# Group cohomology of direct product of Z4 and Z2

From Groupprops

This article gives specific information, namely, group cohomology, about a particular group, namely: direct product of Z4 and Z2.

View group cohomology of particular groups | View other specific information about direct product of Z4 and Z2

## Homology groups for trivial group action

FACTS TO CHECK AGAINST(homology group for trivial group action):

First homology group: first homology group for trivial group action equals tensor product with abelianization

Second homology group: formula for second homology group for trivial group action in terms of Schur multiplier and abelianization|Hopf's formula for Schur multiplier

General: universal coefficients theorem for group homology|homology group for trivial group action commutes with direct product in second coordinate|Kunneth formula for group homology

### Over the integers

The first few homology groups are given below:

### Over an abelian group

Here, represents the quotient of by the subgroup , represents the quotient of by the subgroup , and .

## Cohomology groups for trivial group action

FACTS TO CHECK AGAINST(cohomology group for trivial group action):

First cohomology group: first cohomology group for trivial group action is naturally isomorphic to group of homomorphisms

Second cohomology group: formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization

In general: dual universal coefficients theorem for group cohomology relating cohomology with arbitrary coefficientsto homology with coefficients in the integers. |Cohomology group for trivial group action commutes with direct product in second coordinate | Kunneth formula for group cohomology

### Over the integers

The first few cohomology groups are given below (these are the same as the first few homology groups, shifted over by one):

0 |

### Over an abelian group

Below are the cohomology groups with coefficients in an abelian group :

Here, represents the quotient of by the subgroup , represents the quotient of by the subgroup , and .