GROUP OF ORDER 64 # 86

GROUP # 86

The MAGMA library number for G is 66

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.4 * G.5 * G.6, G.2^2 = G.4, G.3^2 = G.5, G.2^G.1 = G.2 * G.5, 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, 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 a unique maximal elementary abelian subgroup which is normal and has order 16. 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, t ] in degrees [ 1, 1, 1, 2, 2, 2, 2 ] by the ideal generated by the relations
[ z^2 + y*x, z*y + z*x + y*x + x^2, y^2 + y*x, y*v + x*v, z*x^3 + x^4 + y*x*w + x^2*w + z*x*v + x^2*u + y*x*t + 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, z^2 ] of degrees [ 2, 2, 2, 2 ]

The hypercohomolgy spectral sequence has E2 term:

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

Restriction to the Maximal Elementary Abelian Subgroup

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

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 12 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.3 * G.4 * G.5, G.1 * 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, 0, y, y^2 + v, y^2 + z*x, y*x + x^2 + v, y^2 + w + 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 # 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.3 * G.5 * G.6, G.4, G.3 * G.4 ]

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

The images of the generators of the cohomology of G restricted to H are
[ x, x, z + x, y^2 + x^2, z*y + v, z*y + y*x + x^2 + u, y*x + x^2 + w + v + 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 11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.4^$.1 = $.4 * $.5 Generated by [ G.1 * G.2 * G.3 * G.5, G.2 * G.6, G.6 ]

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

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

The Group H is Isomorphic to the Group of Order 32 Number 18 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.2, G.1 ]

The images of the generators of the cohomology of G restricted to H are
[ y, z, 0, w + t, x, t, x + u + t ] 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 32 Number 18 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.2, G.3 ]

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

The group H is abelian of type [ 4, 2, 2, 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, 1, 2 ] by the ideal of relations
[ z^2 ]

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

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

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

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

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

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