3 ] 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, z, z, y^2 + x^2, z^3, 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, x^3 + v ]

This ideal is also the nilradical of the cohomology of G.

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

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.2 * G.3 * G.4 * G.6 ]

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

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

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

Maximal Subgroup H # 3

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.2, G.3 ]

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

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.3, G.2 * G.3 * G.4 * G.5 * G.6 ]

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

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.1, G.3 ]

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

Maximal Subgroup H # 7

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.4 * G.5 * G.6 ]

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

The essential cohomology of G is generated as an ideal by
[ z*y*x + y*x^2 + x^3 ]

The annihilator of the Essential Cohomology has dimension 2 The essential cohomology is a free module over the polynomial subring Q of the cohomology ring of G generated by [ w, u ] .
The essential cohomology is generated as a module over Q by the elements [] [] [ z*y*x + y*x^2 + x^3 ]

CHECKS

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

THE COMPUTATION IS COMPLETE