GROUP OF ORDER 32 #43

GROUP #43

Extraspecial Dihedral(8)*Quaternion(8)

The MAGMA library number for G is 50

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

The center of G is abelian of type [ 2 ] .
The orders of the terms of the lower central series are [ 32, 2, 1 ] .
The orders of the terms of the upper central series are [ 1, 2, 32 ] .
The order of the Frattini subgroup is 2.
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 [ 4, 4, 4, 4, 4 ] .
The orders of the centralizers of the maximal elementary abelian subgroups are [ 16, 16, 16, 16, 16 ] .
The orders of the normalizers of the maximal elementary abelian subgroups are [ 32, 32, 32, 32, 32 ] .

The cohomology ring of G is the quotient of a polynomial ring in the variables
[ z, y, x, w, v ] in degrees [ 1, 1, 1, 1, 8 ] , by the ideal generated by the relations
zy + zx + x2 + zw + xw + w2 ,
zx2 + yx2 + x3 + zxw + yxw + x2w + zw2 + yw2 + xw2 + w3 ,
z2w3 + y2w3 + zw4 + yw4 .

The Hilbert series for the cohomology ring is
t6+ 2t5+ 3t4+ 3t3+ 3t2+ 2t+ 1 / t8 -2t7+ 2t6 -2t5+ 2t4 -2t3+ 2t2 -2t+ 1.
Its numerator factors as ( t2+t+1 ) ( t4+t3+t2+t+1 ) .
Its denominator factors as ( t-1 )2 ( t2+1 ) ( t4+1 ) .

The Krull dimension of the cohomology ring is 2.
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators
v , z2 + y2 + yx + yw + xw .

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E #1
Generated by [ G.5, G.1 * G.4 * 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
z ,
0 ,
0 ,
z ,
z4y4 + y8
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 #2
Generated by [ G.5, G.1 * G.2 * G.3 * G.4 ]

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
z ,
z ,
z ,
z ,
z4y4 + y8
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 + w ,
x + w .

Subgroup E #3
Generated by [ G.5, G.1 * 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
z ,
0 ,
0 ,
0 ,
z4y4 + y8
in the cohomology of E.

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

Subgroup E #4
Generated by [ G.1 * G.3 * G.5, 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
z ,
0 ,
z ,
0 ,
z4y4 + y8
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 #5
Generated by [ G.2 * G.5, 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 ,
0 ,
0 ,
z4y4 + y8
in the cohomology of E.

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

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

It is nilpotent of degree 4.

Restrictions to Maximal Subgroups

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

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.2 * G.3 * G.4 * G.5, G.2 * G.5, G.1 * G.2 * G.3 * G.5 ]

of type AlmostExtraSpecial(16) .

The images of the generators of the cohomology of G restricted to H are
x ,
z + y + x ,
z + x ,
z ,
w2
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 #2 Generated by [ G.1, G.3 * G.4, G.5, G.2 ] .

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.1, G.3 * G.4, G.2 * G.3 * G.4 ]

of type AlmostExtraSpecial(16) .

The images of the generators of the cohomology of G restricted to H are
y ,
x ,
z + x ,
z + x ,
w2
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 #3 Generated by [ G.3, G.1, G.5, 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.3, G.1 * G.3 * G.4 * G.5, G.1 * G.5 ]

of type AlmostExtraSpecial(16) .

The images of the generators of the cohomology of G restricted to H are
z + y ,
0 ,
z + x ,
z ,
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 #4 Generated by [ G.3, G.5, G.4, G.1 * G.2 ] .

The Group H is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.2 * G.3 * G.4 * G.5, G.3, G.3 * G.4 ]

of type Cyclic(2) x Quaternion(8) .

The images of the generators of the cohomology of G restricted to H are
x ,
x ,
z + y + x ,
z + x ,
x4w + w2
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 #5 Generated by [ G.1, G.5, G.4, G.2 ] .

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.2 * G.5, G.2 * G.4 * G.5, G.1 * G.2 * G.4 * G.5 ]

of type AlmostExtraSpecial(16) .

The images of the generators of the cohomology of G restricted to H are
x ,
z + y + x ,
0 ,
z + x ,
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 #6 Generated by [ G.3, G.1, G.5, G.2 ] .

