GROUP OF ORDER 64 #78

GROUP #78

The MAGMA library number for G is 213

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.6, G.2^2 = G.5, G.3^2 = G.6, G.4^2 = G.5 * G.6, G.4^G.1 = G.4 * G.5, G.4^G.2 = G.4 * G.6, G.4^G.3 = G.4 * 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, 1 ] .
The orders of the terms of the upper central series are [ 1, 8, 64 ] .
The order of the Frattini subgroup is 4.
The exponent of G is 4.
G has 5 conjugacy classes of maximal elementary abelian p-subgroups. The orders of the maximal elementary abelian subgroups are [ 8, 8, 8, 8, 8 ] .
The orders of the centralizers of the maximal elementary abelian subgroups are [ 32, 16, 16, 16, 16 ] .
The orders of the normalizers of the maximal elementary abelian subgroups are [ 64, 64, 64, 64, 64 ] .

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, 1, 1, 2, 4 ] , by the ideal generated by the relations
z2 + x2 + yw + xw + w2 ,
y2 + zw + w2 ,
x2w + zw2 + yw2 + xw2 + w3 .

The Hilbert series for the cohomology ring is
-t2 -t -1 / t5 -3t4+ 4t3 -4t2+ 3t -1.
Its numerator factors as ( t2+t+1 ) .
Its denominator factors as ( t-1 )3 ( t2+1 ) .

The Krull dimension of the cohomology ring is 3.
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators
v , u , zy + zx + yx + zw + yw + xw .

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E #1
Generated by [ G.5 * G.6, G.1 * G.3 * G.6, G.5 ]

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 ,
y2 ,
z2y2 + y4 + z2x2 + x4
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.5 * G.6, G.5, 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 ,
z ,
z2 + zy + y2 ,
z2y2 + y4 + z2x2 + x4
in the cohomology of E.

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

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

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 ,
0 ,
z ,
zy + y2 ,
z2y2 + y4 + z2x2 + x4
in the cohomology of E.

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

Subgroup E #4
Generated by [ G.5 * G.6, G.5, G.2 * G.3 * G.4 * G.5 * 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 ,
z ,
z2 + zy + y2 ,
z2y2 + y4 + z2x2 + x4
in the cohomology of E.

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

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

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 ,
0 ,
z ,
zy + y2 ,
z2y2 + y4 + z2x2 + x4
in the cohomology of E.

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

The nilradical of the cohomology of G is generated by
zy , zw + yw + w2 , zx + yx + x2 .

It is nilpotent of degree 3.

Restrictions to Maximal Subgroups

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

The Group H is Isomorphic to the Group of Order 32 Number39 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1, G.4, G.1 * G.2 ] .

The images of the generators of the cohomology of G restricted to H are
z + x ,
x ,
0 ,
y ,
z2 + yx + x2 + w ,
z4 + z2w + y2w + w2 + zv + 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 #2 Generated by [ G.1, G.4, G.5, G.6, G.2 * G.3 ] .

The Group H 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 Generated by [ G.1, G.2 * G.3 * G.4 * G.6, G.4 * G.6 ]

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

The images of the generators of the cohomology of G restricted to H are
y ,
x ,
x ,
z + x ,
x2 + w + v ,
y2w + x2w + v2
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.2 * G.4 * G.6, G.3, G.5, G.6, G.1 * G.2 ] .

The Group H is Isomorphic to the Group of Order 32 Number34 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.2 * G.4 * G.6, G.3, G.1 * G.2 ] .

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

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

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

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

The images of the generators of the cohomology of G restricted to H are
y ,
z + y + x + w ,
z + x + w ,
y + x + w ,
zy + zx + x2 + zw + yw + xw + w2 ,
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 #5 Generated by [ G.1, G.2 * G.4 * G.6, G.3, G.5, G.6 ] .

The Group H 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 Generated by [ G.1, G.1 * G.2 * G.4 * G.5 * G.6, G.3 ] .

The images of the generators of the cohomology of G restricted to H are
y + x ,
y ,
z ,
y ,
y2 + w + v ,
y3x + w2
in the cohomology of H.

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

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

The group H is abelian of type [ 4, 4, 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, 2, 2 ] , by the ideal of relations
z2 , y2 .

The images of the generators of the cohomology of G restricted to H are
z + x ,
y ,
x ,
0 ,
w + v ,
x2v + v2
in the cohomology of H.

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

Maximal Subgroup H #7 Generated by [ G.3 * G.4 * G.6, G.1, G.5, G.6, G.2 * G.3 ] .

The Group H 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 Generated by [ G.3 * G.4 * G.6, G.1, G.2 * G.4 ]

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

The images of the generators of the cohomology of G restricted to H are
y ,
x ,
z ,
z + x ,
y2 + w + v ,
y2w + x2w + v2
in the cohomology of H.

