Its denominator factors as ( t-1 )3 ( t2+1 ) .
The Krull dimension of the cohomology ring is 3.
The longest regular sequence consists of the generators y2 , v .
A homogeneous set of parameters is the set y2 , v , z2 of degrees [ 2, 4, 2 ] .
The hypercohomolgy spectral sequence has E2
term:
ROW
(1)
: []
[ x ]
[ x2, yx ]
[ yx2 ]
ROW (0) : [1] [ x, y, z ] [ zy, x2, yx ] [ w, yx2 ] [ yw, zw ] [ zyw ]
Restrictions to Maximal Elementary Abelian
Subgroups
Subgroup E #1
Generated by
[ G.2 * G.3, G.5 ]
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
,
z
,
z
,
0
,
z2y2
+ y4
in the cohomology of E.
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
,
w
.
Subgroup E #2
Generated by
[ G.2 * G.4 * G.5, 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
,
x
,
0
,
z2y
+ zy2
+ z2x
+ x3
,
z2y2
+ y4
+ z3x
+ z2yx
+ zy2x
+ zyx2
+ y2x2
+ x4
in the cohomology of E.
in the cohomology of E.
The kernel of the restriction to E (Minimal Prime) of the
cohomology of G is generated by
x
.
The nilradical of the cohomology of G is
generated by
yx
+ x2
It is nilpotent of degree 2.
Restrictions to Maximal Subgroups
Maximal Subgroup H #1
Generated by
[ G.3, G.1, G.5, G.4 ]
.
The Group H is Isomorphic to the
Group of Order 16 Number13
GrpPC of order 16 = 2^4
PC-Relations:
$.1^2 = $.3 * $.4,
$.3^2 = $.4,
$.2^$.1 = $.2 * $.3,
$.3^$.2 = $.3 * $.4
Generated by
[ G.1, G.1 * G.3 * G.4 ]
of type
Semidihedral(16)
.
The images of the generators of the cohomology of G
restricted to H are
z
+ y
,
0
,
z
,
x
,
w
in the cohomology of H.
The kernel of the restriction to H of the
cohomology of G is generated by
y
.
Maximal Subgroup H #2
Generated by
[ G.3, G.5, G.2, G.4 ]
.
The Group H is Isomorphic to the
Group of Order 16 Number8
GrpPC of order 16 = 2^4
PC-Relations:
$.1^2 = $.4,
$.3^2 = $.4,
$.3^$.2 = $.3 * $.4
Generated by
[ G.4, G.2 * G.3 * G.5, G.2 ]
of type
AlmostExtraSpecial(16)
.
The images of the generators of the cohomology of G
restricted to H are
0
,
z
+ y
,
y
,
z2x
,
yx3
+ x4
+ w
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
Generated by
[ G.1 * G.3 * G.4, G.5, G.2, G.4 ]
.
The Group H is Isomorphic to the
Group of Order 16 Number11
GrpPC of order 16 = 2^4
PC-Relations:
$.1^2 = $.3,
$.3^2 = $.4,
$.2^$.1 = $.2 * $.4
Generated by
[ G.1 * G.3 * G.4, G.2 * G.4 ]
.
The images of the generators of the cohomology of G
restricted to H are
z
,
y
,
z
,
y3
+ x
,
y4
+ 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
Generated by
[ G.1, G.5, G.2, G.4 ]
.
The Group H is Isomorphic to the
Group of Order 16 Number6
GrpPC of order 16 = 2^4
PC-Relations:
$.3^2 = $.4,
$.3^$.1 = $.3 * $.4,
$.3^$.2 = $.3 * $.4
Generated by
[ G.1 * G.2 * G.5, G.1 * G.4 * G.5, G.2 ]
of type
Cyclic(2) x Dihedral(8)
.
The images of the generators of the cohomology of G
restricted to H are
z
+ y
,
z
+ x
,
0
,
zy2
+ zw
+ yw
,
z3y
+ z2y2
+ zy3
+ zyw
+ w2
in the cohomology of H.
The kernel of the restriction to H of the
cohomology of G is generated by
x
.
Maximal Subgroup H #5
Generated by
[ G.3, G.1 * G.2 * G.5, G.5, G.4 ]
.
The Group H is Isomorphic to the
Group of Order 16 Number13
GrpPC of order 16 = 2^4
PC-Relations:
$.1^2 = $.3 * $.4,
$.3^2 = $.4,
$.2^$.1 = $.2 * $.3,
$.3^$.2 = $.3 * $.4
Generated by
[ G.1 * G.2 * G.5, G.1 * G.2 * G.3 * G.4 * G.5 ]
of type
Semidihedral(16)
.
The images of the generators of the cohomology of G
restricted to H are
z
+ y
,
z
+ y
,
z
,
x
,
w
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
Generated by
[ G.1 * G.2 * G.5, G.1 * G.3 * G.4, G.5, G.4 ]
.
