Restriction to the Maximal Elementary Abelian Subgroup

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

0 ,

0 ,

0 ,

z² + x² ,

z² + y² ,

0 ,

0 ,

0 ,

z⁴ + y⁴ + 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 ,

x ,

u ,

t ,

s .

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.2, 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.2, G.3 * G.4 ]

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

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

0 ,

z ,

y + x ,

y² ,

zy + v ,

y³ + yx² ,

zw ,

zw + yw + xw ,

y³x + yx³ + y²w + x²w + w²
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 #2 Generated by [ G.2 * G.4, G.5, G.3, G.1, 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.2 * G.4, G.2 * G.4 * G.5, G.1 ]

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

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

z ,

y + x ,

y + x ,

w ,

yx ,

zv ,

yv + xv ,

y³ + yx² ,

y²w + x²w + v²
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 #3 Generated by [ G.4, G.5, G.3, G.1, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number18 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.4, G.1 ] .

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

z ,

0 ,

y ,

w ,

t ,

yx + zu ,

yx ,

yx + yu ,

u²
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 #4 Generated by [ G.1 * G.4 * G.6, G.5, G.2, G.3, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number18 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.4 * G.6, G.2 ] .

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

z ,

y ,

z ,

w ,

x + w + u ,

zw + zt ,

yx + yw + yt ,

zw + yw + zt + yt ,

w² + t²
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.1 * G.2, G.1 * G.4 * G.6, G.5, G.3, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number18 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.2, G.1 * G.4 * G.6 ] .

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

z + y ,

y ,

z ,

w + t ,

x + w + u ,

zw + yw + zu + yu + zt + yt ,

yx + yw + yu + yt ,

yx + zw + yw + zu + yu + zt + yt ,

w² + u² + t²
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 #6 Generated by [ G.1 * G.2, G.4, G.5, G.3, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number18 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.2, G.4 ] .

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

y ,

y ,

z ,

t ,

x + w + u ,

yx + yu + yt ,

yu + yt ,

yx + zu + yu + zt + yt ,

u² + t²
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 #7 Generated by [ G.5, G.2, G.3, G.1, G.6 ] .

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

z² , y² .

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

z ,

y ,

0 ,

w ,

yx + x² ,

zx² + zv ,

zyx + yx² + yv ,

yx² + yv ,

x⁴ + v²
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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The essential cohomology of G is generated as an ideal by

zys .

It is nilpotent of degree 2.

The annihilator of the Essential Cohomology has dimension 3 .
The essential cohomology is a free module over the polynomial subring Q of the cohomology ring of G generated by w , v , r .
The essential cohomology is generated as a module over Q by the elements

[] [] [] [] [ zys ]

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Inflations from Maximal Quotient Groups

Maximal Quotient Group Q #1.

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

The Group Q is Isomorphic to the Group of Order 32 Number38 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 .

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

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

y + x ,

z + y ,

z ,

y² + yx + v ,

yv + t ,

yv + xv + s ,

yxw + yxv + v² + xu + xt + r
in the cohomology of G.

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

x² .

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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 Number16 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4, $.3^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 .

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

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

z + y + x ,

y + x ,

z + x ,

yx + w ,

zw + u ,

y²w + w² + r
in the cohomology of G.

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

zy + zx + x² .

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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 Number40 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4 * $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 .

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

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

z + y + x ,

x ,

z + x ,

zw + yw + zv + yv + t + s ,

xw + xv + u + s ,

y²w + y²v + yxv + xt + r ,

y²w + yxv + w² + v² + xu + r
in the cohomology of G.

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

y² + x² .

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Maximal Quotient Group Q #4.

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

The Group Q is Isomorphic to the Group of Order 32 Number14 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4, $.2^$.1 = $.2 * $.5, $.3^$.2 = $.3 * $.5

of type Cyclic(4) x Dihedral(8) .

The generators of G have images [ pcy.1 * pcy.2 * pcy.3 * pcy.4, pcy.1 * 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

z + x ,

x ,

z + y + x ,

w ,

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 ,

x³ .

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Maximal Quotient Group Q #5.

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

The Group Q is Isomorphic to the Group of Order 32 Number37 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 .

The generators of G have images [ pcy.1 * pcy.2 * pcy.3 * pcy.4 * pcy.5, pcy.1 * pcy.3, 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 ,

y + x ,

z + x ,

y² + yx + v ,

yw + xw + yv + xv + s ,

yxw + y²v + w² + v² + xu + xt + r
in the cohomology of G.

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

y² + zx + yx + x² .

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Maximal Quotient Group Q #6.

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

The Group Q is Isomorphic to the Group of Order 32 Number15 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.5, $.4^2 = $.5, $.4^$.2 = $.4 * $.5

of type Cyclic(4) x Quaternion(8) .

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

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

z ,

z + y ,

x ,

w ,

yxw + y²v + w² + v² + xu + xt + r
in the cohomology of G.

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

zx + yx + x² .

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Maximal Quotient Group Q #7.

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 Number41 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.4 * $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 .

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

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

y ,

z + y + x ,

z ,

zw + yw + xv + u + s ,

yw + zv + yv + xv + u + t + s ,

xw + yv + s ,

y²w + w² + v² ,

zyw + y²w + yxv + v² + xt + zs + r
in the cohomology of G.

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

x² .

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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.5
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 ,

u ,

t ,

s ,

r .

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.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 ,

yx + w ,

v ,

u ,

t ,

s ,

r .

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.5
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 ,

zy + v ,

u ,

t ,

s ,

r .

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.2
G.3
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 ,

yx + w ,

v ,

u ,

t ,

s ,

r .

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.6
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 ,

u ,

t ,

s ,

r .

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.2 * G.5
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 ,

v ,

u ,

t ,

s ,

r .

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.3
G.3
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 + yx + w ,

v ,

u ,

t ,

s ,

r .

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CHECKS

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

THE COMPUTATION IS COMPLETE