GROUP OF ORDER 64 # 94

GROUP # 94

The MAGMA library number for G is 88

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.2, G.2^2 = G.5 * G.6, G.4^2 = G.5 * G.6, G.3^G.1 = G.3 * G.5, G.4^G.1 = G.4 * 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 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, s, r ] in degrees [ 1, 1, 1, 2, 2, 3, 3, 4, 4 ] by the ideal generated by the relations
[ z^2, zy + zx, zx^2, zw, y^2x^2 + x^4 + w^2, yx^3 + x^4 + yxw + w^2 + yt + xt, zxv + zu, zt, yx^2w + xwv + y^2u + x^2u + wu + ys + xs, yxt + x^2t + wt, zs, x^6 + x^4v + x^2w^2 + x^2v^2 + ywu + xwu + xwt + y^2r + x^2r + u^2 + ws, yxwv + x^2v^2 + y^2xu + x^3t + w^2v + ywu + xwu + yvt + yxs + y^2r + x^2r + u^2 + ut + ws, x^2wv + x^2v^2 + yx^2u + x^3t + xwu + xwt + xvt + x^2s + y^2r + x^2r + u^2 + ut, t^2, x^5v + yxwu + yxvu + x^2vu + x^2vt + yx^2s + w^2u + yu^2 + xu^2 + xws + xvs + ywr + xwr + us, x^4t + x^2wt + x^2vt + yut + ts, yx^2vu + x^3vu + x^3vt + x^4s + y^4r + x^4r + w^3v + w^2v^2 + xw^2u + xwvu + y^2u^2 + yxu^2 + yw^2t + xw^2t + ywvt + y^2ut + x^2ut + yxws + x^2vs + yxwr + x^2wr + wu^2 + wut + yus + w^2r + s^2 ]

The Hilbert series for the cohomology ring is (t^2 - t + 1)/(t^6 - 4*t^5 + 7*t^4 - 8*t^3 + 7*t^2 - 4*t + 1)
Its numerator factors as ( t^2 - t + 1 )
Its denominator factors as ( t - 1 )^4 ( t^2 + 1 )^1

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

The hypercohomolgy spectral sequence has E2 term:

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

Restriction to the Maximal Elementary Abelian Subgroup

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

The kernel of the restriction to E (Minimal Prime) of the cohomology of G is generated by
[ z, yx + x^2 + w, t, x^4 + x^2w + x^2v + yu + s ]

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 [ 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
[ 0, x + w, y, zy + y^2 + zx + yx + zw + yw, yx + xw, zyx + yx^2 + zyw + y^2w + yxw + x^2w + xw^2 + yv + xv + wv, zyx + zyw, yx^3 + y^3w + zx^2w + x^3w + zyw^2 + y^2w^2 + zxw^2 + yxw^2 + xw^3 + zyv + zxv + yxv + x^2v + zwv + ywv + w^2v, y^3x + y^2x^2 + y^2x*w + x^2w^2 + y^2v + v^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 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.4 ]

The images of the generators of the cohomology of G restricted to H are
[ z, 0, y, x, w, yw + zv + yv, zw + zv, y^2w + y^2v + xv, y^2w + y^2v + v^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 # 3

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 ]

The images of the generators of the cohomology of G restricted to H are
[ z, y, 0, y^2 + x, w, zw + yw + zv + yv, zv, xw + x*v, w^2 + v^2 ] 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 Isomorphic to the Group of Order 32 Number 13 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3 * $.5, $.2^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.3 * G.4, G.1 * G.4, G.1 * G.2 * G.3 ]

of type Cyclic(2) x Group(16)#11

The images of the generators of the cohomology of G restricted to H are
[ z + y, z + x, y + x, zy + y^2, zy + y^2 + zx + yx + x^2, z^2y + x^3, z^2y + zyx + yx^2 + x^3 + w, z^3x + z^2yx + z^2x^2 + zyx^2 + xw, z^3x + z^2yx + z^2x^2 + zyx^2 + 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 # 5

