GROUP OF ORDER 64 # 78

GROUP # 78

The MAGMA library number for G is 213

GrpPC : G of order 64 = 2^6 PC-Relations: G.3^2 = G.5, G.4^2 = G.6, G.2^G.1 = G.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 4
The exponent of G is 4
G has 5 conjugacy classes of maximal elementary abelian p-subgroups. The orders of the maximal elementary abelian subgroups are [ 8, 8, 8, 8, 8 ]
The orders of the centralizers of the maximal elementary abelian subgroups are [ 32, 16, 16, 16, 16 ]
The orders of the normalizers of the maximal elementary abelian subgroups are [ 64, 64, 64, 64, 64 ]

The cohomology ring of G is the quotient of a polynomial ring in the variables [ z, y, x, w, v, u ] in degrees [ 1, 1, 1, 1, 2, 4 ] by the ideal generated by the relations
[ zy + zw + w^2, zx + x^2 + zw + w^2, zw^2 + yw^2 + w^3 ]

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

The Krull dimension of the cohomology ring is 3
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators [ v, u, z^2 + y^2 + z*w + y*w + w^2 ]

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E # 1
Generated by [ G.5 * G.6, G.2 * G.6, G.5 ]

The cohomology ring of E is a polynomial ring in the variables [ z, y, x ]

The images of the generators of the cohomology of G restricted to E are
[ 0, z, 0, 0, z^2 + y^2, z^4 + z^2y^2 + z^2x^2 + x^4 ] in the cohomology of E

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

Subgroup E # 2
Generated by [ G.5 * G.6, G.5, G.1 * G.3 * G.4 * G.5 * G.6 ]

The cohomology ring of E is a polynomial ring in the variables [ z, y, x ]

The images of the generators of the cohomology of G restricted to E are
[ z, 0, z, z, zy + y^2, z^2x^2 + x^4 ] in the cohomology of E

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

Subgroup E # 3
Generated by [ G.5 * G.6, G.5, G.1 * G.3 * G.5 ]

The cohomology ring of E is a polynomial ring in the variables [ z, y, x ]

The images of the generators of the cohomology of G restricted to E are
[ z, 0, z, 0, zy + y^2, z^2x^2 + x^4 ] 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, w ]

Subgroup E # 4
Generated by [ G.5 * G.6, G.5, G.1 ]

The cohomology ring of E is a polynomial ring in the variables [ z, y, x ]

The images of the generators of the cohomology of G restricted to E are
[ z, 0, 0, 0, zy + y^2, z^2x^2 + x^4 ] in the cohomology of E

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

Subgroup E # 5
Generated by [ G.1 * G.4 * G.6, G.5 * G.6, G.5 ]

The cohomology ring of E is a polynomial ring in the variables [ z, y, x ]

The images of the generators of the cohomology of G restricted to E are
[ z, 0, 0, z, zy + y^2, z^2x^2 + x^4 ] in the cohomology of E

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

The nilradical of the cohomology of G is generated by
[ zw + w^2, yx, y*w ]

Restrictions to Maximal Subgroups

Maximal Subgroup H # 1

The Group H is Isomorphic to the Group of Order 32 Number 14 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.2, G.2 * G.3 * G.5 * G.6, G.1 * G.2 * G.4 * G.5 ]

of type Cyclic(4) x Dihedral(8)

The images of the generators of the cohomology of G restricted to H are
[ z + x, z + y + x, y, x, zy + y^2 + zx + w, x^4 + y^2*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 + y + x ]

Maximal Subgroup H # 2

The Group H is Isomorphic to the Group of Order 32 Number 36 GrpPC of order 32 = 2^5 PC-Relations: $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.2 * G.4 * G.6, G.1 * G.4 * G.5 * G.6, G.1 * G.2 * G.3 * G.4 * G.6 ]

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

The Group H is Isomorphic to the Group of Order 32 Number 36 GrpPC of order 32 = 2^5 PC-Relations: $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.2 * G.4, G.1 * G.2 * G.6, G.1 * G.4 * G.6 ]

