GROUP OF ORDER 64 #53

GROUP #53

The MAGMA library number for G is 97

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

The center of G is abelian of type [ 2, 4 ] .
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 8.
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 [ 8, 8 ] .
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, t ] in degrees [ 1, 1, 1, 2, 2, 3, 4 ] , by the ideal generated by the relations
z2 + y2 + x2 ,
zy + y2 + yx ,
y3 + zx2 + x3 ,
zw + yw + xw ,
zx3 + yx3 + x4 + y2w + x2w + zu + yu + xu ,
y2v + zu + xu ,
w2 + zu + xu ,
yx2v + ywv + y2u + zxu + yxu + x2u + wu ,
x4v + zx2u + x3u + zvu + xvu + u2 .

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
x2 , v , t .

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E #1
Generated by [ G.4, G.6, G.1 * G.3 * G.4 * G.6 ]

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
z ,
0 ,
z ,
0 ,
x2 ,
z2x ,
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 ,
y ,
w .

Subgroup E #2
Generated by [ G.2 * G.3 * G.4 * G.6, G.4, G.6 ]

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
0 ,
z ,
z ,
zx ,
x2 ,
zx2 ,
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 ,
xv + u .

The nilradical of the cohomology of G is generated by
z + y + x , y2 + yx , yw + xw .

It is nilpotent of degree 2.

Restrictions to Maximal Subgroups

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

The Group H is Isomorphic to the Group of Order 32 Number16 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4, $.3^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.3 * G.4, G.4 * G.5 * G.6, G.1 * G.4 * G.5 * G.6 ] .

The images of the generators of the cohomology of G restricted to H are
z ,
0 ,
y ,
zy + x2 ,
x2 + w ,
z3 + v ,
u
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.3, G.5, G.4, G.6, G.1 * G.3 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.3 * G.4, G.1 * G.2 * G.4 * G.5 * G.6, G.1 * G.3, G.1 * G.3 * G.4 * G.5 * G.6 ]

of type Cyclic(2) x AlmostExtraSpecial(16) .

The images of the generators of the cohomology of G restricted to H are
z + y + x + w ,
y ,
z + x + w ,
zy + y2 + yw ,
x2 ,
zyx + zx2 + x3 + z2w + zyw + y2w + zw2 + yw2 + xw2 ,
z2y2 + y4 + v
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 Generated by [ G.3, G.5, G.2, G.4, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number27 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.2^2 = $.5, $.4^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.2 * G.3 * G.4, G.2 ] .

The images of the generators of the cohomology of G restricted to H are
0 ,
z + y ,
y ,
w ,
x ,
zx + yx + zv ,
w2 + v2
in the cohomology of H.

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

Maximal Subgroup H #4 Generated by [ G.3, G.5, G.4, G.6, G.1 * G.2 ] .

The Group H is Isomorphic to the Group of Order 32 Number28 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.3^2 = $.5, $.4^2 = $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.3, G.1 * G.2 ] .

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

The Group H is Isomorphic to the Group of Order 32 Number28 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.3^2 = $.5, $.4^2 = $.5, $.3^$.1 = $.3 * $.4, $.4^$.1 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.1, G.1 * G.2 * G.4 * G.5 ] .

The images of the generators of the cohomology of G restricted to H are
z + y ,
y ,
0 ,
y2 + w ,
x ,
zx + yx + yw + v ,
x2 + w2 + u
in the cohomology of H.

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

Maximal Subgroup H #6 Generated by [ G.5, G.2, G.4, G.6, G.1 * G.3 ] .

The group H is abelian of type [ 8, 2, 2 ]

The cohomology ring of H 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 G restricted to H are
y ,
z ,
y ,
zx ,
y2 + x2 ,
zy2 + y3 + y2x + zx2 + zw ,
y2w + w2
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 #7 Generated by [ G.1, G.2 * G.3, G.5, G.4, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number27 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.2, $.2^2 = $.5, $.4^2 = $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.2 * G.3, G.1 * G.2 * G.3 ] .

The images of the generators of the cohomology of G restricted to H are
z ,
z + y ,
z + y ,
y2 + w ,
y2 + x ,
y3 + yx + zv ,
x2 + v2
in the cohomology of H.

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

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 Number11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^$.1 = $.2 * $.5

of type Cyclic(2) x Group(16)#9 .

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

The images of the generators of the cohomology of Q inflated to G are
z + y + x ,
z + x ,
y + x ,
zy + w ,
v ,
z2 + zy + zx + w + v
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
yx + w + v + u ,
xw + xv + xu .

Maximal Quotient Group Q #2.

The kernel of the quotient is generated by G.4 * G.6 .

The Group Q is Isomorphic to the Group of Order 32 Number26 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.4^$.2 = $.4 * $.5 .

The generators of G have images [ pcy.1 * pcy.2 * pcy.3 * pcy.5, pcy.1 * pcy.4, pcy.1 * pcy.2 * pcy.4 * pcy.5 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are
z + y + x ,
z + x ,
x ,
z2v + y2v + v2 + t
in the cohomology of G.

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

Maximal Quotient Group Q #3.

The kernel of the quotient is generated by G.4 .

The Group Q is Isomorphic to the Group of Order 32 Number26 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.4^$.2 = $.4 * $.5 .

The generators of G have images [ pcy.1 * pcy.2 * pcy.5, pcy.1 * pcy.3 * pcy.4 * pcy.5, pcy.1 * pcy.2 * pcy.3 * pcy.4 ] in the quotient group.

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

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

Action of Automorphisms

The groups of outer automorphisms of G has order 64, and is generated by 6 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 ,
t .

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 ,
z2y + u ,
t .

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
z ,
y ,
x ,
w ,
v ,
z2y + u ,
t .

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 ,
x ,
y2 + w ,
y2 + v ,
z2y + zy2 + y2x + u ,
t .

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

The images of the generators of the cohomology of G under the map induced by the automorphism are
z ,
y ,
x ,
zy + w ,
z2 + v ,
z3 + z2y + u ,
t .

Automorphism #6

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
y + x ,
y ,
z + y ,
w ,
v ,
z2y + zv + yv + xv + u ,
z2v + y2v + v2 + t .

CHECKS

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

THE COMPUTATION IS COMPLETE