GROUP OF ORDER 16 #6

GROUP #6

Cyclic(2) x Dihedral(8)

The MAGMA library number for G is 11

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

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

The cohomology ring of G is the quotient of a polynomial ring in the variables
[ z, y, x, w ] in degrees [ 1, 1, 1, 2 ] , by the ideal generated by the relations
zx + yx + x2 .

The Hilbert series for the cohomology ring is
-1 / t3 -3t2+ 3t -1.
Its denominator factors as ( t-1 )3 .

The Krull dimension of the cohomology ring is 3.
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators
z2 , y2 , w .

Restrictions to Maximal Elementary Abelian Subgroups

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

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

The images of the generators of the cohomology of G restricted to E are
y + x ,
z + y + x ,
z ,
zx + x2
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.2, G.1 * G.2 ]

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

The images of the generators of the cohomology of G restricted to E are
y + x ,
z + y + x ,
0 ,
zx + x2
in the cohomology of E.

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

The nilradical of the cohomology of G is zero.

Restrictions to Maximal Subgroups

Maximal Subgroup H #1

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

of type Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are
0 ,
z + y ,
z ,
x
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 abelian of type [ 2, 2, 2 ]

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

The images of the generators of the cohomology of G restricted to H are
y ,
z + y ,
z ,
zx + x2
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 #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.3 * G.4, G.2 ]

of type Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are
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 + x .

Maximal Subgroup H #4

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.3 * G.4, G.1 ]

of type Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are
z + y ,
0 ,
z ,
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 #5

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.3 * G.4 ]

of type Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are
z ,
y ,
y ,
x
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 #6

The group H is abelian of type [ 2, 2, 2 ]

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

The images of the generators of the cohomology of G restricted to H are
z ,
y ,
0 ,
zx + yx + x2
in the cohomology of H.

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

Maximal Subgroup H #7

The group H is abelian of type [ 4, 2 ]

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

The images of the generators of the cohomology of G restricted to H are
y ,
y ,
z ,
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 abelian of type [ 2, 2, 2 ] .

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

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

The kernel of the inflation to G of the cohomology of Q is generated by
z2 + zy + zx .

Maximal Quotient Group Q #2

The kernel of the quotient is generated by G.1 * G.2 * 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, pcy.2 * pcy.3 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are
x ,
z + y + x ,
zy + w
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 .

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, pcy.2 ] in the quotient group.

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

The kernel of the inflation is zero.

Action of Automorphisms

The groups of outer automorphisms of G has order 16, and is generated by 4 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 ,
zy + w .

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
y + x ,
z + x ,
x ,
w .

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

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

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

The images of the generators of the cohomology of G under the map induced by the automorphism are
z ,
y ,
z + y + x ,
w .

CHECKS

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

THE COMPUTATION IS COMPLETE