GROUP OF ORDER 32 # 22

# GROUP # 22

The MAGMA library number for G is 17

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

The center of G is abelian of type [ 8 ]
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, 8, 32 ]
The order of the Frattini subgroup is 8
The exponent of G is 16
G has a unique maximal elementary abelian subgroup which is normal and has order 4. Its centralizer has order 16

### The cohomology ring of G is the quotient of a polynomial ring in the variables [ z, y, x, w ] in degrees [ 1, 1, 3, 4 ] by the ideal generated by the relations [ z^2, z*y^2, z*x, x^2 ]

The Hilbert series for the cohomology ring is 1/(t^4 - 2*t^3 + 2*t^2 - 2*t + 1)
Its denominator factors as ( t - 1 )^2 ( t^2 + 1 )^1

The Krull dimension of the cohomology ring is 2
The longest regular sequence consists of the generators [ w ]
A homogeneous set of parameters is the set [ w, y^2 ] of degrees [ 4, 2 ]

### The hypercohomolgy spectral sequence has E2 term:

ROW (0) [ z, y ] [ z*y ] [ x ] [ y*x ]
ROW (1) [ z ] [ z*y ]
The spectral sequence satisfies Poincaré duality

# Restriction to the Maximal Elementary Abelian Subgroup

Generated by [ G.3, G.5 ]

# Restrictions to Maximal Subgroups

## The essential cohomology of G is generated as an ideal by the elements [ z*y ]

The annihilator of the Essential Cohomology has dimension 1 The essential cohomology is a free module over the polynomial subring Q of the cohomology ring of G generated by [ w ] .
The essential cohomology is generated as a module over Q by the elements [] [ z*y ]

### CHECKS

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