Restrictions to Maximal Elementary Abelian Subgroups

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

The images of the generators of the cohomology of G restricted to E are

y + x + w ,

z + y + x + w ,

z + y + x + w ,

y + x + w ,

z² + zy + y² + zx + zw ,

z² + zy + y² + x² + zw
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 .

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

The images of the generators of the cohomology of G restricted to E are

z + y + x + w ,

y + x + w ,

y + x + w ,

y + x + w ,

y² + zx + zw ,

y² + x² + zw
in the cohomology of E.

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

y + w ,

x + w .

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

The images of the generators of the cohomology of G restricted to E are

z + y + x + w ,

z + y + x + w ,

y + x + w ,

y + x + w ,

z² + zy + y² ,

zy + y² + zx + x²
in the cohomology of E.

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

z + y ,

x + w .

The nilradical of the cohomology of G is generated by

yx + x² + yw + xw

It is nilpotent of degree 2.

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Restrictions to Maximal Subgroups

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

The Group H is Isomorphic to the Group of Order 32 Number8 GrpPC of order 32 = 2^5 PC-Relations: $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.6, G.2 * G.3 * G.4, G.3 * G.4, G.1 ]

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

The images of the generators of the cohomology of G restricted to H are

z + w ,

y ,

y + x ,

y + x ,

zy + y² + x² + zw + xw + v ,

y² + zx + xw + v
in the cohomology of H.

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

x + w .

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Maximal Subgroup H #2 Generated by [ G.4, G.2, G.1, G.5, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number36 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.2 * G.4 * G.5, G.2, G.1 * G.2 * G.4 * G.5 ] .

The images of the generators of the cohomology of G restricted to H are

x ,

z + y + x ,

0 ,

y + x ,

zy + zx + x² + w + v ,

y² + zx + x² + v
in the cohomology of H.

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

x .

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

The Group H is Isomorphic to the Group of Order 32 Number8 GrpPC of order 32 = 2^5 PC-Relations: $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.6, G.2 * G.3 * G.4 * G.5 * G.6, G.1, G.2 * G.3 ]

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

The images of the generators of the cohomology of G restricted to H are

z + w ,

y + x ,

y + x ,

y ,

zy + y² + zx + x² + zw + xw ,

zy + y² + zx + x² + yw + v
in the cohomology of H.

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

y + x .

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

The Group H is Isomorphic to the Group of Order 32 Number12 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5 Generated by [ G.1 * G.3 * G.6, G.1 * G.3 * G.4 * G.6, G.2 * G.6 ]

of type Cyclic(2) x Group(16)#10 .

The images of the generators of the cohomology of G restricted to H are

z + x ,

y ,

z + x ,

x ,

zy + y² + zx + yx + v ,

yx + w + v
in the cohomology of H.

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

z + x .

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

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

z² .

The images of the generators of the cohomology of G restricted to H are

y ,

x ,

z + x ,

y ,

yx + w² + v ,

zy + zw + yw + xw + v
in the cohomology of H.

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

z + w .

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

The Group H is Isomorphic to the Group of Order 32 Number36 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.2 * G.4 * G.6, G.1 * G.2 * G.3 * G.6, G.2 ] .

The images of the generators of the cohomology of G restricted to H are

y + x ,

z + y + x ,

y ,

x ,

y² + zx + x² + w + v ,

zy + y² + zx + x² + v
in the cohomology of H.

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

z + x + w .

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Maximal Subgroup H #7 Generated by [ G.2 * G.4 * G.5, G.1, G.5, G.2 * G.3, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number36 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.3 * G.4 * G.6, G.2 * G.4 * G.5, G.1 * G.2 * G.4 ] .

The images of the generators of the cohomology of G restricted to H are

z + x ,

y + x ,

z ,

z + y + x ,

zy + y² + zx + w + v ,

zx + v
in the cohomology of H.

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

y + x + w .

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Maximal Subgroup H #8 Generated by [ G.4, G.3, G.2, G.5, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number36 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.4, G.3, G.2 ] .

The images of the generators of the cohomology of G restricted to H are

0 ,

z ,

x ,

y ,

zy + w + v ,

v
in the cohomology of H.

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

z .

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Maximal Subgroup H #9 Generated by [ G.4, G.3, G.1 * G.2 * G.5, G.5, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^$.1 = $.2 * $.5 Generated by [ G.4 * G.5 * G.6, G.1 * G.2, G.1 * G.2 * G.3 * G.4 * G.6 ]

of type Cyclic(2) x Group(16)#9 .