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

Maximal Subgroup H #8 Generated by [ G.3 * G.4 * G.6, G.5, G.1 * G.3, G.6, G.2 * 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.2 * G.4 * G.6, G.2 * G.4, G.2 * G.3 * G.6, G.1 * G.2 * 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
y ,
z + y + x + w ,
z + y ,
y + x + w ,
zy + y2 + zx + zw + yw + xw ,
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 + w .

Maximal Subgroup H #9 Generated by [ G.2, G.3, G.5, G.6, G.1 * G.4 ] .

The Group H 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 Generated by [ G.1 * G.2 * G.4 * G.5 * G.6, G.1 * G.3 * G.4, G.2 * G.5 * G.6 ]

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

The images of the generators of the cohomology of G restricted to H are
z + x ,
z + y ,
x ,
z + x ,
zx + x2 + w + v ,
y2w + x2w + w2
in the cohomology of H.

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

Maximal Subgroup H #10 Generated by [ G.1, G.4, G.3, G.5, G.6 ] .

The Group H 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 Generated by [ G.1, G.4, G.3 ] .

The images of the generators of the cohomology of G restricted to H are
x ,
0 ,
z ,
y ,
y2 + w + v ,
y3x + w2
in the cohomology of H.

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

Maximal Subgroup H #11 Generated by [ G.3 * G.4 * G.6, G.1, G.2, G.5, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number39 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4 * $.5, $.2^2 = $.5, $.3^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.3 * G.4 * G.6, G.1, G.1 * G.2 ] .

The images of the generators of the cohomology of G restricted to H are
z + x ,
x ,
y ,
y ,
w ,
z4 + z2w + y2w + w2 + zv + u
in the cohomology of H.

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

Maximal Subgroup H #12 Generated by [ G.3 * G.4 * G.6, G.2, G.5, G.1 * G.3, G.6 ] .

The Group H 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 Generated by [ G.3 * G.4 * G.6, G.2, G.1 * G.2 * G.3 ] .

The images of the generators of the cohomology of G restricted to H are
z ,
z + x ,
z + y ,
y ,
y2 + x2 + w + v ,
zy3 + z2w + zyw + z2v + zyv + v2
in the cohomology of H.

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

Maximal Subgroup H #13 Generated by [ G.2, G.4, G.5, G.1 * G.3, G.6 ] .

The Group H 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 Generated by [ G.2, G.4, G.1 * G.2 * G.3 ] .

The images of the generators of the cohomology of G restricted to H are
z ,
z + x ,
z ,
y ,
x2 + w + v ,
zy3 + z2w + zyw + z2v + zyv + v2
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 #14 Generated by [ G.2, G.4, G.3, G.5, G.6 ] .

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

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

The images of the generators of the cohomology of G restricted to H are
0 ,
y + x ,
z ,
z + x ,
y2 + zx + w + v ,
y2w + x2w + w2
in the cohomology of H.

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

Maximal Subgroup H #15 Generated by [ G.4, G.3, G.5, G.6, G.1 * G.2 ] .

The Group H is Isomorphic to the Group of Order 32 Number35 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.2^2 = $.4 * $.5, $.3^2 = $.4 * $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.5 Generated by [ G.1 * G.2 * G.3 * G.4 * G.5, G.4, G.3 ] .

The images of the generators of the cohomology of G restricted to H are
y ,
y ,
z + y ,
y + x ,
zy + zx + x2 + w ,
zyx2 + zxw + v
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 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5

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

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

The images of the generators of the cohomology of Q inflated to G are
y + w ,
z ,
z + x + w ,
x ,
z4 + z2w2 + u
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
zy + y2 + zx + zw + yw + xw .

Maximal Quotient Group Q #2.

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

The Group Q 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

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

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

The images of the generators of the cohomology of Q inflated to G are
y ,
x ,
z + y + x ,
z + w ,
z2 + x2 + yw + w2 + v
in the cohomology of G.

The kernel of the inflation to G of the cohomology of Q is generated by
z2 + zw + yw + xw + w2 ,
y2w + x2w + w3 .

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 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5

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

The generators of G have images [ pcy.1 * pcy.2 * pcy.3 * pcy.5, pcy.1 * pcy.3 * pcy.5, pcy.3, 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 + x + w ,
w ,
z + y + x + w ,
z ,
z2w2 + z2v + y2v + w2v + v2 + u
in the cohomology of G.

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

Action of Automorphisms

The groups of outer automorphisms of G has order 256, and is generated by 8 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 ,
z2 + y2 + w2 + 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 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 #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 ,
v ,
u .

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 ,
y2 + yw + v ,
u .

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 + w ,
w ,
w2 + v ,
u .

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 + w ,
y + w ,
x + w ,
w ,
w2 + v ,
u .

Automorphism #8

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 ,
z + x ,
x ,
w ,
v ,
x2w2 + z2v + y2v + w2v + v2 + u .

CHECKS

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

THE COMPUTATION IS COMPLETE