GROUP OF ORDER 64 #14

GROUP #14

Abelian(2,2) x AlmostExtraSpecial(16)

The MAGMA library number for G is 263

GrpPC : G of order 64 = 2^6 PC-Relations: G.1^2 = G.6, G.3^2 = G.6, G.4^2 = G.6, G.5^2 = G.6, G.3^G.1 = G.3 * G.6, G.3^G.2 = G.3 * G.6, G.5^G.1 = G.5 * 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, 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 2.
The exponent of G is 4.
G has 3 conjugacy classes of maximal elementary abelian p-subgroups. The orders of the maximal elementary abelian subgroups are [ 16, 16, 16 ] .
The orders of the centralizers of the maximal elementary abelian subgroups are [ 32, 32, 32 ] .
The orders of the normalizers of the maximal elementary abelian subgroups are [ 64, 64, 64 ] .

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, 1, 1, 4 ] , by the ideal generated by the relations

z² + zx + yx + x² + w² + zv + yv + xv + v² ,

y²x + x³ + xw² + y²v + x²v + w²v + xv² + v³ .

The Hilbert series for the cohomology ring is

t²+ t+ 1 / t⁶ -4t⁵+ 7t⁴ -8t³+ 7t² -4t+ 1.
Its numerator factors as ( t²+t+1 ) .
Its denominator factors as ( t-1 )⁴ ( t²+1 ) .

The Krull dimension of the cohomology ring is 4.
The cohomology ring is Cohen-Macaulay and the longest regular sequence consists of the generators

z² , y² , w² , u .

Restrictions to Maximal Elementary Abelian Subgroups

Subgroup E #1
Generated by [ G.1 * G.2 * G.4, G.1 * G.2 * G.3 * G.5 * G.6, G.6, G.1 * 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

z + y + w ,

z + w ,

z + y ,

w ,

z + y ,

z²y² + y⁴ + z²x² + x⁴ + z²w² + w⁴
in the cohomology of E.

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

z + w + v ,

x + v .

Subgroup E #2
Generated by [ G.1 * G.2 * G.5 * G.6, G.1 * G.2 * G.4, G.6, G.1 * 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

z + y + w ,

z + w ,

y ,

w ,

z + y ,

z²y² + y⁴ + z²x² + x⁴ + z²w² + w⁴
in the cohomology of E.

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

z + w + v ,

y + x + w + v .

Subgroup E #3
Generated by [ G.2 * G.5 * G.6, G.1 * G.2 * G.4, G.6, G.1 * 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

y + w ,

z + w ,

y ,

w ,

z + y ,

z²y² + y⁴ + z²x² + x⁴ + z²w² + w⁴
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 ,

y + x + w + v .

The nilradical of the cohomology of G is generated by
yx + x² + xw + yv + wv + v² , zy + zx + zw + yw + xw + w² + zv + yv + xv + v² .

It is nilpotent of degree 2.

Restrictions to Maximal Subgroups

Maximal Subgroup H #1 Generated by [ G.2, G.3, G.1, G.6, G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2, G.1 * G.2 * G.6, G.1 * G.5, G.1 * G.2 * G.3 * G.5 ]

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

The images of the generators of the cohomology of G restricted to H are

z + y + w ,

z + x + w ,

w ,

0 ,

y + w ,

z²y² + y⁴ + z²xw + x²w² + v
in the cohomology of H.

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

w .

Maximal Subgroup H #2 Generated by [ G.2 * G.4, G.3 * G.4, G.6, G.1 * G.4, G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.2 * G.6, G.3 * G.4 * G.5, G.2 * G.4 * G.5, G.1 * G.4 * G.6 ]

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

The images of the generators of the cohomology of G restricted to H are

z + w ,

z + y ,

x ,

y + x + w ,

y + x ,

z²y² + y⁴ + z²xw + x²w² + v
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 #3 Generated by [ G.2, G.1, G.4 * G.5, G.3 * G.4, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2, G.1 * G.2 * G.6, G.1 * G.2 * G.3 * G.4 * G.6, G.1 * G.2 * G.3 * G.5 ]

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

The images of the generators of the cohomology of G restricted to H are

z + y + x ,

z + y + x + w ,

y + x ,

y ,

x ,

z²y² + y⁴ + z²xw + x²w² + v
in the cohomology of H.

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

x + w + v .

