Restrictions to Maximal Elementary Abelian Subgroups

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

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

z + y ,

z + y ,

y ,

0 ,

z⁴ + z²y² + z²x² + x⁴
in the cohomology of E.

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

z + y ,

w .

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

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

y ,

z + y ,

z + y ,

z ,

z⁴ + z²y² + z²x² + x⁴
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 ,

y + x .

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

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

y ,

z + y ,

y ,

0 ,

z³y + z²y² + z²x² + x⁴
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 .

The nilradical of the cohomology of G is generated by

z² + zy + zx + yx + w² , yw + xw .

It is nilpotent of degree 2.

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Restrictions to Maximal Subgroups

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

The group H is abelian of type [ 8, 2, 2 ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables
[ z, y, x, w ] , in degrees [ 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 ,

y ,

z + x ,

z ,

zy³ + y⁴ + zyx² + y²w + x²w + w²
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 #2 Generated by [ G.1 * G.3, G.4, G.5, G.6, G.2 * G.3 ] .

The Group H is Isomorphic to the Group of Order 32 Number26 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.4^$.2 = $.4 * $.5 Generated by [ G.4 * G.5, G.1 * G.3 * G.4, G.1 * G.2 * G.5 * G.6 ] .

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

y + x ,

x ,

y ,

z + y ,

zy²x + zx³ + x⁴ + w
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 #3 Generated by [ G.2 * G.4, G.5, G.6, G.1 * G.4, G.3 ] .

The Group H 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 Generated by [ G.2 * G.4, G.2 * G.3 * G.4 * G.5 * G.6, G.1 * G.4, G.1 * G.2 * 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 + x ,

z + y + w ,

w ,

y + x + w ,

w⁴ + 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 #4 Generated by [ G.1, G.4, G.5, G.6, G.2 * G.3 ] .

The Group H is Isomorphic to the Group of Order 32 Number24 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2 * G.3 * G.6, G.1 * G.4 * G.5, G.1 * G.2 * G.3 * G.5 ]

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

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

y + x ,

z + x ,

z + x ,

y ,

x⁴ + 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 #5 Generated by [ G.5, G.6, G.2, G.1 * G.4, G.3 ] .

The Group H is Isomorphic to the Group of Order 32 Number26 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.3 * G.4, G.1 * G.2 * G.3 * G.4 * G.5, G.3 ] .

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

z + y ,

z ,

z + y + x ,

z + y ,

w
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.4, G.5, G.6, G.2 ] .

The Group H is Isomorphic to the Group of Order 32 Number26 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.3 * G.4 * G.5 * G.6, G.1 * G.2 * G.3 * G.4 * G.6, G.1 * G.2 * G.5 * G.6 ] .

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

z + y + x ,

z + x ,

z + y ,

z + y ,

zy²x + zx³ + x⁴ + w
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 #7 Generated by [ G.1, G.2 * G.4, G.3 * G.4, G.5, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number25 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4, $.3^2 = $.4 * $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5 Generated by [ G.3 * G.4, G.2 * G.4 * G.5, G.1 * G.2 * G.4 ]

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

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

y ,

z + y ,

x ,

z + y + x ,

x⁴ + w
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.5, G.6, G.2, G.3 ] .

The Group H is Isomorphic to the Group of Order 32 Number26 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.4^$.2 = $.4 * $.5 Generated by [ G.2 * G.4, G.3, G.4 * G.6 ] .

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

0 ,

y ,

x ,

z + y ,

y²x² + w
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.3 * G.4, G.5, G.6, G.2, G.1 * G.4 ] .

The Group H is Isomorphic to the Group of Order 32 Number23 GrpPC of order 32 = 2^5 PC-Relations: $.2^2 = $.5, $.3^2 = $.4 * $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5 Generated by [ G.3 * G.4, G.1 * G.2 * G.3 * G.6, G.2 ]

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

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

y ,

z + y ,

y + x ,

x ,

z³y + z²y² + y⁴ + z³x + z²yx + w²
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 #10 Generated by [ G.2 * G.4, G.3 * G.4, G.5, G.6, G.1 * G.4 ] .