The Group H is Isomorphic to the
Group of Order 16 Number12
GrpPC of order 16 = 2^4
PC-Relations:
$.2^2 = $.3 * $.4,
$.3^2 = $.4,
$.2^$.1 = $.2 * $.3,
$.3^$.1 = $.3 * $.4
Generated by
[ G.1 * G.2 * G.5, G.1 * G.3 * G.4 ]
of type
Dihedral(16)
.
The images of the generators of the cohomology of G
restricted to H are
z
+ y
,
z
,
y
,
zx
+ yx
,
x2
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 #7
Generated by
[ G.1, G.2 * G.3, G.5, G.4 ]
.
The Group H is Isomorphic to the
Group of Order 16 Number12
GrpPC of order 16 = 2^4
PC-Relations:
$.2^2 = $.3 * $.4,
$.3^2 = $.4,
$.2^$.1 = $.2 * $.3,
$.3^$.1 = $.3 * $.4
Generated by
[ G.1 * G.2 * G.3, G.2 * G.3 ]
of type
Dihedral(16)
.
The images of the generators of the cohomology of G
restricted to H are
y
,
z
+ y
,
z
+ y
,
yx
,
x2
in the cohomology of H.
The kernel of the restriction to H of the
cohomology of G is generated by
y
+ x
.
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.5
.
The Group Q is Isomorphic to the
Group of Order 16 Number6
GrpPC of order 16 = 2^4
PC-Relations:
$.3^2 = $.4,
$.3^$.1 = $.3 * $.4,
$.3^$.2 = $.3 * $.4
of type
Cyclic(2) x Dihedral(8)
.
The generators of G have images
[ pcy.2 * pcy.3, pcy.1 * pcy.2, pcy.1 * pcy.4 ]
in the quotient group.
The images of the generators of the cohomology of Q
inflated to G are
z
+ y
,
y
+ x
,
x
,
zy
+ y2
+ x2
in the cohomology of G.
The kernel of the inflation to G of the
cohomology of Q is generated by
zy
+ yx
+ w
,
xw
.
The depth-essential cohomology of G
The depth-essential cohomology of G is the intersection
of the restrictions to the centralizers of the maximal elementary abelian
p-subgroups of rank d+1 where d is the depth of the
cohomology ring. For this group the depth is 2.
There is only one conjugacy class of subgroups which
are centralizers of elementary abelian subgroups of rank 3. It is represented by
the subgroup generated by
[ G.1 * G.2, G.1 * G.4 * G.5, G.1 * G.4 ]
.
The depth-essential cohomology of G is
generated as an ideal by
x
.
The annihilator of the depth-essential cohomology has dimension
2
.
The depth-essential cohomology is a free module over
the polynomial subring Q of the cohomology ring of G generated by
y2
,
v
.
The depth-essential cohomology is generated as a module
over Q by the elements
[ x ]
[ x2, yx ]
[ yx2 ]
Action of Automorphisms
The groups of outer automorphisms of G has order 4, and is generated by 2
elements.
Automorphism #1
G.1 * G.5
G.2 * G.5
G.3 * G.5
G.4
G.5
.
The images of the generators of the cohomology of G
under the map induced by the automorphism are
z
,
y
,
x
,
z2y
+ zy2
+ w
,
v
.
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
G.1 * G.2 * G.4
G.2 * G.5
G.3
G.4 * G.5
G.5
.
The images of the generators of the cohomology of G
under the map induced by the automorphism are
z
,
z
+ y
,
x
,
z2y
+ zy2
+ w
,
v
.
CHECKS
paramflag =
true
qregflagflag =
true
cmflag =
false
essflag =
true
centflag =
true
THE COMPUTATION IS COMPLETE
Restrictions to Maximal Subgroups
Maximal Subgroup H #1
Generated by
[ G.3, G.1, G.5, G.4 ]
.
The Group H is Isomorphic to the Group of Order 16 Number13 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3 * $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.2 = $.3 * $.4 Generated by [ G.1, G.1 * G.3 * G.4 ]
of type
Semidihedral(16)
.
The images of the generators of the cohomology of G
restricted to H are
z
+ y
,
0
,
z
,
x
,
w
in the cohomology of H.
in the cohomology of H.
The kernel of the restriction to H of the
cohomology of G is generated by
y
.
Maximal Subgroup H #2
Generated by
[ G.3, G.5, G.2, G.4 ]
.
The Group H is Isomorphic to the Group of Order 16 Number8 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.3^2 = $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.4, G.2 * G.3 * G.5, G.2 ]
of type
AlmostExtraSpecial(16)
.
The images of the generators of the cohomology of G
restricted to H are
0
,
z
+ y
,
y
,
z2x
,
yx3
+ x4
+ w
in the cohomology of H.
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
Generated by
[ G.1 * G.3 * G.4, G.5, G.2, G.4 ]
.
The Group H is Isomorphic to the Group of Order 16 Number11 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3, $.3^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.3 * G.4, G.2 * G.4 ] .
The images of the generators of the cohomology of G
restricted to H are
z
,
y
,
z
,
y3
+ x
,
y4
+ w
in the cohomology of H.