Maximal Subgroup H #4 Generated by [ G.2, G.4 * G.5, G.3 * G.4, G.6, G.1 * G.4 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2, G.1 * G.4, G.1 * G.5 * G.6, G.2 * G.3 * G.5 ]

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

The images of the generators of the cohomology of G restricted to H are

y + x ,

z + w ,

z ,

x ,

z + y ,

z²y² + y⁴ + z²xw + x²w² + v
in the cohomology of H.

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

z + x + w + v .

Maximal Subgroup H #5 Generated by [ G.3, G.6, G.1 * G.2, G.5, G.4 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.3 * G.5, G.1 * G.2 * G.3, G.3 * G.4, G.4 ]

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

The images of the generators of the cohomology of G restricted to H are

w ,

w ,

y + x + w ,

z + x ,

y ,

z²y² + y⁴ + 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 #6 Generated by [ G.2, G.3, G.4 * G.5, G.6, G.1 * G.4 ] .

The Group H is Isomorphic to the Group of Order 32 Number8 GrpPC of order 32 = 2^5 PC-Relations: $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.2, G.3 * G.4 * G.5, G.4 * G.5, G.1 * G.4 ]

of type Abelian(2,2) x Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are

z ,

w ,

y ,

z + y + x ,

y + x ,

zyx² + y²x² + zx²w + z²w² + y²w² + v²
in the cohomology of H.

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

z + w + v .

Maximal Subgroup H #7 Generated by [ G.2, G.3 * G.5 * G.6, G.6, G.1 * G.3, G.4 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2 * G.6, G.3 * G.4 * G.5 * G.6, G.1 * G.5 * G.6, G.2 * G.3 * G.5 * G.6 ]

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

The images of the generators of the cohomology of G restricted to H are

y ,

z + w ,

z + x ,

x ,

z + y + x ,

z²y² + y⁴ + z²xw + x²w² + v
in the cohomology of H.

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

z + x + v .

Maximal Subgroup H #8 Generated by [ G.3 * G.5 * G.6, G.2 * G.3, G.6, G.1 * G.3, G.4 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.2 * G.6, G.2 * G.3, G.1 * G.3 * G.4, G.1 * G.2 * G.3 * G.4 * G.5 * G.6 ]

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

The images of the generators of the cohomology of G restricted to H are

z + y + x ,

z + y + w ,

y + x + w ,

y + x ,

y ,

z²y² + y⁴ + z²xw + x²w² + v
in the cohomology of H.

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

z + y + x + v .

Maximal Subgroup H #9 Generated by [ G.3, G.1, G.2 * G.4, G.4 * G.5, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.2 * G.3 * G.4, G.3 * G.4 * G.5 * G.6, G.1 * G.2 * G.3 * G.5 * G.6, G.2 * G.3 * G.5 * G.6 ]

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

The images of the generators of the cohomology of G restricted to H are

y + x ,

z + y + x ,

z + y + x + w ,

y + w ,

z + x + w ,

z²y² + y⁴ + v
in the cohomology of H.

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

y + w + v .

Maximal Subgroup H #10 Generated by [ G.3, G.1, G.2 * G.5, G.6, G.4 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2 * G.5 * G.6, G.3 * G.4, G.1 * G.2 * G.4 * G.5 * G.6, G.2 * G.3 * G.5 * G.6 ]

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

The images of the generators of the cohomology of G restricted to H are

y ,

z + y + x ,

z + w ,

y + w ,

z + y + x ,

z²y² + y⁴ + z²xw + x²w² + v
in the cohomology of H.

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

y + v .

Maximal Subgroup H #11 Generated by [ G.2, G.3, G.1, G.6, G.4 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2, G.1 * G.2 * G.3 * G.4, G.1 * G.2 * G.6, G.1 * G.4 ]

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

The images of the generators of the cohomology of G restricted to H are

z + y + x ,

z + y + w ,

y ,

y + x ,

0 ,

z²y² + y⁴ + z²xw + x²w² + v
in the cohomology of H.

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

v .

Maximal Subgroup H #12 Generated by [ G.3, G.2 * G.4, G.4 * G.5, G.6, G.1 * G.4 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.2 * G.6, G.3 * G.4 * G.5 * G.6, G.1 * G.5, G.1 * G.4 * G.6 ]

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

The images of the generators of the cohomology of G restricted to H are

z + y + x ,

z ,

w ,

x + w ,

y + w ,

z²y² + y⁴ + v
in the cohomology of H.

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

z + y + w + v .

