GROUP OF ORDER 64 # 14

GROUP # 14

Abelian(2,2) x AlmostExtraSpecial(16)

The MAGMA library number for G is 263

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

The center of G is abelian of type [ 2, 2, 4 ]
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 2
The exponent of G is 4
G has 3 conjugacy classes of maximal elementary abelian p-subgroups. The orders of the maximal elementary abelian subgroups are [ 16, 16, 16 ]
The orders of the centralizers of the maximal elementary abelian subgroups are [ 32, 32, 32 ]
The orders of the normalizers of the maximal elementary abelian subgroups are [ 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, 1, 4 ] by the ideal generated by the relations
[ x^2 + w^2 + y*v + v^2, y^2*v + y*v^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 cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators [ z^2, x^2, u, y^2 + y*v + v^2 ]

Restrictions to Maximal Elementary Abelian Subgroups

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

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
[ y, 0, y + w, z + y + w, z, z^2*y^2 + y^4 + z^2*x^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 + v ]

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

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
[ y, z, y + w, y + w, 0, z^2*y^2 + y^4 + z^2*x^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
[ x + w, v ]

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

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
[ z + y, z, z + y + w, z + y + w, z, z^2*y^2 + y^4 + z^2*x^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 + v, x + w ]

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

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
[ y + x, y, w, z + w, y, 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 + v ]

Maximal Subgroup H # 2

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.3 * G.5 * G.6, G.1 * G.5, G.2 * G.3 * G.4 * G.6, G.1 * G.2 * G.3 ]

of type Cyclic(2) x AlmostExtraSpecial(16)

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

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

Maximal Subgroup H # 3

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.5, G.1 * G.2 * G.3 * G.4 * G.5, G.2 * G.4 * G.5 * G.6, G.3 * G.5 * G.6 ]

of type Cyclic(2) x AlmostExtraSpecial(16)

The images of the generators of the cohomology of G restricted to H are
[ w, z + w, x + w, z + w, z + y + x + w, z^2*y*x + z^2*y*w + y^3*w + y^2*w^2 + y*x*w^2 + v ] 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 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.3 * G.5 * G.6, G.2 * G.3 * G.4, G.3 * G.4 * G.5 * G.6, G.1 * G.2 * 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 + x, z + w, y + x + w, z + y + w, y + x, z^2*y*x + z^2*y*w + y^3*w + y^2*w^2 + y*x*w^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 # 5

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
[ y, y + x, w, z + w, 0, y^2*v + x^2*v + v^2 ] in the cohomology of H

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

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.5, G.4 * G.5 * G.6, G.2 * G.3 * G.4 * G.6, G.1 * G.2 * G.3 ]

of type Cyclic(2) x AlmostExtraSpecial(16)

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

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.3 * G.4 * G.5 * G.6, G.1 * G.2 * G.4, G.4 * G.5 * G.6, G.1 * G.2 ]

of type Cyclic(2) x AlmostExtraSpecial(16)

The images of the generators of the cohomology of G restricted to H are
[ z + y + w, z + w, y, z + y + x, y + x, z^2*y*x + z^2*y*w + y^3*w + y^2*w^2 + y*x*w^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 # 8

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
[ x, 0, w, z + y + w, y, 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 # 9

The Group H is Isomorphic to the Group of Order 32 Number 8 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.5, G.1 * G.2 * G.3 * G.4 * G.5, G.1, G.1 * G.3 * G.4 * G.5 ]

of type Abelian(2,2) x Dihedral(8)

The images of the generators of the cohomology of G restricted to H are
[ y + x + w, x, y + x, y + x, z + y + x, z^2*y^2 + z*y^2*x + y^3*x + v^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 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.3 * G.5 * G.6, G.1 * G.3 * G.4 * G.5 * G.6, G.2 * G.3 * G.4 * G.6, G.1 * G.2 * G.3 ]

of type Cyclic(2) x AlmostExtraSpecial(16)

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

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

Maximal Subgroup H # 11

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.5 * G.6, G.3 * G.5, G.2 * G.6, G.1 * G.2 * G.3 ]

of type Cyclic(2) x AlmostExtraSpecial(16)

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

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

Maximal Subgroup H # 12

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

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

Maximal Subgroup H # 13

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.5, G.1 * G.2 * G.3 * G.5, G.2 * G.3 * G.4 * G.6, G.1 * G.2 * G.3 ]

of type Cyclic(2) x AlmostExtraSpecial(16)

The images of the generators of the cohomology of G restricted to H are
[ z + y + w, z + y + x + w, z + y + x, x, z + w, z^2*y*x + z^2*y*w + y^3*w + y^2*w^2 + y*x*w^2 + v ] 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 # 14

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.5, G.1 * G.2 * G.4, G.2, G.4 * G.5 * G.6 ]

of type Cyclic(2) x AlmostExtraSpecial(16)

The images of the generators of the cohomology of G restricted to H are
[ z, z + w, 0, z + x, y + x, z^2*y*x + z^2*y*w + y^3*w + y^2*w^2 + y*x*w^2 + 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 # 15

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.3 * G.5 * G.6, G.1 * G.2, G.2 * G.3, G.3 * G.4 * G.5 * G.6 ]

of type Cyclic(2) x AlmostExtraSpecial(16)

