Restrictions to Maximal Elementary Abelian Subgroups

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

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

x ,

x ,

0 ,

z + x ,

z²y² + y⁴ + z²yx + zy²x + z²x² + zyx² + y²x²
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 .

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

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

x ,

x ,

z ,

z + x ,

z⁴ + z²y² + y⁴ + z³x + z²yx + zy²x + zyx² + y²x²
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 + w .

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

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

y ,

z ,

y ,

z + y ,

z²yx + zy²x + z²x² + zyx² + 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 + x ,

y + x + w .

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

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

y ,

z ,

z + y ,

z + y ,

z⁴ + z³y + z²y² + z²yx + zy²x + z²x² + zyx² + 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 + w ,

x + w .

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

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

0 ,

z ,

0 ,

z + x ,

z²y² + y⁴ + z²yx + zy²x + zyx² + y²x²
in the cohomology of E.

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

z ,

x .

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

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

0 ,

x ,

z + x ,

z + x ,

z⁴ + z²y² + y⁴ + z³x + z²yx + zy²x + zyx² + y²x² + zx³ + 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 zero.

Restrictions to Maximal Subgroups

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

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.2 * G.4, G.3 * G.4, G.1 ]

of type Cyclic(2) x Dihedral(8) .

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

z + x ,

z ,

y ,

z + y ,

zy³ + y⁴ + zy²x + y²x² + zyw + w²
in the cohomology of H.

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

y + x + w .

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

The Group H is Isomorphic to the Group of Order 16 Number8 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.3^2 = $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.2 * G.3 * G.5, G.1 * G.4, G.1 ]

of type AlmostExtraSpecial(16) .

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

z + y + x ,

y ,

y ,

x ,

zy³ + zy²x + yx³ + w
in the cohomology of H.

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

y + x .

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

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.2 * G.3 * G.4, G.2 * G.3, G.3 * G.4 ]

of type Cyclic(2) x Dihedral(8) .

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

0 ,

z + x ,

z + y + x ,

z + y ,

z⁴ + z³y + zy³ + y⁴ + z³x + zy²x + y²x² + zyw + w²
in the cohomology of H.

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

z .

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

The Group H is Isomorphic to the Group of Order 16 Number8 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.3^2 = $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.3 * G.4, G.2 * G.3, G.3 ]

of type AlmostExtraSpecial(16) .

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

y ,

x ,

z + y + x ,

y ,

w
in the cohomology of H.

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

z + w .

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

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.2 * G.4, G.1 * G.4, G.1 * G.2 ]

of type Cyclic(2) x Dihedral(8) .

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

z + y + x ,

z + y ,

0 ,

z + x ,

z²y² + y⁴ + y²x² + zyw + w²
in the cohomology of H.

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

x .

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

The Group H is Isomorphic to the Group of Order 16 Number8 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.3^2 = $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.4 * G.5, G.4 * G.5, G.3 ]

of type AlmostExtraSpecial(16) .

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

z ,

0 ,

x ,

z + y ,

yx³ + w
in the cohomology of H.

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

y .

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

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.3 * G.5, G.2, G.4 ]

of type Cyclic(2) x Dihedral(8) .

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

x ,

y ,

x ,

z ,

zyw + w²
in the cohomology of H.

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

z + x .

Maximal Subgroup H #8 Generated by [ G.1 * G.2 * G.5, G.4, G.5, G.3 ] .

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.4 * G.5, G.1 * G.2, G.3 ]

of type Cyclic(2) x Dihedral(8) .

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

y ,

y ,

x ,

z ,

z²y² + y⁴ + z³x + zy²x + y²x² + zyw + w²
in the cohomology of H.

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

z + y .

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

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.2 * G.4, G.1 * G.2 * G.3, G.3 ]

of type Cyclic(2) x Dihedral(8) .

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

z ,

z + y ,

z + x ,

y ,

z³x + zy²x + y²x² + zyw + w²
in the cohomology of H.

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

z + y + w .

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

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.2 * G.4 * G.5, G.1 * G.2 * G.3, G.1 * G.3 * G.5 ]

of type Cyclic(2) x Dihedral(8) .

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

z + y + x ,

z + y ,

z + x ,

y ,

z³y + z²yx + zy²x + y²x² + zyw + w²
in the cohomology of H.

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

z + x + w .

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

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.2 * G.3 * G.4, G.2 * G.3, G.1 * G.3 * G.4 * G.5 ]

of type Cyclic(2) x Dihedral(8) .

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

y ,

z + x ,

z + y + x ,

z + y ,

z⁴ + z³y + z²y² + z³x + zyw + w²
in the cohomology of H.

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

z + y + x .

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

The Group H is Isomorphic to the Group of Order 16 Number8 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.3^2 = $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.2 * G.3, G.3 * G.4, G.1 * G.3 * G.5 ]

of type AlmostExtraSpecial(16) .

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

z ,

x ,

z + y + x ,

y ,

y⁴ + z³x + yx³ + 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 .

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

The Group H is Isomorphic to the Group of Order 16 Number8 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.3^2 = $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.2 * G.4 * G.5, G.1 * G.2 * G.3 * G.4, G.3 ]

of type AlmostExtraSpecial(16) .

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

z ,

z + y ,

z + x ,

z + y ,

yx³ + w
in the cohomology of H.

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

y + w .

Maximal Subgroup H #14 Generated by [ G.2, G.5, G.3, G.1 ] .

The Group H is Isomorphic to the Group of Order 16 Number8 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.3^2 = $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.2 * G.3 * G.5, G.2 * G.5, G.1 ]

of type AlmostExtraSpecial(16) .

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

x ,

z + y ,

z ,

0 ,

z³x + zy²x + x⁴ + w
in the cohomology of H.

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

w .

Maximal Subgroup H #15 Generated by [ G.2, G.5, G.3 * G.4 * G.5, G.1 ] .

The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.3 * G.4, G.2 * G.5, G.1 ]

of type Cyclic(2) x Dihedral(8) .

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

x ,

z ,

y ,

y ,

z²y² + zy³ + y⁴ + z³x + zy²x + y²x² + zyw + w²
in the cohomology of H.

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

x + w .

The essential cohomology of G is zero.

Inflations from Maximal Quotient Groups

Maximal Quotient Group Q #1

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

The group Q is abelian of type [ 2, 2, 2, 2 ] .

The cohomology ring of Q is a polynomial ring with variables z , y , x , w in degrees [ 1, 1, 1, 1 ]

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

w ,

x ,

y ,

z
in the cohomology of G.

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

zy + y² + yw + xw + w² ,

zxw + x²w + zw² + w³ .

Action of Automorphisms

The groups of outer automorphisms of G has order 72, and is generated by 5 elements.

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

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

x ,

y + w ,

z + x ,

y ,

z³x + y²x² + zy²w + z²xw + zyxw + x³w + yxw² + v .

Automorphism #2

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

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

z + y + w ,

y + w ,

x + w ,

y ,

zy³ + z²x² + y²x² + zyxw + y²xw + zyw² + zw³ + yw³ + v .

Automorphism #3

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

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

x + w ,

y ,

z + y + w ,

y + w ,

y⁴ + z³x + zy²w + y³w + z²xw + zyxw + y²w² + v .

Automorphism #4

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

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

z + y ,

y ,

x ,

y + w ,

zy³ + y²w² + v .

Automorphism #5

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

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

z ,

z + y + x + w ,

z + x ,

z + w ,

zy³ + y²x² + z²xw + y²xw + x³w + zyw² + xw³ + v .

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CHECKS

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

THE COMPUTATION IS COMPLETE