GROUP OF ORDER 64 #24

GROUP #24

Cyclic(2) x Group(32)#20

The MAGMA library number for G is 87

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

The center of G is abelian of type [ 2, 2, 4 ] .
The orders of the terms of the lower central series are [ 64, 2, 1 ] .
The orders of the terms of the upper central series are [ 1, 16, 64 ] .
The order of the Frattini subgroup is 8.
The exponent of G is 8.
G has a unique maximal elementary abelian subgroup which is normal and has order 16. Its centralizer 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 ,
zw ,
w2 .

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 x2 , v , u .
A homogeneous set of parameters is the set x2 , v , u , y2 of degrees [ 2, 2, 2, 2 ] .

The hypercohomolgy spectral sequence has E2 term:

ROW (1) : [ z ] [ zx ]
ROW (0) : [ z, y, x ] [ yx, w, zx ] [ xw, yw ] [ yxw ]

Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.5, G.4 * G.5 * G.6, G.6, G.3 * G.4 * G.6 ] 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 + x ,
0 ,
x2 + w2 ,
zy + y2 + zx + x2
in the cohomology of E.

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

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

Restrictions to Maximal Subgroups

Maximal Subgroup H #1

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
0 ,
y ,
w ,
zy ,
v ,
yx + x2
in the cohomology of H

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

Maximal Subgroup H #2

The Group H is Isomorphic to the Group of Order 32 Number20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1, G.2 * G.3 ]

The images of the generators of the cohomology of G restricted to H are
z ,
y ,
0 ,
y2 + x ,
v ,
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 #3

The Group H is Isomorphic to the Group of Order 32 Number20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.1 * G.4, G.2 * G.3 ]

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

The Group H is Isomorphic to the Group of Order 32 Number20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.2 * G.3 * G.4, G.1 * G.4 ]

The images of the generators of the cohomology of G restricted to H are
z ,
y ,
z + y ,
y2 + x ,
v ,
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 #5

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 ,
z ,
y + x ,
zy ,
x2 + w ,
y2
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 #6

The Group H is Isomorphic to the Group of Order 32 Number20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.5 Generated by [ G.2 * G.3 * G.4, G.1 ]

The images of the generators of the cohomology of G restricted to H are
z ,
y ,
y ,
y2 + x ,
v ,
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 #7

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 ,
0 ,
y + x ,
zy ,
x2 + w ,
zy + y2
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 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, 26, 4, 27)(2, 25, 3, 28)(5, 30, 8, 31)(6, 29, 7, 32)(9, 18, 12, 19)(10, 17, 11, 20)(13, 22, 16, 23)(14, 21, 15, 24), (1, 7)(2, 8)(3, 5)(4, 6)(9, 15)(10, 16)(11, 13)(12, 14)(17, 23)(18, 24)(19, 21)(20, 22)(25, 31)(26, 32)(27, 29)(28, 30), (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 + x ,
z + y ,
y2 ,
y2 + w ,
w + u
in the cohomology of G

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

Maximal Quotient Group Q #2

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

The cohomology ring of Q 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 Q inflated to G are
z ,
x ,
y ,
v
in the cohomology of G

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

Maximal Quotient Group Q #3

The Group Q is Isomorphic to the Group of Order 32 Number13 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3 * $.5, $.2^2 = $.3, $.3^2 = $.5, $.4^2 = $.5, $.2^$.1 = $.2 * $.5, $.4^$.1 = $.4 * $.5, $.4^$.2 = $.4 * $.5

of type Cyclic(2) x Group(16)#11 .

The generators of G have images [ (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), (1, 6)(2, 5)(3, 8)(4, 7)(9, 14)(10, 13)(11, 16)(12, 15)(17, 22)(18, 21)(19, 24)(20, 23)(25, 30)(26, 29)(27, 32)(28, 31), (1, 10)(2, 9)(3, 12)(4, 11)(5, 14)(6, 13)(7, 16)(8, 15)(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
y + x ,
z + y + x ,
x ,
y3 + zx2 + yw + zv + zu ,
y2x2 + x4 + y2v + v2 + u2
in the cohomology of G

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

Maximal Quotient Group Q #4

The Group Q is Isomorphic to the Group of Order 32 Number20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.5

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

The images of the generators of the cohomology of Q inflated to G are
z ,
y ,
y2 + w ,
u ,
v
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 Number20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.5

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

The images of the generators of the cohomology of Q inflated to G are
z ,
y ,
y2 + w ,
u ,
x2 + v
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 Number20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.5

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

The images of the generators of the cohomology of Q inflated to G are
z ,
y ,
y2 + zx + w ,
zx + yx + x2 + u ,
v
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 Number20 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.3, $.2^2 = $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.5

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

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