Maximal Subgroup H #13 Generated by [ G.2, G.3, G.6, G.1 * G.4, G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2, G.3, G.1 * G.4, G.2 * G.3 * G.5 ]

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

The images of the generators of the cohomology of G restricted to H are

w ,

z + x ,

z + y ,

w ,

z ,

z²y² + y⁴ + z²xw + x²w² + v
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.1, G.6, G.5, G.4 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.4 * G.5, G.3 * G.4, G.5, G.4 ]

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

The images of the generators of the cohomology of G restricted to H are

w ,

0 ,

x ,

z + x + w ,

y + w ,

z²y² + y⁴ + z²xw + x²w² + 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 #15 Generated by [ G.3, G.1, G.2 * G.4, G.6, G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number9 GrpPC of order 32 = 2^5 PC-Relations: $.2^2 = $.5, $.3^2 = $.5, $.4^2 = $.5, $.4^$.2 = $.4 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.3 * G.5 * G.6, G.1, G.1 * G.2 * G.4 * G.6, G.5 ]

of type Abelian(2,2) x Quaternion(8) .

The images of the generators of the cohomology of G restricted to H are

z + y ,

z ,

x ,

z ,

x + w ,

y⁴ + v
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 #16 Generated by [ G.3, G.2 * G.5, G.1 * G.5, G.6, G.4 ] .

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

z + x + w ,

y ,

x ,

w ,

z + y + x + w ,

y²x² + x⁴ + x²w² + y²v + w²v + v²
in the cohomology of H.

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

z + y + v .

Maximal Subgroup H #17 Generated by [ G.5, G.4, G.2 * G.3, G.1 * G.3, G.6 ] .

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

z + x + w ,

y ,

z + y + x + w ,

w ,

x ,

y²v + w²v + v²
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 #18 Generated by [ G.3, G.2 * G.4, G.6, G.1 * G.4, G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2 * G.4 * G.5 * G.6, G.1 * G.4, G.1 * G.3 * G.4 * G.5, G.1 * G.2 * G.3 * G.5 ]

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

The images of the generators of the cohomology of G restricted to H are

z + x + w ,

y + x ,

z + x ,

z + y + w ,

z + y + x ,

z²y² + y⁴ + v
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 #19 Generated by [ G.2, G.3, G.6, G.5, G.4 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.5 * G.6, G.2 * G.5, G.3 * G.4, G.2 * G.3 * G.5 ]

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

The images of the generators of the cohomology of G restricted to H are

0 ,

z + x ,

z + w ,

w ,

z + y + x ,

z²y² + y⁴ + 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 #20 Generated by [ G.2, G.3 * G.5 * G.6, G.1, G.6, G.4 ] .

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

z + x + w ,

y ,

x ,

w ,

x ,

y²x² + x⁴ + x²w² + y²v + w²v + v²
in the cohomology of H.

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

x + v .

Maximal Subgroup H #21 Generated by [ G.2, G.3, G.1 * G.5, G.6, G.4 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.4 * G.5 * G.6, G.3 * G.4, G.2 * G.3 * G.4 * G.6, G.4 ]

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

The images of the generators of the cohomology of G restricted to H are

w ,

y ,

y + x ,

z + y + x + w ,

w ,

z²y² + y⁴ + v
in the cohomology of H.

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

z + v .

Maximal Subgroup H #22 Generated by [ G.1, G.2 * G.3, G.6, G.5, G.4 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2 * G.3 * G.6, G.4 * G.5, G.1 * G.5 * G.6, G.2 * G.3 * G.5 ]

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

The images of the generators of the cohomology of G restricted to H are

y ,

z + x ,

z + x ,

w ,

z + y + w ,

z²y² + y⁴ + v
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 #23 Generated by [ G.2, G.1, G.3 * G.4, G.6, G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2, G.1 * G.2 * G.3 * G.4 * G.6, G.3 * G.4 * G.5, G.1 * G.3 * G.4 * G.5 * G.6 ]

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

The images of the generators of the cohomology of G restricted to H are

z + y ,

y + w ,

z + y + x ,

z + y + x ,

z + x ,

z²y² + y⁴ + z²xw + x²w² + v
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 #24 Generated by [ G.2, G.3 * G.4, G.6, G.1 * G.4, G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number8 GrpPC of order 32 = 2^5 PC-Relations: $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.2, G.1 * G.4 * G.5 * G.6, G.2 * G.5 * G.6, G.3 * G.4 * G.6 ]

of type Abelian(2,2) x Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are

z ,

x + w ,

y ,

z + y ,

z + w ,

z²y² + zyx² + zx²w + z²w² + v²
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 #25 Generated by [ G.2, G.3, G.1, G.4 * G.5, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2, G.3 * G.4 * G.5 * G.6, G.3 * G.6, G.1 * G.3 * G.4 * G.5 ]

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

The images of the generators of the cohomology of G restricted to H are

z ,

w ,

z + y + x ,

z + x ,

z + x ,

z²y² + y⁴ + v
in the cohomology of H.

