Restrictions to Maximal Elementary Abelian Subgroups

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

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

x ,

x ,

x ,

0 ,

z² + x² ,

z²x ,

z²y² + y⁴
in the cohomology of E.

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

z + x ,

y + x ,

w ,

x³ + xv + u .

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

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

x ,

x ,

z + x ,

0 ,

x² ,

zx² ,

z²y² + y⁴
in the cohomology of E.

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

z + y ,

y² + v ,

w ,

yv + xv + u .

The nilradical of the cohomology of G is generated by

z + y , w , y²x + yv + u .

It is nilpotent of degree 3.

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Restrictions to Maximal Subgroups

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

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

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

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

x ,

x ,

y + x ,

zy + y² + yx + yw ,

yx + yw + w² ,

zx² + x³ + zyw + y²w + zw² + yw² + xw² ,

z²y² + y⁴ + z²yx + zyx² + zx³ + z³w + zxw² + x²w² + 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 #2 Generated by [ G.5, G.3, G.6, G.4, G.1 ] .

The Group H is Isomorphic to the Group of Order 32 Number31 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.3, G.3 ] .

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

z ,

0 ,

z + y ,

x ,

w ,

yw + v ,

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

The Group H is Isomorphic to the Group of Order 32 Number13 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.3, G.1 * G.3 * G.5, G.1 * G.2 * G.3 * G.4 ]

of type Cyclic(2) x Group(16)#11 .

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

z + y + x ,

x ,

z + y + x ,

zy + y² ,

zy + zx + yx ,

z²x + x³ + w ,

zy³ + zyx² + y²x² + xw + 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 #4 Generated by [ G.5, G.6, G.2 * G.3 * G.5 * G.6, G.4, G.1 ] .

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 + x ,

x ,

x ,

zy + zx ,

w ,

zyx + yx² + x³ + yw + xw + zv ,

yx³ + zyw + zxw + yxw + zxv + x²v + wv + 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 #5 Generated by [ G.5, G.6, G.1 * G.2, G.2 * G.3 * G.5 * G.6, G.4 ] .

The Group H is Isomorphic to the Group of Order 32 Number31 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.3 * G.4, G.1 * G.2 ] .

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

z + y ,

y ,

z ,

x ,

y² + w ,

y³ + yx + v ,

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

The Group H is Isomorphic to the Group of Order 32 Number31 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2 * G.5, G.1 * G.2 ] .

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

y ,

z + y ,

0 ,

x ,

y² + w ,

y³ + yx + yw + v ,

yv + 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 #7 Generated by [ G.5, G.3, G.6, G.2, G.4 ] .

The Group H is Isomorphic to the Group of Order 32 Number31 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.3, G.2 ] .

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

0 ,

z ,

y ,

x ,

w ,

v ,

u
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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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 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.1 * pcy.2 * pcy.4, pcy.2 * pcy.3 * pcy.5, pcy.1 * pcy.3 ] in the quotient group.

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

z + y ,

y + x ,

y ,

zy + y² + yx + x² + w ,

y² + x² + v ,

zy + y² + w
in the cohomology of G.

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

y² + yx + w + u ,

yw + xw + yv + yu + xu .

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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 .

The Group Q is Isomorphic to the Group of Order 32 Number31 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 .

The generators of G have images [ pcy.1, pcy.2, 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 + x ,

zy + y² + w ,

y² + v ,

z²y + zy² + y²x + zw + zv + u ,

y³x + zx³ + yx³ + y²w + wv + xu + t
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 * G.6 .

The Group Q is Isomorphic to the Group of Order 32 Number31 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 .

The generators of G have images [ pcy.1, pcy.2, 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 + x ,

zy + y² + w ,

y² + v ,

y²x + zw + zv + u ,

zy³ + y³x + z²x² + y²w + y²v + wv + 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 32, and is generated by 5 elements.

Automorphism #1

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

u ,

t .

Automorphism #2

G.1
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 ,

z²y + zy² + u ,

zy³ + z²x² + y²v + t .

Automorphism #3

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

Automorphism #4

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

x ,

z + y + x ,

z ,

w ,

y² + x² + v ,

zx² + x³ + zw + xw + zv + xv + u ,

zx³ + yx³ + y²w + zyv + y²v + wv + t .

Automorphism #5

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

w ,

y² + x² + v ,

zy² + y²x + zx² + x³ + zw + xw + u ,

t .

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CHECKS

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

THE COMPUTATION IS COMPLETE