The hypercohomolgy spectral sequence has E2 term:

Restrictions to Maximal Elementary Abelian Subgroups

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

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

z ,

z ,

0 ,

z² + zx + x² ,

z²y + zy² + z²x + zx² ,

z³y + y⁴ + z³x + 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 .

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

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

z ,

z ,

z ,

z² + zx + x² ,

z²y + zy² + z²x + zx² ,

z³y + y⁴ + z³x + 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 .

The nilradical of the cohomology of G is generated by

z + y

It is nilpotent of degree 3.

---

---

Restrictions to Maximal Subgroups

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

The Group H is Isomorphic to the Group of Order 16 Number9 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3 * $.4, $.2^2 = $.3, $.2^$.1 = $.2 * $.4 Generated by [ G.1, G.2 * G.3 * G.5 ] .

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

z ,

y ,

y ,

w ,

zx + zv ,

zyx + v²
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 #2 Generated by [ G.5, G.2, G.4, G.1 * G.3 ] .

The Group H is Isomorphic to the Group of Order 16 Number9 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3 * $.4, $.2^2 = $.3, $.2^$.1 = $.2 * $.4 Generated by [ G.2, G.1 * G.3 ] .

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

y ,

z ,

y ,

x + w + v ,

zy² + yw ,

y²w + 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 #3 Generated by [ G.1, G.5, G.2, G.4 ] .

The Group H is Isomorphic to the Group of Order 16 Number9 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3 * $.4, $.2^2 = $.3, $.2^$.1 = $.2 * $.4 Generated by [ G.1, G.2 ] .

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

y ,

z ,

0 ,

x + w + v ,

zx + yv ,

zyx + v²
in the cohomology of H.

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

x .

---

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

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.1 * G.2, G.1 * 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

z + y ,

z + y ,

x ,

z² + zy + y² ,

zw + yw ,

z²w + y²w + 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 #5 Generated by [ G.5, G.2 * G.3 * G.5, G.1 * G.2, G.4 ] .

The Group H is Isomorphic to the Group of Order 16 Number9 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3 * $.4, $.2^2 = $.3, $.2^$.1 = $.2 * $.4 Generated by [ G.2 * G.3 * G.5, G.1 * G.3 ] .

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

z ,

y ,

z + y ,

w ,

zy² + zx + zw + zv ,

zyx + y²w + w² + v²
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 #6 Generated by [ G.3, G.5, G.2, G.4 ] .

The Group H is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.3, G.2, G.4 ]

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

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

0 ,

z ,

y ,

zx + x² ,

z²x ,

w
in the cohomology of H.

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

z .

---

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

The group H is abelian of type [ 4, 4 ]

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

0 ,

z + y ,

x ,

zw ,

w²
in the cohomology of H.

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

y .

---

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.4 .

The Group Q 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

of type AlmostExtraSpecial(16) .

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

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

z + y ,

y ,

x ,

zv + u
in the cohomology of G.

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

zy + yx + x² .

---

---

Maximal Quotient Group Q #2

The kernel of the quotient is generated by G.4 * G.5 .

The Group Q 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

of type AlmostExtraSpecial(16) .

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

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

x ,

z + x ,

y ,

z⁴ + z²w + y²w + w² + zv + u
in the cohomology of G.

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

y² + zx + x² .

---

---

Maximal Quotient Group Q #3

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

The Group Q 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

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

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

z + x ,

z + y ,

z² + w
in the cohomology of G.

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

zy + x² ,

x³ .

---

---

The depth-essential cohomology of G

The depth-essential cohomology of G is the intersection of the restrictions to the centralizers of the maximal elementary abelian p-subgroups of rank d+1 where d is the depth of the cohomology ring. For this group the depth is 2.

There are 2 conjugacy classes of subgroups which are centralizers of elementary abelian subgroups of rank 3. They are represented by the subgroups generated by

[ G.1 * G.2, G.1 * G.2 * G.5, G.1 * G.2 * G.4 ]

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

The depth-essential cohomology of G is generated as an ideal by

z + y .

The annihilator of the depth-essential cohomology has dimension 2 .

The depth-essential cohomology is a free module over the polynomial subring Q of the cohomology ring of G generated by w , u .
The depth-essential cohomology is generated as a module over Q by the elements

[ z+ y ] [ zx+ yx, yx+ x² ] [ yx²+ x³ ]

---

---

Action of Automorphisms

The groups of outer automorphisms of G has order 32, and is generated by 5 elements.

Automorphism #1

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

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

z ,

y ,

x ,

yx + x² + w ,

zyx + yx² + v ,

u .

Automorphism #2

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

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

z ,

y ,

x ,

yx + x² + w ,

zyx + yx² + v ,

u .

Automorphism #3

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

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

z ,

y ,

x ,

yx + x² + w ,

zyx + yx² + v ,

u .

Automorphism #4

G.1
G.2 * G.3
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 ,

w ,

v ,

u .

Automorphism #5

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

z ,

y ,

z + y + x ,

w ,

z³ + zw + v ,

y²w + w² + u .

---

---

CHECKS

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

THE COMPUTATION IS COMPLETE