GROUP OF ORDER 16 #12

GROUP #12

Dihedral(16)

The MAGMA library number for G is 7

GrpPC : G of order 16 = 2^4 PC-Relations: G.2^2 = G.3 * G.4, G.3^2 = G.4, G.2^G.1 = G.2 * G.3, G.3^G.1 = G.3 * G.4

The center of G is abelian of type [ 2 ] .
The orders of the terms of the lower central series are [ 16, 4, 2, 1 ] .
The orders of the terms of the upper central series are [ 1, 2, 4, 16 ] .
The order of the Frattini subgroup is 4.
The exponent of G is 8.
G has 2 conjugacy classes of maximal elementary abelian p-subgroups. The orders of the maximal elementary abelian subgroups are [ 4, 4 ] .
The orders of the centralizers of the maximal elementary abelian subgroups are [ 4, 4 ] .
The orders of the normalizers of the maximal elementary abelian subgroups are [ 8, 8 ] .

The cohomology ring of G is the quotient of a polynomial ring in the variables
[ z, y, x ] in degrees [ 1, 1, 2 ] , by the ideal generated by the relations

zy + y² .

The Hilbert series for the cohomology ring is

1 / t² -2t+ 1.
Its denominator factors as ( t-1 )² .

The Krull dimension of the cohomology ring is 2.
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators

z² , x .

Restrictions to Maximal Elementary Abelian Subgroups

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

The cohomology ring of E is a polynomial ring in the variables z , y .

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

z ,

z ,

zy + y²
in the cohomology of E.

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

z + y .

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

The cohomology ring of E is a polynomial ring in the variables z , y .

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

z ,

0 ,

zy + y²
in the cohomology of E.

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

y .

The nilradical of the cohomology of G is zero.

Restrictions to Maximal Subgroups

Maximal Subgroup H #1

The group H is abelian of type [ 8 ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables
[ z, y ] , in degrees [ 1, 2 ] , by the ideal of relations
z² .

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

0 ,

z ,

y
in the cohomology of H.

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

z .

Maximal Subgroup H #2

The Group H is Isomorphic to the Group of Order 8 Number4 GrpPC of order 8 = 2^3 PC-Relations: $.2^$.1 = $.2 * $.3 Generated by [ G.1, G.1 * G.3 * G.4 ]

of type Dihedral(8) .

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

z + y ,

0 ,

x
in the cohomology of H.

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

y .

Maximal Subgroup H #3

The Group H is Isomorphic to the Group of Order 8 Number4 GrpPC of order 8 = 2^3 PC-Relations: $.2^$.1 = $.2 * $.3 Generated by [ G.1 * G.2, G.1 * G.2 * G.3 * G.4 ]

of type Dihedral(8) .

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

z + y ,

z + y ,

x
in the cohomology of H.

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

z + 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 8 Number4 GrpPC of order 8 = 2^3 PC-Relations: $.2^$.1 = $.2 * $.3

of type Dihedral(8) .

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

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

y ,

z + y ,

0
in the cohomology of G.

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

x .

Action of Automorphisms

The groups of outer automorphisms of G has order 4, and is generated by 2 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

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

z ,

y ,

x .

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

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

z ,

z + y ,

x .

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

THE COMPUTATION IS COMPLETE

GROUP #12

Dihedral(16)

GrpPC : G of order 16 = 2^4 PC-Relations: G.2^2 = G.3 * G.4, G.3^2 = G.4, G.2^G.1 = G.2 * G.3, G.3^G.1 = G.3 * G.4

The cohomology ring of G is the quotient of a polynomial ring in the variables [ z, y, x ] in degrees [ 1, 1, 2 ] , by the ideal generated by the relations zy + y2 .

Restrictions to Maximal Elementary Abelian Subgroups

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

The images of the generators of the cohomology of G restricted to E are z , z , zy + y2 in the cohomology of E.

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

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

The images of the generators of the cohomology of G restricted to E are z , 0 , zy + y2 in the cohomology of E.

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

The nilradical of the cohomology of G is zero.

Restrictions to Maximal Subgroups

Maximal Subgroup H #1

The group H is abelian of type [ 8 ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables [ z, y ] , in degrees [ 1, 2 ] , by the ideal of relations z2 .

The images of the generators of the cohomology of G restricted to H are 0 , z , y in the cohomology of H.

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

Maximal Subgroup H #2

The Group H is Isomorphic to the Group of Order 8 Number4 GrpPC of order 8 = 2^3 PC-Relations: $.2^$.1 = $.2 * $.3 Generated by [ G.1, G.1 * G.3 * G.4 ]

of type Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are z + y , 0 , x in the cohomology of H.

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

Maximal Subgroup H #3

The Group H is Isomorphic to the Group of Order 8 Number4 GrpPC of order 8 = 2^3 PC-Relations: $.2^$.1 = $.2 * $.3 Generated by [ G.1 * G.2, G.1 * G.2 * G.3 * G.4 ]

of type Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are z + y , z + y , x in the cohomology of H.

The kernel of the restriction to H of the cohomology of G is generated by z + 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 8 Number4 GrpPC of order 8 = 2^3 PC-Relations: $.2^$.1 = $.2 * $.3

of type Dihedral(8) .

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

The images of the generators of the cohomology of Q inflated to G are y , z + y , 0 in the cohomology of G.

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

Action of Automorphisms

The groups of outer automorphisms of G has order 4, and is generated by 2 elements.

Automorphism #1

The images of the generators of the cohomology of G under the map induced by the automorphism are z , y , x . Automorphism #2

Automorphism #2

The images of the generators of the cohomology of G under the map induced by the automorphism are z , z + y , x . CHECKS paramflag = true qregflagflag = true cmflag = true essflag = true centflag = true THE COMPUTATION IS COMPLETE

CHECKS

THE COMPUTATION IS COMPLETE

The cohomology ring of G is the quotient of a polynomial ring in the variables
[ z, y, x ] in degrees [ 1, 1, 2 ] , by the ideal generated by the relations

zy + y² .

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

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

z ,

z ,

zy + y²
in the cohomology of E.

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

z + y .

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

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

z ,

0 ,

zy + y²
in the cohomology of E.

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

y .

The cohomology ring of H is a the quotient of a polynomial ring in the variables
[ z, y ] , in degrees [ 1, 2 ] , by the ideal of relations
z² .

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

0 ,

z ,

y
in the cohomology of H.

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

z .

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

z + y ,

0 ,

x
in the cohomology of H.

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

y .

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

z + y ,

z + y ,

x
in the cohomology of H.

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

z + y .

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

y ,

z + y ,

0
in the cohomology of G.

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

x .

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

z ,

y ,

x .

Automorphism #2

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

z ,

z + y ,

x .

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

THE COMPUTATION IS COMPLETE