The Group H is Isomorphic to the Group of Order 32 Number26 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.4^$.2 = $.4 * $.5 Generated by [ G.2 * G.3 * G.5, G.1 * G.2 * G.3 * G.4 * G.6, G.1 * G.2 * G.5 * G.6 ] .

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

z + x ,

z + y + x ,

z + y ,

z ,

zx³ + yx³ + x⁴ + w
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 #11 Generated by [ G.1 * G.3, G.4, G.5, G.6, G.2 ] .

The Group H is Isomorphic to the Group of Order 32 Number24 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.3 * G.5 * G.6, G.2 * G.4 * G.6, G.1 * G.2 * G.3 * G.5 * G.6 ]

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

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

y + x ,

z + x ,

y + x ,

z ,

y³x + y²x² + 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 #12 Generated by [ G.1, G.4, G.5, G.6, G.3 ] .

The Group H is Isomorphic to the Group of Order 32 Number26 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.3 * G.4 * G.6, G.4, G.3 ] .

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

y ,

0 ,

y + x ,

z + y ,

w
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 #13 Generated by [ G.1, G.4, G.5, G.6, G.2 ] .

The Group H is Isomorphic to the Group of Order 32 Number26 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.4^$.2 = $.4 * $.5 Generated by [ G.4 * G.5 * G.6, G.2 * G.4 * G.6, G.1 * G.2 * G.5 * G.6 ] .

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

x ,

y + x ,

0 ,

z + y ,

zy²x + y²x² + zx³ + x⁴ + w
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 #14 Generated by [ G.1, G.5, G.6, G.2, G.3 ] .

The Group H 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 Generated by [ G.2 * G.5, G.1 * G.3, G.2 * G.3, G.1 * G.2 * 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 + w ,

z + y + x ,

y + w ,

0 ,

z²y² + zy³ + z²yx + y²x² + zx³ + y²xw + zx²w + x³w + zxw² + x²w² + zw³ + xw³ + 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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Maximal Subgroup H #15 Generated by [ G.1, G.2 * G.4, G.5, G.6, G.3 ] .

The Group H is Isomorphic to the Group of Order 32 Number26 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.4^$.2 = $.4 * $.5 Generated by [ G.1, G.1 * G.2 * G.3 * G.4 * G.6, G.3 ] .

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

z + y ,

z ,

z + x ,

z ,

z²y² + w
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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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.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.3 * pcy.4 * pcy.5, pcy.2 * pcy.4, pcy.1, pcy.4 ] in the quotient group.

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

y ,

x ,

w ,

z + x + w ,

z² + zy + x² + w²
in the cohomology of G.

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

zy + yx + zw + xw + w² + v ,

zw² + yw² + xw² + w³ .

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Maximal Quotient Group Q #2.

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

The Group Q is Isomorphic to the Group of Order 32 Number26 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.4^$.2 = $.4 * $.5 .

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

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

w ,

z + y ,

y + x ,

zy³ + zyx² + y³w + zx²w + v
in the cohomology of G.

The kernel of the inflation is zero.

Maximal Quotient Group Q #3.

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

The Group Q is Isomorphic to the Group of Order 32 Number26 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.4^$.2 = $.4 * $.5 .

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

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

w ,

z + y ,

y + x ,

zy³ + zyx² + y²x² + y³w + yx²w + v
in the cohomology of G.

The kernel of the inflation is zero.

Action of Automorphisms

The groups of outer automorphisms of G has order 128, and is generated by 7 elements.

Automorphism #1

G.1
G.2
G.3
G.4 * 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 .

Automorphism #2

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

Automorphism #3

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

y²x² + zx²w + yx²w + v .

Automorphism #4

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

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

y ,

z ,

z + y + x ,

w ,

w⁴ + v .

Automorphism #5

G.1
G.2
G.3
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 + w ,

y + w ,

x + w ,

w ,

w⁴ + v .

Automorphism #6

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

x + w ,

z + y + x + w ,

z + w ,

w ,

zy³ + z³x + z²yx + zy²x + y³x + zyx² + y²x² + x⁴ + y³w + z²xw + yx²w + x³w + v .

Automorphism #7

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

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

y ,

z ,

x ,

z + y + w ,

z³y + zy³ + z²x² + y²x² + v .

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CHECKS

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

THE COMPUTATION IS COMPLETE