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

w + v .

Maximal Subgroup H #26 Generated by [ G.1, G.2 * G.4, G.3 * G.4, G.6, G.5 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.3 * G.4 * G.5, G.2 * G.4 * G.5, G.1 * G.2 * G.3 * G.5, G.2 * G.3 * G.5 ]

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

The images of the generators of the cohomology of G restricted to H are

x ,

z + y + x ,

z + x + w ,

y + w ,

z + y + x + w ,

z²y² + y⁴ + v
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 #27 Generated by [ G.1, G.2 * G.4, G.4 * G.5, G.3 * G.4, G.6 ] .

The Group H is Isomorphic to the Group of Order 32 Number8 GrpPC of order 32 = 2^5 PC-Relations: $.3^$.1 = $.3 * $.5, $.3^$.2 = $.3 * $.5, $.4^$.3 = $.4 * $.5 Generated by [ G.4 * G.5, G.2 * G.5, G.1 * G.2 * G.5, G.2 * G.3 ]

of type Abelian(2,2) x Dihedral(8) .

The images of the generators of the cohomology of G restricted to H are

z ,

z + y + x ,

y ,

w ,

z + x + w ,

z²y² + y²x² + y²w² + v²
in the cohomology of H.

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

y + x + w + v .

Maximal Subgroup H #28 Generated by [ G.2 * G.4, G.4 * G.5, G.3 * G.4, G.6, G.1 * G.4 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.2 * G.3 * G.4 * G.6, G.1 * G.4, G.1 * G.3 * G.4 * G.5, G.1 * G.2 * G.3 * G.5 ]

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

The images of the generators of the cohomology of G restricted to H are

z + y + x + w ,

y + x ,

z + y + x ,

z + y + w ,

z + x ,

z²y² + y⁴ + v
in the cohomology of H.

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

z + y + x + w + v .

Maximal Subgroup H #29 Generated by [ G.2, G.6, G.1 * G.3, G.5, G.4 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.1 * G.2 * G.3 * G.4, G.4 * G.5, G.1 * G.3 * G.4 * G.6, G.4 ]

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

The images of the generators of the cohomology of G restricted to H are

y + x ,

y ,

y + x ,

z + y + x + w ,

w ,

z²y² + y⁴ + z²xw + x²w² + 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 #30 Generated by [ G.2, G.1, G.6, G.5, G.4 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2, G.1 * G.2 * G.4 * G.5, G.1 * G.2 * G.6, G.1 * G.4 ]

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

The images of the generators of the cohomology of G restricted to H are

z + y + x ,

z + y + w ,

0 ,

y + x ,

y ,

z²y² + y⁴ + z²xw + x²w² + v
in the cohomology of H.

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

x .

Maximal Subgroup H #31 Generated by [ G.3 * G.5 * G.6, G.1, G.2 * G.3, G.6, G.4 ] .

The Group H is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5 Generated by [ G.2 * G.4 * G.5 * G.6, G.3 * G.4 * G.5 * G.6, G.1 * G.4 * G.6, G.4 ]

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

The images of the generators of the cohomology of G restricted to H are

x ,

y ,

w ,

z + y + x + w ,

y + w ,

z²y² + y⁴ + v
in the cohomology of H.

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

y + x + v .

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.6 .

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

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

The images of the generators of the cohomology of Q inflated to G are

v ,

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 + y² + zx + x² + zw + xw + zv + xv + v² ,

y²x + x³ + y²w + x²w + y²v + x²v + xv² + wv² + v³ .

Maximal Quotient Group Q #2.

The kernel of the quotient is generated by G.1 * G.2 * G.4 .

