GROUP OF ORDER 64 # 81

GROUP # 81

The MAGMA library number for G is 60

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.4 * G.5, G.2^2 = G.4, G.2^G.1 = G.2 * G.5, G.3^G.1 = G.3 * G.6, 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 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, s, r, q ] in degrees [ 1, 1, 1, 2, 2, 2, 2, 2, 2, 3 ] by the ideal generated by the relations
[ z^2 + y^2, zy + y^2, zx + yx, y^2x + xw + yu + yt, zw + yw, zu + yu, zt + yt, yxw + yxs + wt + yq, x^2w + yxt + x^2s + ut + t^2 + xq, x^2v + x^2s + y^2r + u^2 + ut + t^2 + xq, yxu + u^2 + ut + xq, y^2s + w^2, wu + yq, zq + yq, y^3r + xvu + yu^2 + xwt + xvt + yut + yt^2 + ywr + yxq + uq, yus + wq, xus + yxq + uq + tq, uts + xwq + xsq + q^2 ]

The Hilbert series for the cohomology ring is (-t^2 - 1)/(t^7 - 3*t^6 + t^5 + 5*t^4 - 5*t^3 - t^2 + 3*t - 1)
Its numerator factors as ( t^2 + 1 )
Its denominator factors as ( t - 1 )^5 ( t + 1 )^2

The Krull dimension of the cohomology ring is 5
The longest regular sequence consists of the generators [ v, s, r ]
A homogeneous set of parameters is the set [ v, s, r, z^2, x^2 ] of degrees [ 2, 2, 2, 2, 2 ]

The hypercohomolgy spectral sequence has E2 term:

ROW (0 0) [ y, x, z ] [ t, u, w, y*x ] [ y*u, y*w, x*u, q, y*t, x*t ] [ y*q, x*q, t^2 ] [ u*q ]
ROW (1 0) [ z + y ]
ROW (0 1) [ z + y ] [] [ y*w ]
ROW (1 1) [ z + y ]

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.1 * G.2 * G.3 * G.6, G.1 * G.2, G.4 * G.5, G.5, 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
[ z + v, z + v, v, z^2 + zy + zw + yv + wv + v^2, z^2 + zx + x^2 + zw + w^2 + xv + wv + v^2, zy + yv + xv + wv, zy + xv, z^2 + y^2 + w^2 + v^2, y^2 + yv, z^2y + zy^2 + zyw + y^2v + zxv + yxv + zwv + xwv + w^2v + yv^2 + xv^2 + wv^2 ] 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 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.5, G.2 * G.5 ]

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, zy + x^2 + v, zy + y^2 + yx + x^2 + u, yx, yx + v, x^2 + w, u, yx^2 + yv + zu ] 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 # 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.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, 0, x, zy, y^2 + w, yx + v, yx, w, u, xw + y*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 # 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.6, G.1 * G.2 * 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
[ x, z + x, 0, x^2 + v, x^2 + u, yx, zy + yx, x^2 + w, zy + y^2, yx^2 + yv ] in the cohomology of H

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

Maximal Subgroup H # 4

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
[ z, z, y, z^2 + zw, z^2 + zx + x^2, zy + yx + zv, zy + yx + yw + zv, z^2 + w^2, yv + v^2, z^2y + zyx + zyw + yxw + z^2v + zwv ] 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 # 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.2 * G.3 * G.6, G.1 * G.2 * G.4 * G.6, 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, z + x, z, yx + v, x^2 + u, x^2 + v, zy + x^2 + v, y^2 + w, x^2 + w, yx^2 + xw + yv + xv ] 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 11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.4^$.1 = $.4 * $.5 Generated by [ G.3, G.5, 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, zy, zy + y^2, yx, yx + v, w, u, y*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 # 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.1 * G.2 * G.3 * G.6, G.2 * G.3 * G.5 * G.6, G.5 ]

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, z + x, zy + x^2 + v, zy + y^2 + yx + x^2 + u, zy + yx + v, zy + yx, x^2 + w, w + v + u, yx^2 + xw + yv + xv + zu ] in the cohomology of H

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

GROUP # 81

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

The hypercohomolgy spectral sequence has E2 term:

Restriction to the Maximal Elementary Abelian Subgroup

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 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.5, G.2 * G.5 ]

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, z*y + x^2 + v, z*y + y^2 + y*x + x^2 + u, y*x, y*x + v, x^2 + w, u, y*x^2 + y*v + z*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 # 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.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, 0, x, z*y, y^2 + w, y*x + v, y*x, w, u, x*w + y*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 # 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.6, G.1 * G.2 * 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 [ x, z + x, 0, x^2 + v, x^2 + u, y*x, z*y + y*x, x^2 + w, z*y + y^2, y*x^2 + y*v ] in the cohomology of H

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

Maximal Subgroup H # 4

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 [ z, z, y, z^2 + z*w, z^2 + z*x + x^2, z*y + y*x + z*v, z*y + y*x + y*w + z*v, z^2 + w^2, y*v + v^2, z^2*y + z*y*x + z*y*w + y*x*w + z^2*v + z*w*v ] 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 # 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.2 * G.3 * G.6, G.1 * G.2 * G.4 * G.6, 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, z + x, z, y*x + v, x^2 + u, x^2 + v, z*y + x^2 + v, y^2 + w, x^2 + w, y*x^2 + x*w + y*v + 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 ]

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.3, G.5, 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, z*y + y^2, y*x, y*x + v, w, u, y*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 # 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.1 * G.2 * G.3 * G.6, G.2 * G.3 * G.5 * G.6, G.5 ]

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

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

The essential cohomology of G is zero

CHECKS

THE COMPUTATION IS COMPLETE

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

The images of the generators of the cohomology of G restricted to H are
[ x, z + x, x, zy + x^2 + v, zy + y^2 + yx + x^2 + u, yx, yx + v, x^2 + w, u, yx^2 + yv + zu ] in the cohomology of H

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

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

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

The images of the generators of the cohomology of G restricted to H are
[ x, z + x, 0, x^2 + v, x^2 + u, yx, zy + yx, x^2 + w, zy + y^2, yx^2 + yv ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ z, z, y, z^2 + zw, z^2 + zx + x^2, zy + yx + zv, zy + yx + yw + zv, z^2 + w^2, yv + v^2, z^2y + zyx + zyw + yxw + z^2v + zwv ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ x, z + x, z, yx + v, x^2 + u, x^2 + v, zy + x^2 + v, y^2 + w, x^2 + w, yx^2 + xw + yv + xv ] in the cohomology of H

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

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

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

The images of the generators of the cohomology of G restricted to H are
[ x, z + x, z + x, zy + x^2 + v, zy + y^2 + yx + x^2 + u, zy + yx + v, zy + yx, x^2 + w, w + v + u, yx^2 + xw + yv + xv + zu ] in the cohomology of H

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