The images of the generators of the cohomology of G restricted to H are

y + x ,

y + x ,

x ,

z + x ,

y² + zx + yx + x² + u ,

zx + yx + x² + w + v
in the cohomology of H.

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

z + y .

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Maximal Subgroup H #10 Generated by [ G.4, G.1 * G.3 * G.6, G.1 * G.2 * G.5, G.5, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number36 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.4, G.1 * G.3 * G.4 * G.5, G.1 * G.2 * G.4 ] .

The images of the generators of the cohomology of G restricted to H are

z + x ,

x ,

z ,

z + y + x ,

zx + w + v ,

y² + x² + v
in the cohomology of H.

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

z + y + x .

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Maximal Subgroup H #11 Generated by [ G.3, G.2 * G.4 * G.5, G.1, G.5, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^$.1 = $.2 * $.5 Generated by [ G.2 * G.4 * G.5 * G.6, G.1, G.1 * G.2 * G.3 * G.4 * G.6 ]

of type Cyclic(2) x Group(16)#9 .

The images of the generators of the cohomology of G restricted to H are

y + x ,

z + x ,

x ,

z + x ,

x² + u ,

x² + w + v
in the cohomology of H.

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

y + w .

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Maximal Subgroup H #12 Generated by [ G.3, G.1 * G.2 * G.5, G.5, G.6, G.1 * G.4 * G.5 * G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number36 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.3, G.2 * G.4 * G.5 * G.6, G.1 * G.3 * G.4 * G.5 ] .

The images of the generators of the cohomology of G restricted to H are

z ,

y ,

z + x ,

z + y ,

zy + w + v ,

zy + zx + x² + v
in the cohomology of H.

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

z + y + w .

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Maximal Subgroup H #13 Generated by [ G.1 * G.3 * G.6, G.1 * G.2 * G.5, G.5, G.6, G.1 * G.4 * G.5 * G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number8 GrpPC of order 32 = 2^5 PC-Relations: $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.2 * G.5 * G.6, G.2 * G.3 * G.5 * G.6, G.2 * G.3 * G.5, G.1 * G.4 ]

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

The images of the generators of the cohomology of G restricted to H are

y + x ,

z + x + w ,

z + w ,

y ,

zy + zx + x² + yw + w² ,

zy + zx + zw + xw + w² + 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 .

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Maximal Subgroup H #14 Generated by [ G.4, G.3, G.1, G.5, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number36 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.4, G.3, G.1 * G.3 * G.4 * G.6 ] .

The images of the generators of the cohomology of G restricted to H are

z ,

0 ,

z + x ,

z + y ,

w + v ,

zy + v
in the cohomology of H.

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

y .

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Maximal Subgroup H #15 Generated by [ G.3, G.2, G.1, G.5, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number36 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.3, G.2, G.1 * G.3 ] .

The images of the generators of the cohomology of G restricted to H are

y ,

z ,

y + x ,

0 ,

y² + w + v ,

zy + v
in the cohomology of H.

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

w .

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The essential cohomology of G is zero.

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Inflations from Maximal Quotient Groups

Maximal Quotient Group Q #1.

The kernel of the quotient is generated by G.5 * G.6 .

The Group Q is Isomorphic to the Group of Order 32 Number8 GrpPC of order 32 = 2^5 PC-Relations: $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.3 = $.4 * $.5

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

The generators of G have images [ pcy.2 * pcy.3 * pcy.4, pcy.1 * pcy.3, pcy.3 * pcy.4 * pcy.5, pcy.3 * pcy.5 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are

x ,

w ,

z + y + x + w ,

y + w ,

y² + x² + yw + v
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by

z² + y² + zw + yw + xw .

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Maximal Quotient Group Q #2.

The kernel of the quotient is generated by G.5 .

The Group Q is Isomorphic to the Group of Order 32 Number8 GrpPC of order 32 = 2^5 PC-Relations: $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.3 = $.4 * $.5

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

The generators of G have images [ pcy.2 * pcy.3, pcy.1, pcy.1 * pcy.4, pcy.3 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are

y + x ,

w ,

z + w ,

y ,

y² + zx + x² + v + u
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by

z² + yx + xw .

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Maximal Quotient Group Q #3.

The kernel of the quotient is generated by G.6 .

The Group Q is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5

of type Cyclic(2) x AlmostExtraSpecial(16) .