The Group Q is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5

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

The generators of G have images [ pcy.1 * pcy.3 * pcy.5, pcy.1 * pcy.3 * pcy.4, pcy.1 * pcy.2 * pcy.4 * pcy.5, pcy.1 * pcy.2 * pcy.3 * pcy.4 * pcy.5, pcy.2 * pcy.5 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are

y + x + w + v ,

z + y + x ,

y + w + v ,

y + x + w ,

z²x² + y²w² + x²w² + z²v² + x²v² + w²v² + v⁴ + u
in the cohomology of G.

The kernel of the inflation is zero.

Maximal Quotient Group Q #3.

The kernel of the quotient is generated by G.1 * G.2 * G.4 * G.6 .

The Group Q is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5

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

The generators of G have images [ pcy.1 * pcy.3 * pcy.5, pcy.1 * pcy.3 * pcy.4, pcy.1 * pcy.2 * pcy.3 * pcy.5, pcy.1 * pcy.2 * pcy.5, pcy.2 * pcy.3 * pcy.4 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are

y + x + w + v ,

z + y + x ,

z + x + w + v ,

z + w ,

y³x + y²x² + zx³ + yxw² + x²w² + y³v + z²xv + yx²v + yw²v + xw²v + yxv² + zv³ + u
in the cohomology of G.

The kernel of the inflation is zero.

Maximal Quotient Group Q #4.

The kernel of the quotient is generated by G.1 * G.3 * G.5 * G.6 .

The Group Q is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5

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

The generators of G have images [ pcy.1 * pcy.3 * pcy.5, pcy.1 * pcy.3 * pcy.4, pcy.1 * pcy.2 * pcy.4 * pcy.5, pcy.1 * pcy.2 * pcy.3 * pcy.4 * pcy.5, pcy.2 * pcy.3 * pcy.4 * pcy.5 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are

y + x + w + v ,

z + y + x ,

z + y + w + v ,

z + y + x + w ,

z²x² + y²w² + x²w² + z²v² + x²v² + w²v² + v⁴ + u
in the cohomology of G.

The kernel of the inflation is zero.

Maximal Quotient Group Q #5.

The kernel of the quotient is generated by G.1 * G.3 * G.5 .

The Group Q is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5

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

The generators of G have images [ pcy.1 * pcy.3 * pcy.5, pcy.1 * pcy.3 * pcy.4, pcy.1 * pcy.2 * pcy.3 * pcy.5, pcy.1 * pcy.2 * pcy.5, pcy.2 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are

y + x + w + v ,

z + y + x ,

x + w + v ,

w ,

y³x + y²x² + zx³ + yxw² + x²w² + y³v + z²xv + yx²v + yw²v + xw²v + yxv² + zv³ + u
in the cohomology of G.

The kernel of the inflation is zero.

Maximal Quotient Group Q #6.

The kernel of the quotient is generated by G.2 * G.3 * G.4 * G.5 .

The Group Q is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5

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

The generators of G have images [ pcy.1 * pcy.4 * pcy.5, pcy.1, pcy.1 * pcy.2 * pcy.3 * pcy.5, pcy.1 * pcy.2 * pcy.3 * pcy.4 * pcy.5, pcy.2 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are

y + x + w + v ,

z + y + x ,

y + x ,

y + v ,

y³x + y²x² + zx³ + yxw² + x²w² + y³v + z²xv + yx²v + yw²v + xw²v + yxv² + zv³ + u
in the cohomology of G.

The kernel of the inflation is zero.

Maximal Quotient Group Q #7.

The kernel of the quotient is generated by G.2 * G.3 * G.4 * G.5 * G.6 .

The Group Q is Isomorphic to the Group of Order 32 Number10 GrpPC of order 32 = 2^5 PC-Relations: $.1^2 = $.5, $.2^2 = $.5, $.3^$.2 = $.3 * $.5, $.4^$.2 = $.4 * $.5

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

The generators of G have images [ pcy.1 * pcy.3 * pcy.5, pcy.1, pcy.1 * pcy.2 * pcy.3 * pcy.5, pcy.1 * pcy.2, pcy.2 * pcy.3 * pcy.4 ] in the quotient group.

The images of the generators of the cohomology of Q inflated to G are

y + x + w + v ,

z + y + x ,

z + x + v ,

z ,

y³x + y²x² + zx³ + y²w² + yxw² + x²w² + y³v + z²xv + yx²v + yw²v + xw²v + yxv² + zv³ + u
in the cohomology of G.

The kernel of the inflation is zero.

Action of Automorphisms

The groups of outer automorphisms of G has order 18432, and is generated by 13 elements.

Automorphism #1