The images of the generators of the cohomology of G restricted to H are
[ z + y + x, y + x, 0, z + x, zy + zx + w, z^3x + z^2y*x + 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 14 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.2 * G.4, G.1 * G.2 * G.3 * G.4 * G.6, G.1 * G.2 * G.4 * G.5 ]

of type Cyclic(4) x Dihedral(8)

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

Maximal Subgroup H # 5

The Group H is Isomorphic to the Group of Order 32 Number 10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.2 * G.3 * G.4 * G.6, G.2 * G.5 * G.6, G.1 * G.5, G.3 * G.4 * G.6 ]

of type Cyclic(2) x AlmostExtraSpecial(16)

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

The Group H is Isomorphic to the Group of Order 32 Number 10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.2 * G.3 * G.6, G.1 * G.4 * G.6, G.2 * G.6, G.3 * G.4 * G.5 ]

of type Cyclic(2) x AlmostExtraSpecial(16)

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

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

Maximal Subgroup H # 7

The Group H is Isomorphic to the Group of Order 32 Number 14 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.2 * G.3 * G.6, G.1 * G.2 * G.5, G.2 * G.4 * G.6 ]

of type Cyclic(4) x Dihedral(8)

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

Maximal Subgroup H # 8

The Group H is Isomorphic to the Group of Order 32 Number 34 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.4, G.1, G.3 ]

The images of the generators of the cohomology of G restricted to H are
[ x, 0, y, z, y^2 + zx + w + v, z^2x^2 + w^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 # 9

The Group H is Isomorphic to the Group of Order 32 Number 35 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^2 = $.4 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.4, G.1 * G.2 * G.5 * G.6, G.3 ]

The images of the generators of the cohomology of G restricted to H are
[ x, x, y, z, y^2 + w, x^2*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 # 10

The Group H is Isomorphic to the Group of Order 32 Number 39 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4 * $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.1 * G.2 * G.4, G.2 * G.3, G.3 * G.4 ]

The images of the generators of the cohomology of G restricted to H are
[ z, z + y, y + x, z + x, w, z^2y^2 + zyw + zxw + x^2w + w^2 + 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 # 11

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
[ 0, x, z, y, w + v, x^2*w + 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 # 12

The Group H is Isomorphic to the Group of Order 32 Number 36 GrpPC of order 32 = 2^5 PC-Relations: $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.2 * G.3, G.1 * G.2, G.1 * G.5 ]

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

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

Maximal Subgroup H # 13

The Group H is Isomorphic to the Group of Order 32 Number 39 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4 * $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5 Generated by [ G.1 * G.2 * G.5, G.3 * G.4 * G.6, G.2 * G.4 * G.5 * G.6 ]

The images of the generators of the cohomology of G restricted to H are
[ z, z + y, x, y + x, zx + x^2 + w, z^2y^2 + zyw + 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 # 14

The Group H is Isomorphic to the Group of Order 32 Number 36 GrpPC of order 32 = 2^5 PC-Relations: $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.2 * G.3 * G.5, G.1 * G.3 * G.5 * G.6, G.1 * G.2 * G.3 * G.4 * G.5 ]

The images of the generators of the cohomology of G restricted to H are
[ z + y + x, y + x, z + y + x, y, zy + zx + w, z^3x + z^2y*x + 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 # 15

The Group H is Isomorphic to the Group of Order 32 Number 14 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.2 * G.3 * G.5 * G.6, G.2 * G.3, G.1 * G.2 * G.3 * G.4 ]

of type Cyclic(4) x Dihedral(8)

The images of the generators of the cohomology of G restricted to H are
[ z + x, z + y + x, z + y + x, z, zy + y^2 + zx + w, x^4 + y^2*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
[ y + x ]

The essential cohomology of G is zero

CHECKS

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

THE COMPUTATION IS COMPLETE