GROUP OF ORDER 16 #7

GROUP #7

Cyclic(2) x Quaternion(8)

The MAGMA library number for G is 12

GrpPC : G of order 16 = 2^4 PC-Relations: G.1^2 = G.4, G.2^2 = G.4, G.2^G.1 = G.2 * 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 a unique maximal elementary abelian subgroup which is central in G and has order 4.

The cohomology ring of G is the quotient of a polynomial ring in the variables
[ z, y, x, w ] in degrees [ 1, 1, 1, 4 ] , by the ideal generated by the relations
z2 + zy + y2 ,
y3 .

The Hilbert series for the cohomology ring is
t2+ t+ 1 / t4 -2t3+ 2t2 -2t+ 1.
Its numerator factors as ( t2+t+1 ) .
Its denominator factors as ( t-1 )2 ( t2+1 ) .

The Krull dimension of the cohomology ring is 2.
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators
x2 , w .

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.3, G.3 * G.4 ] 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
0 ,
0 ,
z + y ,
z4
in the cohomology of E.

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

This ideal is also the nilradical of the cohomology of G.

It is nilpotent of degree 4.

Restrictions to Maximal Subgroups

Maximal Subgroup H #1

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
z ,
0 ,
y ,
x2
in the cohomology of H.

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

Maximal Subgroup H #2

The Group H is Isomorphic to the Group of Order 8 Number5 GrpPC of order 8 = 2^3 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.2^$.1 = $.2 * $.3 Generated by [ G.1 * G.3, G.2 ]

of type Quaternion(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 #3

The Group H is Isomorphic to the Group of Order 8 Number5 GrpPC of order 8 = 2^3 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.2^$.1 = $.2 * $.3 Generated by [ G.2, G.1 ]

of type Quaternion(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
x .

Maximal Subgroup H #4

The Group H is Isomorphic to the Group of Order 8 Number5 GrpPC of order 8 = 2^3 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.2^$.1 = $.2 * $.3 Generated by [ G.2 * G.3, G.1 ]

of type Quaternion(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 #5

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
0 ,
z ,
y ,
x2
in the cohomology of H.

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

Maximal Subgroup H #6

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
z ,
z ,
y ,
x2
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 #7

The Group H is Isomorphic to the Group of Order 8 Number5 GrpPC of order 8 = 2^3 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.2^$.1 = $.2 * $.3 Generated by [ G.1 * G.3, G.1 * G.2 * G.4 ]

of type Quaternion(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 + x .

The essential cohomology of G is generated as an ideal by
zy2x ,
zyx2 ,
y2x2 .

It is nilpotent of degree 2.
The annihilator of the Essential Cohomology has dimension 2 .

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
y2 + yx + x2 ,
x3 .

Maximal Quotient Group Q #2

The kernel of the quotient is generated by G.3 .

The Group Q is Isomorphic to the Group of Order 8 Number5 GrpPC of order 8 = 2^3 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.2^$.1 = $.2 * $.3

of type Quaternion(8) .

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

The images of the generators of the cohomology of Q inflated to G are
z ,
y ,
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.3 * G.4 .

The Group Q is Isomorphic to the Group of Order 8 Number5 GrpPC of order 8 = 2^3 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.2^$.1 = $.2 * $.3

of type Quaternion(8) .

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

The images of the generators of the cohomology of Q inflated to G are
z ,
y ,
x4 + 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 48, and is generated by 5 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

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

The images of the generators of the cohomology of G under the map induced by the automorphism are
z ,
y ,
x ,
x4 + 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

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

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

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

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

Automorphism #4

The order of the class of the automorphism in the outer automorphism group is 3. The images of the generators of G are

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

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

Automorphism #5

The order of the class of the automorphism in the outer automorphism group is 2. The images of the generators of G are

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

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

CHECKS

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

THE COMPUTATION IS COMPLETE