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

Maximal Subgroup H # 16

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.2 * G.6, G.3 * G.5 * G.6, G.3 * G.4 * G.5 * G.6, G.1 * G.2 * 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, z + w, y + x, z + y, y + x, z^2*y*x + z^2*y*w + y^3*w + y^2*w^2 + y*x*w^2 + v ] in the cohomology of H

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

Maximal Subgroup H # 17

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.2 * G.3 * G.5, G.1 * G.2 * G.4, G.1 * G.2 * G.3 * G.4 * G.6, G.1 * G.2 * G.3 * G.4 * G.5 * G.6 ]

of type Cyclic(2) x AlmostExtraSpecial(16)

The images of the generators of the cohomology of G restricted to H are
[ y + x + w, z + y + x + w, z + x + w, y + x + w, z + w, z^2*y*x + z^2*y*w + y^3*w + y^2*w^2 + y*x*w^2 + v ] 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 # 18

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.4, G.1 * G.2 * G.3 * G.4 * G.6, G.1 * G.5, G.1 * G.4 * G.5 * G.6 ]

of type Cyclic(2) x AlmostExtraSpecial(16)

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

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

Maximal Subgroup H # 19

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.3 * G.5, G.1 * G.5 * G.6, G.2 * G.3 * G.4 * G.6, G.2 * G.3 ]

of type Cyclic(2) x AlmostExtraSpecial(16)

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

Maximal Subgroup H # 20

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.5 * G.6, G.1 * G.2 * G.3 * G.5 * G.6, G.4 * G.5, G.1 * 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 + y + x, y + x, z + y, z + w, z + y + x + w, z^2*y*x + z^2*y*w + y^3*w + y^2*w^2 + y*x*w^2 + v ] in the cohomology of H

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

Maximal Subgroup H # 21

The Group H is Isomorphic to the Group of Order 32 Number 8 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.1, G.4 * G.5 * G.6, G.1 * G.2 * G.3, G.2 * G.4 * G.6 ]

of type Abelian(2,2) x Dihedral(8)

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

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

Maximal Subgroup H # 22

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.2 * G.3 * G.4, G.1 * G.5, G.4 * G.5 * G.6, G.1 * G.2 * 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 + y, z + w, w, z + x + w, y + x, z^2*y*x + z^2*y*w + y^3*w + y^2*w^2 + y*x*w^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 + v ]

Maximal Subgroup H # 23

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.2 * G.3 * G.6, G.1 * G.2, G.1 * G.4 * G.5, G.1 * 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 + x + w, y + x, z + y, z + w, z + w, z^2*y*x + z^2*y*w + y^3*w + y^2*w^2 + y*x*w^2 + x^2*w^2 + v ] in the cohomology of H

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

Maximal Subgroup H # 24

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.2 * G.3 * G.6, G.3 * G.5 * G.6, G.1 * G.3 * G.4 * G.5 * G.6, G.1 * G.2 * G.6 ]

of type Cyclic(2) x AlmostExtraSpecial(16)

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

Maximal Subgroup H # 25

The Group H is Isomorphic to the Group of Order 32 Number 9 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.5, G.1 * G.5, G.2 * G.3, G.2 * G.4 * G.6 ]

of type Abelian(2,2) x Quaternion(8)

The images of the generators of the cohomology of G restricted to H are
[ y, z + w, z, w, y + x, z^3*w + z^2*w^2 + z*w^3 + v ] 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 # 26

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.3 * G.5 * G.6, G.5, G.1 * G.2 * G.3 * G.4, G.1 * G.2 * G.3 * G.6 ]

of type Cyclic(2) x AlmostExtraSpecial(16)

The images of the generators of the cohomology of G restricted to H are
[ z + x + w, z + w, z + x + w, w, y + x, z^2*y*x + z^2*y*w + y^3*w + y^2*w^2 + y*x*w^2 + v ] 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 # 27

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.2 * G.4, G.2 * G.5, G.2 * G.3 * G.4 * G.6, G.2 * 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
[ 0, z + y + x + w, x, z + y + x, z + w, z^2*y*x + z^2*y*w + y^3*w + y^2*w^2 + y*x*w^2 + 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 # 28

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.3 * G.5, G.1 * G.2 * G.3 * G.4 * G.6, G.1 * G.5 * G.6, G.1 * G.2 * G.3 ]

of type Cyclic(2) x AlmostExtraSpecial(16)

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

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

Maximal Subgroup H # 29

The Group H is Isomorphic to the Group of Order 32 Number 8 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.1, G.2, G.1 * G.2 * G.3 * G.5 * G.6, G.2 * G.4 * G.5 ]

of type Abelian(2,2) x Dihedral(8)

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

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

Maximal Subgroup H # 30

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.5, G.1 * G.2 * G.4, G.1 * G.2 * G.3 * G.4 * G.6, G.2 * 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
[ y + x + w, z + y + x + w, x, z + y + x, z + w, z^2*y*x + z^2*y*w + y^3*w + y^2*w^2 + y*x*w^2 + v ] in the cohomology of H

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

Maximal Subgroup H # 31

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.4 * G.5, G.1 * G.2 * G.5, G.1 * G.2 * G.4, G.1 * G.2 * 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 + y + x + w, z + y + x + w, x, z + y + x, z + w, z^2*y*x + z^2*y*w + y^3*w + y^2*w^2 + y*x*w^2 + v ] 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 = true
essflag = true
centflag = true
bigflag = true
restrictflag = true

THE COMPUTATION IS COMPLETE