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.2 * G.3 * G.5, G.2 * G.5, G.1 * G.2 * G.3 * G.5 ]

of type AlmostExtraSpecial(16) .

The images of the generators of the cohomology of G restricted to H are
x ,
z + y + x ,
z + x ,
0 ,
w2
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.2 * G.3, G.1, G.5, 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.1, G.1 * G.2 * G.3, G.4 ]

of type AlmostExtraSpecial(16) .

The images of the generators of the cohomology of G restricted to H are
z + y ,
z ,
z ,
x ,
w2
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 #8 Generated by [ G.3, G.1, G.2 * G.4, G.5 ] .

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.3, G.1 * G.5, G.1 * G.2 * G.4 * G.5 ]

of type AlmostExtraSpecial(16) .

The images of the generators of the cohomology of G restricted to H are
z + y ,
z ,
x ,
z ,
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 #9 Generated by [ G.3, G.2 * G.4, G.5, G.1 * G.2 ] .

The Group H is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.3, G.2 * G.4, G.1 * G.4 * G.5 ]

of type Cyclic(2) x Quaternion(8) .

The images of the generators of the cohomology of G restricted to H are
x ,
z ,
y ,
z + x ,
x4w + w2
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 #10 Generated by [ G.2 * G.3, G.5, G.4, G.1 * G.2 ] .

The Group H is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.1 * G.3 * G.5, G.4, G.1 * G.2 ]

of type Cyclic(2) x Quaternion(8) .

The images of the generators of the cohomology of G restricted to H are
z + x ,
z ,
x ,
y ,
x4w + w2
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 #11 Generated by [ G.2 * G.3, G.1, G.2 * G.4, G.5 ] .

The Group H is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.2 * G.3, G.1, G.2 * G.4 ]

of type Cyclic(2) x Quaternion(8) .

The images of the generators of the cohomology of G restricted to H are
x ,
z + y ,
y ,
z ,
x4w + w2
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 #12 Generated by [ G.2 * G.3, G.2 * G.4, G.5, G.1 * G.2 ] .

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.2 * G.3, G.1 * G.4 * G.5, G.1 * G.2 ]

of type AlmostExtraSpecial(16) .

The images of the generators of the cohomology of G restricted to H are
z + y ,
z + x ,
x ,
y ,
w2
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 #13 Generated by [ G.3, G.1 * G.4, G.5, G.2 ] .

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.3, G.2 * G.3, G.1 * G.4 ]

of type AlmostExtraSpecial(16) .

The images of the generators of the cohomology of G restricted to H are
y ,
x ,
z + x ,
y ,
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 #14 Generated by [ G.3, G.5, G.4, G.2 ] .

The Group H is Isomorphic to the Group of Order 16 Number7 GrpPC of order 16 = 2^4 PC-Relations: $.1^2 = $.4, $.2^2 = $.4, $.2^$.1 = $.2 * $.4 Generated by [ G.3, G.2, G.4 ]

of type Cyclic(2) x Quaternion(8) .

The images of the generators of the cohomology of G restricted to H are
0 ,
x ,
y ,
z ,
x4w + 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.5, G.1 * G.3, G.4, G.2 ] .

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.2 * G.4, G.1 * G.3, G.4 ]

of type AlmostExtraSpecial(16) .

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

The kernel of the restriction to H of the cohomology of G is generated by
z + 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 abelian of type [ 2, 2, 2, 2 ] .

The cohomology ring of Q is a polynomial ring with variables z , y , x , w in degrees [ 1, 1, 1, 1 ]

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

The kernel of the inflation to G of the cohomology of Q is generated by
z2 + zy + y2 + zw + yw + xw ,
y3 + y2w + yxw + x2w ,
x4w + x3w2 .

Action of Automorphisms

The groups of outer automorphisms of G has order 120, and is generated by 3 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 + w ,
w ,
x ,
y ,
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

The images of the generators of the cohomology of G under the map induced by the automorphism are
z ,
y + w ,
x + w ,
w ,
v .

Automorphism #3

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
z ,
y + x ,
x + w ,
z + w ,
v .

CHECKS

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

THE COMPUTATION IS COMPLETE