GROUP OF ORDER 64 # 15

GROUP # 15

Abelian(2,2) x Group(16)#9

The MAGMA library number for G is 193

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

The center of G is abelian of type [ 2, 2, 2, 2 ]
The orders of the terms of the lower central series are [ 64, 2, 1 ]
The orders of the terms of the upper central series are [ 1, 16, 64 ]
The order of the Frattini subgroup is 4
The exponent of G is 4
G has a unique maximal elementary abelian subgroup which is normal and has order 32. 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, 1, 2, 2, 2 ] by the ideal generated by the relations
[ z^2, z*x + z*w, z*v, x^2*u + w^2*u + v^2 ]

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

The Krull dimension of the cohomology ring is 5
The longest regular sequence consists of the generators [ y^2, x^2, u, t ]
A homogeneous set of parameters is the set [ y^2, x^2, u, t, w^2 ] of degrees [ 2, 2, 2, 2, 2 ]

The hypercohomolgy spectral sequence has E2 term:

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

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.2, G.2 * G.3 * G.4 * G.5, G.2 * G.3 * G.4, G.2 * G.3 * G.5 * G.6, G.2 * G.3 * G.4 * G.6 ] The cohomology ring of E is a polynomial ring in the variables [ z, y, x, w, v ]

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

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

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

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

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

The cohomology ring of H is a polynomial ring with variables [ z, y, x, w, v ] in degrees [ 1, 1, 1, 1, 1 ]

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

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.2 * G.4, G.1 * G.2, 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
[ z, z + x, y, y + x, v, w, u ] in the cohomology of H

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

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.2 * G.4, G.2 * G.3 * G.4, 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, y + x, y, y + x, v, w, u ] in the cohomology of H

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

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.2, 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, y, x, z, v, w, w + v + u ] in the cohomology of H

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

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.5, G.1, 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
[ z, 0, y, y + x, x^2 + v, x^2 + w, u ] 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.2 * G.3 * G.4, G.1 * G.2, G.2 * 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 + y + x, y, y + x, x^2 + v, x^2 + w, u ] in the cohomology of H

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

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.2 * G.4, G.1, 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
[ z, x, y, y + x, v, w, u ] in the cohomology of H

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

Maximal Subgroup H # 8

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.2 * G.3 * G.4, 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, y, y, y + x, x^2 + v, x^2 + w, u ] 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 # 9

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 + x, y, y, z*w, v, z*w + w^2 ] in the cohomology of H

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

Maximal Subgroup H # 10

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.2, G.1 * G.3, G.4 * G.5 * 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, y, z, x, x^2 + v, x^2 + w, w + 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 # 11

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.2 * G.3 * G.4, 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, z + y, y, y + x, v, w, u ] 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 # 12

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.2, 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, y, 0, x, v, w, u ] in the cohomology of H

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

Maximal Subgroup H # 13

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.2, 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
[ z, z, y, y + x, v, w, 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 # 14

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.2, G.3 * 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, y, x, 0, x^2 + v, x^2 + w, u ] in the cohomology of H

The kernel of the restriction to H of the cohomology of G is generated by
[ w ]

Maximal Subgroup H # 15

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 + x, z + y, y, z*w, w^2 + v, v ] in the cohomology of H

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

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