GROUP OF ORDER 64 #47

GROUP #47

Cyclic(2) x Group(32)#27

The MAGMA library number for G is 95

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.4, G.4^2 = G.6, G.5^2 = G.6, G.2^G.1 = G.2 * G.5 * G.6, G.3^G.1 = G.3 * G.5 * G.6, G.3^G.2 = G.3 * G.6, G.4^G.2 = G.4 * G.6, G.4^G.3 = G.4 * G.6, G.5^G.2 = G.5 * G.6, G.5^G.3 = G.5 * 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, 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 one conjugate class of maximal elementary abelian subgroups. Any element of the class has order 16. Its centralizer has order 16 and its normalizer has order 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, 1, 2, 2, 2 ] , by the ideal generated by the relations
z2 ,
zy + zx ,
zx2 + zw + zv ,
y2x2 + y2w + x2w + w2 + v2 .

The Hilbert series for the cohomology ring is
1 / t5 -3t4+ 2t3+ 2t2 -3t+ 1.
Its denominator factors as ( t-1 )4 ( t+1 ) .

The Krull dimension of the cohomology ring is 4.
The longest regular sequence consists of the generators y2 , w , u .
A homogeneous set of parameters is the set y2 , w , u , x2 of degrees [ 2, 2, 2, 2 ] .

The hypercohomolgy spectral sequence has E2 term:

ROW (1) : [ z ] [ zx ]
ROW (0) : [ z, y, x ] [ yx, v, zx ] [ xv, yv ] [ yxv ]

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.2 * G.3 * G.4 * G.6, G.2 * G.3 * G.4, G.3 * G.4, G.2 * G.3 * G.5 ] 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 ,
y + x + w ,
z + y + x + w ,
z2 + x2 + w2 ,
zy + y2 ,
z2 + x2 + zw
in the cohomology of E.

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

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

Restrictions to Maximal Subgroups

Maximal Subgroup H #1

The Group H is Isomorphic to the Group of Order 32 Number12 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.2^$.1 = $.2 * $.5 Generated by [ G.2 * G.3 * G.4 * G.6, G.1 * G.2 * G.4, G.1 * G.3 * G.5 * G.6 ]

of type Cyclic(2) x Group(16)#10 .

The images of the generators of the cohomology of G restricted to H are
z + y ,
z + x ,
y + x ,
y2 + x2 + v ,
y2 + v ,
zx + yx + w
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 #2

The Group H is Isomorphic to the Group of Order 32 Number27 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.2 * G.5 * G.6, G.3 ]

The images of the generators of the cohomology of G restricted to H are
z ,
z ,
y ,
w ,
x + w ,
x + 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 #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
z ,
y ,
y ,
y2 + x2 ,
zx + x2 ,
zx + x2 + w
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 #4

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

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

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

Maximal Subgroup H #5

The Group H is Isomorphic to the Group of Order 32 Number8 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5 Generated by [ G.2, G.2 * G.3 * G.6, G.4 * G.5 * G.6, 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 ,
y + x ,
y ,
y2 + zx + yx + w2 ,
xw + w2 ,
y2 + zx + w2 + v
in the cohomology of H

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

Maximal Subgroup H #6

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

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

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

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

Inflations from Maximal Quotient Groups

Maximal Quotient Group Q #1

The Group Q is Isomorphic to the Group of Order 32 Number11 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.4^$.1 = $.4 * $.5

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

The generators of G have images [ (1, 23, 2, 24)(3, 21, 4, 22)(5, 19, 6, 20)(7, 17, 8, 18)(9, 31, 10, 32)(11, 29, 12, 30)(13, 27, 14, 28)(15, 25, 16, 26), (1, 14)(2, 13)(3, 16)(4, 15)(5, 10)(6, 9)(7, 12)(8, 11)(17, 32)(18, 31)(19, 30)(20, 29)(21, 28)(22, 27)(23, 26)(24, 25), (1, 12)(2, 11)(3, 10)(4, 9)(5, 16)(6, 15)(7, 14)(8, 13)(17, 26)(18, 25)(19, 28)(20, 27)(21, 30)(22, 29)(23, 32)(24, 31) ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are
z ,
y ,
y + x ,
y2 + x2 + w ,
y2 + yx + x2 + w + v ,
v
in the cohomology of G

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

Maximal Quotient Group Q #2

The Group Q is Isomorphic to the Group of Order 32 Number23 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.4, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5

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

The generators of G have images [ (1, 26, 4, 27, 2, 25, 3, 28)(5, 30, 8, 31, 6, 29, 7, 32)(9, 18, 11, 20, 10, 17, 12, 19)(13, 22, 15, 24, 14, 21, 16, 23), (1, 22)(2, 21)(3, 24)(4, 23)(5, 18)(6, 17)(7, 20)(8, 19)(9, 30)(10, 29)(11, 32)(12, 31)(13, 26)(14, 25)(15, 28)(16, 27), (1, 19)(2, 20)(3, 18)(4, 17)(5, 23)(6, 24)(7, 22)(8, 21)(9, 28)(10, 27)(11, 25)(12, 26)(13, 32)(14, 31)(15, 29)(16, 30) ] 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 ,
yx + w + v + u
in the cohomology of G