The generators of G have images [ pcy.1 * pcy.3 * pcy.5, pcy.1 * pcy.3 * pcy.4, pcy.1 * pcy.2 * pcy.4 * pcy.5, pcy.1 * pcy.2 * pcy.3 * pcy.4 * pcy.5 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are

z + y + x + w ,

z + y ,

z + x + w ,

z + y + x ,

z³y + zy³ + z²yx + x⁴ + z³w + z²w² + yxw² + x²w² + zw³ + x²v + w²v + u²
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by

zy + y² + zx + yx + zw + yw .

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Maximal Quotient Group Q #4.

The kernel of the quotient is generated by G.1 * G.2 * G.3 * G.4 * G.5 .

The Group Q is Isomorphic to the Group of Order 32 Number36 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 .

The generators of G have images [ pcy.2, pcy.3, pcy.1, pcy.1 * pcy.2 * pcy.3 * pcy.4 * pcy.5 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are

z + y ,

z + w ,

z + x ,

z² + y² + zx + x² + w² + v + u ,

z² + y² + zx + x² + yw + w² + u
in the cohomology of G.

The kernel of the inflation is zero.

Maximal Quotient Group Q #5.

The kernel of the quotient is generated by G.1 * G.2 * G.3 * G.4 * G.6 .

The Group Q is Isomorphic to the Group of Order 32 Number36 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 .

The generators of G have images [ pcy.2, pcy.3, pcy.1, pcy.1 * pcy.2 * pcy.3 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are

z + y ,

z + w ,

z + x ,

z² + y² + x² + w² + v + u ,

z² + y² + x² + yw + w² + u
in the cohomology of G.

The kernel of the inflation is zero.

Maximal Quotient Group Q #6.

The kernel of the quotient is generated by G.1 * G.2 * G.3 * G.4 .

The Group Q is Isomorphic to the Group of Order 32 Number36 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 .

The generators of G have images [ pcy.2, pcy.3, pcy.1, pcy.1 * pcy.2 * pcy.3 * pcy.5 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are

z + y ,

z + w ,

z + x ,

z² + y² + zx + x² + w² + v + u ,

z² + y² + x² + yw + xw + u
in the cohomology of G.

The kernel of the inflation is zero.

Maximal Quotient Group Q #7.

The kernel of the quotient is generated by G.1 * G.2 * G.3 * G.4 * G.5 * G.6 .

The Group Q is Isomorphic to the Group of Order 32 Number36 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 .

The generators of G have images [ pcy.1 * pcy.2, pcy.3, pcy.1, pcy.2 * pcy.3 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are

y + w ,

z + w ,

z + x ,

z² + y² + x² + w² + v + u ,

y² + zx + x² + yw + xw + w² + u
in the cohomology of G.

The kernel of the inflation is zero.

Action of Automorphisms

The groups of outer automorphisms of G has order 512, and is generated by 9 elements.

Automorphism #1

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

The images of the generators of the cohomology of G under the map induced by the automorphism are

z ,

y ,

x ,

w ,

v ,

yx + x² + yw + xw + u .

Automorphism #2

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

The images of the generators of the cohomology of G under the map induced by the automorphism are

z ,

y ,

x ,

w ,

x² + w² + v ,

x² + w² + u .

Automorphism #3

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

The images of the generators of the cohomology of G under the map induced by the automorphism are

z ,

y ,

x ,

w ,

v ,

u .

Automorphism #4

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

The images of the generators of the cohomology of G under the map induced by the automorphism are

z ,

y ,

x ,

w ,

zx + xw + w² + v ,

zx + xw + w² + u .

Automorphism #5

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

The images of the generators of the cohomology of G under the map induced by the automorphism are

z ,

y ,

x ,

w ,

v ,

zx + u .

Automorphism #6

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

The images of the generators of the cohomology of G under the map induced by the automorphism are

w ,

z + y + w ,

z + x + w ,

z ,

z² + y² + x² + w² + v ,

z² + y² + x² + w² + u .

Automorphism #7

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

The images of the generators of the cohomology of G under the map induced by the automorphism are

z + y + x ,

x ,

y ,

y + x + w ,

yx + x² + yw + xw + v ,

yx + x² + yw + xw + u .

Automorphism #8

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

The images of the generators of the cohomology of G under the map induced by the automorphism are

z + x + w ,

y + x + w ,

w ,

x ,

yx + x² + yw + xw + v ,

yx + x² + yw + xw + u .

Automorphism #9

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

The images of the generators of the cohomology of G under the map induced by the automorphism are

z ,

z + y + w ,

x ,

w ,

z² + zx + xw + w² + v ,

zx + xw + u .

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CHECKS

paramflag = true
qregflagflag = true
cmflag = true
essflag = true
centflag = true

THE COMPUTATION IS COMPLETE