GROUP OF ORDER 64 #84

GROUP #84

The MAGMA library number for G is 67

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

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

The cohomology ring of G is the quotient of a polynomial ring in the variables
[ z, y, x, w, v, u, t, s ] in degrees [ 1, 1, 1, 2, 2, 2, 2, 2 ] , by the ideal generated by the relations
z2 ,
zy ,
y2 + yx ,
zv ,
yv + xv + zt ,
yt ,
yxw + v2 ,
zxu + vt ,
yxu + x2u + zxt + t2 .

The Hilbert series for the cohomology ring is
t2+ t+ 1 / t6 -2t5 -t4+ 4t3 -t2 -2t+ 1.
Its numerator factors as ( t2+t+1 ) .
Its denominator factors as ( t-1 )4 ( t+1 )2 .

The Krull dimension of the cohomology ring is 4.
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators
x2 , w , u , s .

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E #1
Generated by [ G.2 * G.4 * G.5 * G.6, G.2 * G.6, G.5, G.2 ]

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

The images of the generators of the cohomology of G restricted to E are
0 ,
0 ,
z ,
z2 + y2 + w2 ,
0 ,
z2 + x2 ,
z2 + zx ,
zw + w2
in the cohomology of E.

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

Subgroup E #2
Generated by [ G.2 * G.6, G.5, G.2 * G.3 * G.4 * G.6, G.2 ]

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

The images of the generators of the cohomology of G restricted to E are
0 ,
z ,
z ,
z2 + y2 + w2 ,
z2 + zy + zw ,
zx + x2 ,
0 ,
z2 + zw + w2
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 ,
t .

The nilradical of the cohomology of G is generated by
z

It is nilpotent of degree 2.

Restrictions to Maximal Subgroups

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

The Group H is Isomorphic to the Group of Order 32 Number11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.2 * G.3 * G.4 * G.5 * G.6, G.4 * G.5 * G.6, G.2 ]

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

The images of the generators of the cohomology of G restricted to H are
z ,
z ,
z + y ,
x2 + v ,
zx ,
y2 + v ,
y2 + w ,
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 #2 Generated by [ G.1, G.6, G.5, G.3 * G.4 * G.6, G.2 ] .

The Group H is Isomorphic to the Group of Order 32 Number11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^$.1 = $.2 * $.5 Generated by [ G.6, G.1 * G.3 * G.4 * G.6, G.3 * G.4 ]

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

The images of the generators of the cohomology of G restricted to H are
z ,
z + y ,
z + y ,
v ,
w ,
w + v + u ,
zx ,
y2 + zx + yx + 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.6, G.5, G.1 * G.3, G.3 * G.4 * G.6, G.2 ] .

The Group H is Isomorphic to the Group of Order 32 Number11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.2 * G.4 * G.5 * G.6, G.2 * G.3 * G.4, G.5 ]

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

The images of the generators of the cohomology of G restricted to H are
z ,
y ,
z + y ,
y2 + v ,
y2 + zx + w ,
zx + yx + x2 + u ,
zx ,
y2 + u
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 #4 Generated by [ G.6, G.5, G.1 * G.4, G.3, G.2 ] .

The Group H is Isomorphic to the Group of Order 32 Number18 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.4, G.3 ] .

The images of the generators of the cohomology of G restricted to H are
y ,
z ,
y ,
t ,
v ,
u ,
x ,
x + w + u
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 #5 Generated by [ G.1, G.6, G.5, G.3, G.2 ] .

The Group H is Isomorphic to the Group of Order 32 Number18 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1, G.3 ] .

The images of the generators of the cohomology of G restricted to H are
y ,
z ,
0 ,
t ,
v ,
u ,
x ,
w
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.4, G.1, G.6, G.5, G.2 ] .

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

The cohomology ring of H is a the quotient of a polynomial ring in the variables
[ z, y, x, w, v ] , in degrees [ 1, 1, 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 + v ,
zw ,
zw + w2 ,
zx + yw ,
yx + 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 #7 Generated by [ G.4, G.6, G.5, G.3, G.2 ] .

The Group H is Isomorphic to the Group of Order 32 Number8 GrpPC of order 32 = 2^5 PC-Relations: $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.3 * G.4 * G.5 * G.6, G.2 * G.4 * G.5, G.4 * G.5, G.2 * G.4 ]

of type Abelian(2,2) x Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are
0 ,
x ,
z + y + x + w ,
z2 + w2 ,
zx + xw ,
z2 + y2 + xw ,
z2 + y2 + zw + yw ,
x2 + v
in the cohomology of H.

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

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.2 * G.5 * G.6 .

The Group Q is Isomorphic to the Group of Order 32 Number36 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 .

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

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

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

Maximal Quotient Group Q #2.

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.4, pcy.2, pcy.1 ] in the quotient group.

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

The kernel of the inflation to G of the cohomology of Q is generated by
y2 + yx .

Maximal Quotient Group Q #3.

The kernel of the quotient is generated by G.2 * G.5 .

The Group Q is Isomorphic to the Group of Order 32 Number36 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 .

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

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

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

Maximal Quotient Group Q #4.

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

The Group Q is Isomorphic to the Group of Order 32 Number14 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4, $.2^$.1 = $.2 * $.5, $.3^$.2 = $.3 * $.5

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

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

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

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

Maximal Quotient Group Q #5.

The kernel of the quotient is generated by G.2 .

The Group Q is Isomorphic to the Group of Order 32 Number33 GrpPC of order 32 = 2^5 PC-Relations: $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 .

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

The images of the generators of the cohomology of Q inflated to G are
y + x ,
z ,
y ,
y2 + s ,
t ,
u
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
y2 .

Maximal Quotient Group Q #6.

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

The Group Q is Isomorphic to the Group of Order 32 Number14 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.4, $.2^$.1 = $.2 * $.5, $.3^$.2 = $.3 * $.5

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

The generators of G have images [ pcy.1 * pcy.2 * pcy.4 * pcy.5, pcy.1 * pcy.2 * pcy.3 * pcy.4, pcy.1 * pcy.3 * 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 ,
y + x ,
z + y ,
w ,
y2 + w + s
in the cohomology of G.

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

Maximal Quotient Group Q #7.

The kernel of the quotient is generated by G.2 * G.6 .

The Group Q is Isomorphic to the Group of Order 32 Number38 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 .

The generators of G have images [ pcy.2 * pcy.3 * pcy.5, pcy.2, pcy.1 * 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 ,
u ,
y3 + zw + yw + xv + xt + zs + ys ,
zx2 + zw + xt + zs ,
x4 + x2w + w2 + s2
in the cohomology of G.

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

Action of Automorphisms

The groups of outer automorphisms of G has order 128, 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

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 ,
u ,
t ,
s .

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 ,
t ,
zx + s .

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 ,
y2 + w ,
y2 + v ,
u ,
zx + t ,
zx + s .

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 ,
w ,
zx + v ,
y2 + zx + x2 + u ,
y2 + x2 + t ,
zx + s .

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 ,
w ,
v ,
u ,
zx + t ,
zx + s .

Automorphism #6

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 + x2 + w ,
zx + v ,
y2 + zx + x2 + u ,
y2 + x2 + t ,
zx + s .

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

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

CHECKS

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

THE COMPUTATION IS COMPLETE