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
Generated by
[ G.1, G.5, G.2, G.4 ]
.
The Group H is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.2 * G.5, G.1 * G.4 * G.5, G.2 ]
of type
Cyclic(2) x Dihedral(8)
.
The images of the generators of the cohomology of G
restricted to H are
z
+ y
,
z
+ x
,
0
,
zy2
+ zw
+ yw
,
z3y
+ z2y2
+ zy3
+ zyw
+ w2
in the cohomology of H.
in the cohomology of H.
The kernel of the restriction to H of the
cohomology of G is generated by
x
.
Maximal Subgroup H #5
Generated by
[ G.3, G.1 * G.2 * G.5, G.5, G.4 ]
.
The Group H is Isomorphic to the Group of Order 16 Number13 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.3 * $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.2 = $.3 * $.4 Generated by [ G.1 * G.2 * G.5, G.1 * G.2 * G.3 * G.4 * G.5 ]
of type
Semidihedral(16)
.
The images of the generators of the cohomology of G
restricted to H are
z
+ y
,
z
+ y
,
z
,
x
,
w
in the cohomology of H.
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
Generated by
[ G.1 * G.2 * G.5, G.1 * G.3 * G.4, G.5, G.4 ]
.
The Group H is Isomorphic to the Group of Order 16 Number12 GrpPC of order 16 = 2^4 PC-Relations: $.2^2 = $.3 * $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.1 = $.3 * $.4 Generated by [ G.1 * G.2 * G.5, G.1 * G.3 * G.4 ]
of type
Dihedral(16)
.
The images of the generators of the cohomology of G
restricted to H are
z
+ y
,
z
,
y
,
zx
+ yx
,
x2
in the cohomology of H.
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 #7
Generated by
[ G.1, G.2 * G.3, G.5, G.4 ]
.
The Group H is Isomorphic to the Group of Order 16 Number12 GrpPC of order 16 = 2^4 PC-Relations: $.2^2 = $.3 * $.4, $.3^2 = $.4, $.2^$.1 = $.2 * $.3, $.3^$.1 = $.3 * $.4 Generated by [ G.1 * G.2 * G.3, G.2 * G.3 ]
of type
Dihedral(16)
.
The images of the generators of the cohomology of G
restricted to H are
y
,
z
+ y
,
z
+ y
,
yx
,
x2
in the cohomology of H.
in the cohomology of H.
The kernel of the restriction to H of the
cohomology of G is generated by
y
+ x
.
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.5 .
The Group Q is Isomorphic to the Group of Order 16 Number6 GrpPC of order 16 = 2^4 PC-Relations: $.3^2 = $.4, $.3^$.1 = $.3 * $.4, $.3^$.2 = $.3 * $.4
of type
Cyclic(2) x Dihedral(8)
.
The generators of G have images [ pcy.2 * pcy.3, pcy.1 * pcy.2, pcy.1 * pcy.4 ] in the quotient group.
The images of the generators of the cohomology of Q
inflated to G are
z
+ y
,
y
+ x
,
x
,
zy
+ y2
+ x2
in the cohomology of G.
in the cohomology of G.
The kernel of the inflation to G of the
cohomology of Q is generated by
zy
+ yx
+ w
,
xw
.
The depth-essential cohomology of G
The depth-essential cohomology of G is the intersection of the restrictions to the centralizers of the maximal elementary abelian p-subgroups of rank d+1 where d is the depth of the cohomology ring. For this group the depth is 2.
There is only one conjugacy class of subgroups which
are centralizers of elementary abelian subgroups of rank 3. It is represented by
the subgroup generated by
[ G.1 * G.2, G.1 * G.4 * G.5, G.1 * G.4 ]
.
The depth-essential cohomology of G is
generated as an ideal by
x
.
The annihilator of the depth-essential cohomology has dimension
2
.
The depth-essential cohomology is a free module over
the polynomial subring Q of the cohomology ring of G generated by
y2
,
v
.
The depth-essential cohomology is generated as a module
over Q by the elements
[ x ]
[ x2, yx ]
[ yx2 ]
Action of Automorphisms
The groups of outer automorphisms of G has order 4, and is generated by 2 elements.
Automorphism #1
G.1 * G.5
G.2 * G.5
G.3 * G.5
G.4
G.5 .
The images of the generators of the cohomology of G
under the map induced by the automorphism are
z
,
y
,
x
,
z2y
+ zy2
+ w
,
v
.
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 Automorphism #2
G.1 * G.2 * G.4
G.2 * G.5
G.3
G.4 * G.5
G.5 .
The images of the generators of the cohomology of G
under the map induced by the automorphism are
z
,
z
+ y
,
x
,
z2y
+ zy2
+ w
,
v
.
CHECKS
paramflag =
true
qregflagflag =
true
cmflag =
false
essflag =
true
centflag =
true
THE COMPUTATION IS COMPLETE
CHECKS
paramflag =
true
qregflagflag = true
cmflag = false
essflag = true
centflag = true