GROUP OF ORDER 64 #35

GROUP #35

The MAGMA library number for G is 126

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.3, G.2^2 = G.3, G.3^2 = G.5 * G.6, G.4^2 = G.5, G.2^G.1 = G.2 * G.6, G.4^G.1 = G.4 * G.6

The center of G is abelian of type [ 2, 8 ] .
The orders of the terms of the lower central series are [ 64, 2, 1 ] .
The orders of the terms of the upper central series are [ 1, 16, 64 ] .
The order of the Frattini subgroup is 8.
The exponent of G is 8.
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, v ] in degrees [ 1, 1, 1, 2, 4 ] , by the ideal generated by the relations
z2 + y2 ,
zy + zx + x2 ,
zx2 + yx2 + x3 .

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
w , v .

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.5 * G.6, G.6 ] 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 ,
0 ,
y2 ,
z4 + y4
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 .

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

It is nilpotent of degree 5.

Restrictions to Maximal Subgroups

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

The Group H is Isomorphic to the Group of Order 32 Number21 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1, G.4 ] .

The images of the generators of the cohomology of G restricted to H are
z ,
0 ,
y ,
w ,
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 Generated by [ G.2 * G.4, G.1, G.3, G.5 * G.6, G.5 ] .

The group H is abelian of type [ 8, 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 + y ,
y ,
y ,
x + w ,
x2
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.3, G.5 * G.6, G.1 * G.2, G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number15 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.5, $.4^2 = $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.2 * G.3 * G.5 * G.6, G.3, G.3 * G.4 ]

of type Cyclic(4) x Quaternion(8) .

The images of the generators of the cohomology of G restricted to H are
x ,
x ,
y ,
y2 + w ,
x2w + w2 + v
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 #4 Generated by [ G.2 * G.4, G.3, G.5 * G.6, G.1 * G.4, G.5 ] .

The group H is abelian of type [ 8, 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 + y ,
y ,
z ,
x + w ,
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 #5 Generated by [ G.3, G.2, G.5 * G.6, G.1 * G.4, G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number21 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.2 * G.3 * G.4 * G.6, G.1 * G.4 ] .

The images of the generators of the cohomology of G restricted to H are
z + y ,
y ,
z + y ,
w ,
y2w + x2
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 #6 Generated by [ G.4, G.3, G.2, G.5 * G.6, G.5 ] .

The group H is abelian of type [ 8, 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
0 ,
z ,
y ,
x + w ,
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 #7 Generated by [ G.1, G.3, G.2, G.5 * G.6, G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number21 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1, G.1 * G.2 * G.6 ] .

The images of the generators of the cohomology of G restricted to H are
z + y ,
y ,
0 ,
w ,
y2w + x2
in the cohomology of H.

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

The essential cohomology of G is generated as an ideal by
y2x + yx2 ,
x3 .

It is nilpotent of degree 2.
The annihilator of the Essential Cohomology has dimension 2 .
The essential cohomology is a free module over the polynomial subring Q of the cohomology ring of G generated by w , v .
The essential cohomology is generated as a module over Q by the elements
[] [] [ y2x+ yx2, x3 ] [ yx3 ]

Inflations from Maximal Quotient Groups

Maximal Quotient Group Q #1.

The kernel of the quotient is generated by G.5 * G.6 .

The Group Q is Isomorphic to the Group of Order 32 Number15 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.5, $.4^2 = $.5, $.4^$.2 = $.4 * $.5

of type Cyclic(4) x Quaternion(8) .

The generators of G have images [ pcy.2, pcy.1 * pcy.2, pcy.1 * pcy.4 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are
z + y ,
y + x ,
z ,
zy + zx ,
y2w + w2 + v
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
yx + w .

Maximal Quotient Group Q #2.

The kernel of the quotient is generated by G.6 .

The group Q is abelian of type [ 8, 2, 2 ] .

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

The images of the generators of the cohomology of Q inflated to G are
z + y ,
x ,
y ,
zy + y2 + zx + w
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
zy + y2 + zx + yx + x2 ,
zx2 + x3 .

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 32 Number17 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.3^$.1 = $.3 * $.5 .

The generators of G have images [ pcy.3 * pcy.4, pcy.2 * pcy.4, pcy.1 * pcy.4 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are
z ,
y ,
x ,
y2w + v
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
zy + zx + x2 .

Action of Automorphisms

The groups of outer automorphisms of G has order 192, and is generated by 7 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.5 * G.6
    G.3
    G.4 * G.5 * G.6
    G.5
    G.6 .

The images of the generators of the cohomology of G under the map induced by the automorphism are
z ,
y ,
x ,
zy + y2 + zx + w ,
v .

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.5 * G.6
    G.2
    G.3
    G.4
    G.5
    G.6 .

The images of the generators of the cohomology of G under the map induced by the automorphism are
z ,
y ,
x ,
y2 + w ,
v .

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.5 * G.6
    G.2
    G.3
    G.4 * G.5 * G.6
    G.5
    G.6 .

The images of the generators of the cohomology of G under the map induced by the automorphism are
z ,
y ,
x ,
zy + y2 + zx + w ,
v .

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

    G.1 * G.6
    G.2
    G.3
    G.4 * G.6
    G.5
    G.6 .

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

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.3
    G.2 * G.3
    G.3 * G.5 * G.6
    G.4 * G.5 * G.6
    G.5
    G.6 .

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

Automorphism #6

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

    G.2 * G.5 * G.6
    G.1 * G.3 * G.4 * G.6
    G.3
    G.1 * G.2 * G.5 * G.6
    G.5
    G.6 .

The images of the generators of the cohomology of G under the map induced by the automorphism are
y + x ,
z + x ,
y ,
zy + zx + w ,
x2w + v .

Automorphism #7

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.4 * G.5
    G.2
    G.3
    G.4
    G.5
    G.6 .

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

CHECKS

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

THE COMPUTATION IS COMPLETE