The Group H is Isomorphic to the Group of Order 32 Number 13 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3 * $.5, $.2^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1, G.1 * G.2 * G.3 * G.4 * G.6, G.3 * G.4 ]

of type Cyclic(2) x Group(16)#11

The images of the generators of the cohomology of G restricted to H are
[ z + y, z + x, z + x, zy + y^2, zy, zyx + yx^2 + w, z^2y + zyx + yx^2 + x^3 + w, z^3y + z^2x^2 + zyx^2 + x^4 + xw, z^3x + z^2yx + z^2x^2 + zyx^2 + 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 # 6

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

The images of the generators of the cohomology of G restricted to H are
[ z, y, z, y^2 + x, y^2 + x + w, y^3 + yx + zw + yw + yv, zv, y^2x + x^2 + xw + xv, w^2 + v^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 # 7

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.3, G.4 ]

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

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

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 # 94

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.2, G.2^2 = G.5 * G.6, G.4^2 = G.5 * G.6, G.3^G.1 = G.3 * G.5, G.4^G.1 = G.4 * 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*x + x^2 + w, t, x^4 + x^2*w + x^2*v + y*u + s ]

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 [ 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 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 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.4 ]

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

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 ]

The images of the generators of the cohomology of G restricted to H are [ z, y, 0, y^2 + x, w, z*w + y*w + z*v + y*v, z*v, x*w + x*v, w^2 + v^2 ] 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 Isomorphic to the Group of Order 32 Number 13 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3 * $.5, $.2^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.3 * G.4, G.1 * G.4, G.1 * G.2 * G.3 ]

of type Cyclic(2) x Group(16)#11

The images of the generators of the cohomology of G restricted to H are [ z + y, z + x, y + x, z*y + y^2, z*y + y^2 + z*x + y*x + x^2, z^2*y + x^3, z^2*y + z*y*x + y*x^2 + x^3 + w, z^3*x + z^2*y*x + z^2*x^2 + z*y*x^2 + x*w, z^3*x + z^2*y*x + z^2*x^2 + z*y*x^2 + 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 # 5

The Group H is Isomorphic to the Group of Order 32 Number 13 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3 * $.5, $.2^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1, G.1 * G.2 * G.3 * G.4 * G.6, G.3 * G.4 ]

of type Cyclic(2) x Group(16)#11

The images of the generators of the cohomology of G restricted to H are [ z + y, z + x, z + x, z*y + y^2, z*y, z*y*x + y*x^2 + w, z^2*y + z*y*x + y*x^2 + x^3 + w, z^3*y + z^2*x^2 + z*y*x^2 + x^4 + x*w, z^3*x + z^2*y*x + z^2*x^2 + z*y*x^2 + 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 # 6

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

The images of the generators of the cohomology of G restricted to H are [ z, y, z, y^2 + x, y^2 + x + w, y^3 + y*x + z*w + y*w + y*v, z*v, y^2*x + x^2 + x*w + x*v, w^2 + v^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 # 7

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.3, G.4 ]

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

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

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, yx + x^2 + w, t, x^4 + x^2w + x^2v + yu + s ]

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 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
[ z, 0, y, x, w, yw + zv + yv, zw + zv, y^2w + y^2v + xv, y^2w + y^2v + v^2 ] 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
[ z, y, 0, y^2 + x, w, zw + yw + zv + yv, zv, xw + x*v, w^2 + v^2 ] 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 + y, z + x, y + x, zy + y^2, zy + y^2 + zx + yx + x^2, z^2y + x^3, z^2y + zyx + yx^2 + x^3 + w, z^3x + z^2yx + z^2x^2 + zyx^2 + xw, z^3x + z^2yx + z^2x^2 + zyx^2 + 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 images of the generators of the cohomology of G restricted to H are
[ z + y, z + x, z + x, zy + y^2, zy, zyx + yx^2 + w, z^2y + zyx + yx^2 + x^3 + w, z^3y + z^2x^2 + zyx^2 + x^4 + xw, z^3x + z^2yx + z^2x^2 + zyx^2 + v ] in the cohomology of H

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

The images of the generators of the cohomology of G restricted to H are
[ z, y, z, y^2 + x, y^2 + x + w, y^3 + yx + zw + yw + yv, zv, y^2x + x^2 + xw + xv, w^2 + v^2 ] 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, z, y, x, y^2 + x + w, yx + yw + yv, zw + zv, y^2w + y^2v + xv, y^2x + y^2w + y^2*v + x^2 + v^2 ] in the cohomology of H

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