GROUP OF ORDER 64 #64

GROUP #64

The MAGMA library number for G is 16

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.2, G.2^2 = G.3, G.4^2 = G.5 * G.6, G.5^2 = G.6, G.4^G.1 = G.4 * G.5, G.5^G.1 = 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 16.
The exponent of G is 8.
G has a unique maximal elementary abelian subgroup which is central in G and has order 4.

The cohomology ring of G is the quotient of a polynomial ring in the variables
[ z, y, x, w ] in degrees [ 1, 1, 2, 2 ] , by the ideal generated by the relations
z2 ,
zy + y2 .

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
x , w .

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.6, G.3 ] 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 ,
y2 ,
z2
in the cohomology of E.

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

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 Number21 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.4, G.1 * G.4 * G.5 * G.6 ]

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

The group H is abelian of type [ 8, 4 ]

The cohomology ring of H is a the quotient of a polynomial ring in the variables
[ z, y, x, w ] , in degrees [ 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
0 ,
z ,
w ,
x
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

The Group H is Isomorphic to the Group of Order 32 Number21 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.5 * G.6, G.1 ]

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

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

The essential cohomology of G is generated as an ideal by
y2 .

The annihilator of the Essential Cohomology has dimension 2 The essential cohomology is a free module over the polynomial subring Q of the cohomology ring of G generated by x , w .
The essential cohomology is generated as a module over Q by the elements
[] [ y2 ]

Inflations from Maximal Quotient Groups

Maximal Quotient Group Q #1

The Group Q is Isomorphic to the Group of Order 32 Number29 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5

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

The images of the generators of the cohomology of Q inflated to G are
y ,
z + y ,
w ,
0
in the cohomology of G

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

Maximal Quotient Group Q #2

The Group Q is Isomorphic to the Group of Order 32 Number21 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.3, $.3^2 = $.4, $.2^$.1 = $.2 * $.5

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

The images of the generators of the cohomology of Q inflated to G are
y ,
z + y ,
0 ,
x
in the cohomology of G

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

Maximal Quotient Group Q #3

The Group Q is Isomorphic to the Group of Order 32 Number32 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3 * $.5, $.2^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.4, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5

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

The images of the generators of the cohomology of Q inflated to G are
y ,
z + y ,
0 ,
zx + zw ,
x2 + w2
in the cohomology of G

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

CHECKS

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

THE COMPUTATION IS COMPLETE