GROUP OF ORDER 64 #66

GROUP #66

The MAGMA library number for G is 31

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

The center of G is abelian of type [ 8 ] .
The orders of the terms of the lower central series are [ 64, 4, 2, 1 ] .
The orders of the terms of the upper central series are [ 1, 8, 16, 64 ] .
The order of the Frattini subgroup is 16.
The exponent of G is 16.
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 [ 32, 16 ] .
The orders of the normalizers of the maximal elementary abelian subgroups are [ 64, 32 ] .

The cohomology ring of G is the quotient of a polynomial ring in the variables
[ z, y, x, w, v, u ] in degrees [ 1, 1, 2, 2, 3, 4 ] , by the ideal generated by the relations
z2 ,
zy ,
zx ,
yw ,
x2 ,
xw + zv ,
yv ,
zw2 + xv ,
w3 + v2 .

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
u , y2 + w .

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E #1
Generated by [ G.6, G.4 * G.5 ]

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 ,
z2 ,
z3 ,
z2y2 + 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 .

Subgroup E #2
Generated by [ G.3 * G.5, 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 ,
z ,
0 ,
0 ,
0 ,
z2y2 + y4
in the cohomology of E.

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

The nilradical of the cohomology of G is generated by
z , x .

It is nilpotent of degree 2.

Restrictions to Maximal Subgroups

Maximal Subgroup H #1 Generated by [ G.2, G.5, G.4, G.1, G.6 ] .

The group H is abelian of type [ 16, 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 ,
zy ,
y2 ,
zy2 + y3 + zx ,
y2x + 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.3, G.2, G.5, G.4, G.6 ] .

The Group H 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 Generated by [ G.2, G.5, G.2 * G.3 * G.4 * G.6 ] .

The images of the generators of the cohomology of G restricted to H are
0 ,
z ,
z2 + zy ,
z2 + zy + zx + x2 ,
z2y + zy2 + zyx + x3 ,
z2y2 + 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 #3 Generated by [ G.2, G.5, G.4, G.6, G.1 * G.3 * G.5 * G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number22 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.2^2 = $.3, $.3^2 = $.5, $.4^$.1 = $.4 * $.5 Generated by [ G.1 * G.3 * G.5 * G.6, G.4 * G.5 ] .

The images of the generators of the cohomology of G restricted to H are
z ,
z ,
zy ,
zy + y2 ,
y3 + x ,
w
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.6 .

The Group Q is Isomorphic to the Group of Order 32 Number20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.2^2 = $.4, $.3^$.1 = $.3 * $.5 .

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

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

The kernel of the inflation to G of the cohomology of Q is generated by
x + w + v ,
yw .

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 ,
w ,
v ,
u .

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 ,
y ,
x ,
w ,
v ,
u .

Automorphism #3

The order of the class of the automorphism in the outer automorphism group is 4. 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 ,
w ,
v ,
u .

Automorphism #4

The order of the class of the automorphism in the outer automorphism group is 4. 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 ,
w ,
v ,
u .

CHECKS

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

THE COMPUTATION IS COMPLETE