GROUP OF ORDER 64 # 91

GROUP # 91

The MAGMA library number for G is 69

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.3, G.2^2 = G.3, G.2^G.1 = G.2 * G.5, G.4^G.1 = G.4 * G.5, G.4^G.2 = G.4 * 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 16

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, 3 ] by the ideal generated by the relations
[ z^2 + y^2, z*y + z*x, y*x, z*t, y*t, x^4*v + t^2 ]

The Hilbert series for the cohomology ring is (-t^2 + t + 1)/(t^6 - 2*t^5 - t^4 + 4*t^3 - t^2 - 2*t + 1)
Its numerator factors as ( t^2 - t - 1 )
Its denominator factors as ( t - 1 )^4 ( t + 1 )^2

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

The hypercohomolgy spectral sequence has E2 term:

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

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.3 * G.5, G.5, G.4 * G.6, G.3 * 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
[ 0, 0, z, z*x + x^2 + z*w + w^2, y^2 + w^2, z*y + y^2, z^2*y + z^2*w ] in the cohomology of E

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by
[ z, y ]

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 18 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.4 * G.5, G.1 * G.2 * G.5 ]

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

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
[ z + y, z, z, z*x + y*x + x^2, v, w, z*y*x ] 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 # 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.5, G.4, G.2 ]

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

The images of the generators of the cohomology of G restricted to H are
[ 0, z, x, z*y + y^2 + y*x, w, u, 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 # 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.1 * G.4 * G.5, G.2 ]

The images of the generators of the cohomology of G restricted to H are
[ z, y, z, x + w + u, w + t, u, y*x ] 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.4, G.1, G.6 ]

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, u, w, y^2 + y*x, x*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 # 6

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.5, G.4, G.3 ]

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 + u, y^2, u, y*x^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 # 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.1, G.2, 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, w, v, y*x + x^2, z^2*x ] 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 zero

CHECKS

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

THE COMPUTATION IS COMPLETE