Restriction to the Maximal Elementary Abelian Subgroup

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

0 ,

0 ,

z + y + x ,

x² ,

z² + y² ,

0 ,

y⁴ + z²x² + y²x² + 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 ,

x² + w + v ,

u .

This ideal is also the nilradical of the cohomology of G.

It is nilpotent of degree 4.

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Restrictions to Maximal Subgroups

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

z ,

x ,

w ,

y² + w ,

y³ + yx² + zv ,

y³x + y²x² + yx³ + x²w + zyv + y²v + x²v + v²
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.4, G.5, G.2, G.3, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number9 GrpPC of order 32 = 2^5 PC-Relations: $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.4, G.5, G.3 * G.5 * G.6, G.2 ]

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

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

0 ,

w ,

x ,

z² ,

z² + y² + zw + yw ,

zy² + zx² + zyw + y²w + zw² ,

z²x² + y²x² + y³w + z²xw + zxw² + 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 #3 Generated by [ G.2 * G.4, G.5, G.1, G.3, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number28 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.3^2 = $.5, $.4^2 = $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.2 * G.4, G.1 * G.2 * G.4 * G.6 ] .

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

z ,

z + y ,

z + y ,

x ,

y² + x + w ,

yw + v ,

w² + u
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.5, G.1, G.2, G.3, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number28 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.3^2 = $.5, $.4^2 = $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.2 * G.6, G.2 ] .

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

z ,

z + y ,

0 ,

x ,

y² + x + w ,

v ,

w² + u
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 #5 Generated by [ G.1 * G.4, G.5, G.2, G.3, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number28 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.3^2 = $.5, $.4^2 = $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1 * G.2 * G.4, G.2 ] .

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

z ,

z + y ,

z ,

x ,

y² + x + w ,

v ,

y²w + w² + u
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 #6 Generated by [ G.1 * G.4, G.5, G.1 * G.2 * G.5 * G.6, G.3, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number28 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.3^2 = $.5, $.4^2 = $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.2 * G.4, G.1 * G.2 * G.5 * G.6 ] .

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

z ,

z + y ,

y ,

x ,

y² + x + w ,

yw + v ,

y²w + w² + u
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 #7 Generated by [ G.4, G.5, G.1, G.3, G.6 ] .

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

z ,

0 ,

z + y + x ,

x² ,

zy + y² ,

zy² + zyx + zw ,

zy³ + y⁴ + zy²x + y²x² + zyw + zxw + w²
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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The essential cohomology of G is generated as an ideal by

zxu .

It is nilpotent of degree 2.

The annihilator of the Essential Cohomology has dimension 3 .

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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 Number11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^$.1 = $.2 * $.5

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

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

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

z ,

y ,

x ,

y² + yx + x² + w + v ,

w ,

v
in the cohomology of G.

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

y² + yx + x² + w + v + u ,

x³ + yw + xw + yv + xv + yu + xu .

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Maximal Quotient Group Q #2.

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

The Group Q 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

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

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

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

z + y ,

z ,

z + x ,

y²x + zx² + x³ + xw + xv + u ,

y²x² + zx³ + yx³ + zxw + yxw + y²v + x²v + w² + v² + yu + xu + t
in the cohomology of G.

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

y² .

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Maximal Quotient Group Q #3.

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

The Group Q 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

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

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

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

y ,

z + x ,

y + x ,

y²x² + zx³ + yx³ + zxw + yxw + y²v + x²v + v² + yu + xu + t
in the cohomology of G.

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

z² + y² + x² .

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Maximal Quotient Group Q #4.

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

The Group Q is Isomorphic to the Group of Order 32 Number28 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.3^2 = $.5, $.4^2 = $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5, $.4^$.3 = $.4 * $.5 .

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

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

z ,

z + y ,

x² + w ,

y² + x² + w + v ,

y²x + x³ + xw + xv + u ,

y²x² + zx³ + yx³ + zxw + yxw + y²v + x²v + yu + xu + t
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.3 * G.4 .

The Group Q is Isomorphic to the Group of Order 32 Number28 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.3^2 = $.5, $.4^2 = $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5, $.4^$.3 = $.4 * $.5 .

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

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

z ,

z + y ,

x² + w ,

y² + x² + w + v ,

y²x + zx² + x³ + xw + xv + u ,

y²x² + zx³ + yx³ + zxw + yxw + x²v + w² + v² + zu + yu + xu + t
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.4 * G.5 * G.6 .

The Group Q is Isomorphic to the Group of Order 32 Number28 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.3^2 = $.5, $.4^2 = $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5, $.4^$.3 = $.4 * $.5 .

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

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

z ,

z + y ,

w ,

y² + zx + yx + x² + w + v ,

y²x + x³ + xw + xv + u ,

zx³ + yx³ + zxw + yxw + y²v + x²v + yu + xu + t
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.4 * G.5 .

The Group Q is Isomorphic to the Group of Order 32 Number28 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.3^2 = $.5, $.4^2 = $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5, $.4^$.3 = $.4 * $.5 .

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

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

z ,

z + y ,

w ,

y² + zx + yx + x² + w + v ,

y²x + zx² + x³ + xw + xv + u ,

zx³ + yx³ + zxw + yxw + x²v + w² + v² + zu + yu + xu + t
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

The order of the class of the automorphism in the outer automorphism group is 2. The images of the generators of G are

G.1 * G.6
G.2 * 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

z ,

y ,

x ,

w ,

v ,

u ,

t .

Automorphism #2

The order of the class of the automorphism in the outer automorphism group is 2. The images of the generators of G are

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 ,

zx² + u ,

zx³ + y²v + w² + v² + zu + t .

Automorphism #3

The order of the class of the automorphism in the outer automorphism group is 2. The images of the generators of G are

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

z ,

y ,

x ,

w ,

v ,

u ,

t .

Automorphism #4

The order of the class of the automorphism in the outer automorphism group is 2. The images of the generators of G are

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

z ,

y ,

x ,

y² + w ,

v ,

y²x + u ,

y²x² + t .

Automorphism #5

The order of the class of the automorphism in the outer automorphism group is 2. The images of the generators of G are

G.1
G.2
G.3
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 ,

x² + w ,

zx + yx + x² + v ,

zx² + yx² + u ,

y²x² + zx³ + yx³ + y²v + w² + v² + zu + t .

Automorphism #6

The order of the class of the automorphism in the outer automorphism group is 2. The images of the generators of G are

G.1 * G.4 * G.5
G.2 * G.4 * G.5
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

z ,

y ,

z + y + x ,

w ,

v ,

y²x + yx² + yw + yv + u ,

y²x² + zu + t .

Automorphism #7

The order of the class of the automorphism in the outer automorphism group is 2. The images of the generators of G are

G.1
G.2 * G.4 * G.5
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

z ,

y ,

y + x ,

w ,

v ,

y²x + yx² + yw + yv + u ,

y²x² + t .

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CHECKS

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

THE COMPUTATION IS COMPLETE