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

Maximal Quotient Group Q #3

The Group Q is Isomorphic to the Group of Order 32 Number24 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5

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

The generators of G have images [ (1, 28, 4, 26, 2, 27, 3, 25)(5, 32, 8, 30, 6, 31, 7, 29)(9, 19, 11, 18, 10, 20, 12, 17)(13, 23, 15, 22, 14, 24, 16, 21), (1, 9)(2, 10)(3, 11)(4, 12)(5, 13)(6, 14)(7, 15)(8, 16)(17, 27)(18, 28)(19, 26)(20, 25)(21, 31)(22, 32)(23, 30)(24, 29), (1, 16)(2, 15)(3, 13)(4, 14)(5, 12)(6, 11)(7, 9)(8, 10)(17, 30)(18, 29)(19, 32)(20, 31)(21, 26)(22, 25)(23, 28)(24, 27) ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are
z ,
z + y + x ,
z + x ,
zyx + y2x + yx2 + zw + yw + xw + zu + yu + xu ,
y2x2 + w2 + u2
in the cohomology of G

The kernel of the inflation to G of the cohomology of Q is generated by
z2 .

Maximal Quotient Group Q #4

The Group Q is Isomorphic to the Group of Order 32 Number27 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5

The generators of G have images [ (1, 26, 8, 31, 2, 25, 7, 32)(3, 28, 6, 29, 4, 27, 5, 30)(9, 18, 15, 24, 10, 17, 16, 23)(11, 20, 13, 22, 12, 19, 14, 21), (1, 11)(2, 12)(3, 9)(4, 10)(5, 15)(6, 16)(7, 13)(8, 14)(17, 31)(18, 32)(19, 29)(20, 30)(21, 28)(22, 27)(23, 26)(24, 25), (1, 15)(2, 16)(3, 13)(4, 14)(5, 12)(6, 11)(7, 10)(8, 9)(17, 27)(18, 28)(19, 25)(20, 26)(21, 31)(22, 32)(23, 29)(24, 30) ] in the quotient group.

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

The kernel of the inflation is zero.

Maximal Quotient Group Q #5

The Group Q is Isomorphic to the Group of Order 32 Number27 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5

The generators of G have images [ (1, 26, 8, 31, 2, 25, 7, 32)(3, 28, 6, 29, 4, 27, 5, 30)(9, 18, 15, 24, 10, 17, 16, 23)(11, 20, 13, 22, 12, 19, 14, 21), (1, 11)(2, 12)(3, 9)(4, 10)(5, 15)(6, 16)(7, 13)(8, 14)(17, 31)(18, 32)(19, 29)(20, 30)(21, 28)(22, 27)(23, 26)(24, 25), (1, 16)(2, 15)(3, 14)(4, 13)(5, 11)(6, 12)(7, 9)(8, 10)(17, 28)(18, 27)(19, 26)(20, 25)(21, 32)(22, 31)(23, 30)(24, 29) ] in the quotient group.

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

The kernel of the inflation is zero.

Maximal Quotient Group Q #6

The Group Q is Isomorphic to the Group of Order 32 Number27 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5

The generators of G have images [ (1, 26, 8, 31, 2, 25, 7, 32)(3, 28, 6, 29, 4, 27, 5, 30)(9, 18, 15, 24, 10, 17, 16, 23)(11, 20, 13, 22, 12, 19, 14, 21), (1, 11)(2, 12)(3, 9)(4, 10)(5, 15)(6, 16)(7, 13)(8, 14)(17, 31)(18, 32)(19, 29)(20, 30)(21, 28)(22, 27)(23, 26)(24, 25), (1, 13)(2, 14)(3, 15)(4, 16)(5, 10)(6, 9)(7, 12)(8, 11)(17, 25)(18, 26)(19, 27)(20, 28)(21, 29)(22, 30)(23, 31)(24, 32) ] in the quotient group.

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

The kernel of the inflation is zero.

Maximal Quotient Group Q #7

The Group Q is Isomorphic to the Group of Order 32 Number27 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.4^2 = $.5, $.2^$.1 = $.2 * $.4 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5

The generators of G have images [ (1, 26, 8, 31, 2, 25, 7, 32)(3, 28, 6, 29, 4, 27, 5, 30)(9, 18, 15, 24, 10, 17, 16, 23)(11, 20, 13, 22, 12, 19, 14, 21), (1, 11)(2, 12)(3, 9)(4, 10)(5, 15)(6, 16)(7, 13)(8, 14)(17, 31)(18, 32)(19, 29)(20, 30)(21, 28)(22, 27)(23, 26)(24, 25), (1, 14)(2, 13)(3, 16)(4, 15)(5, 9)(6, 10)(7, 11)(8, 12)(17, 26)(18, 25)(19, 28)(20, 27)(21, 30)(22, 29)(23, 32)(24, 31) ] in the quotient group.

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

The kernel of the inflation is zero.

CHECKS

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

THE COMPUTATION IS COMPLETE