Group cohomology of groups of order 27
This article gives specific information, namely, group cohomology, about a family of groups, namely: groups of order 27.
View group cohomology of group families | View group cohomology of groups of a particular order |View other specific information about groups of order 27
With the exception of the zeroth homology group and cohomology group, all homology and cohomology groups over all possible abelian groups are 3-groups.
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 table below lists the first few homology groups with coefficients in the integers. We use to denote the cyclic group of order .
We use 0 to denote the trivial group.
|Group||GAP ID 2nd part||Nilpotency class||(= abelianization)||(= Schur multiplier)|
|direct product of Z9 and Z3||2||1|
|prime-cube order group:U(3,3)||3||2||?||?||?|
|elementary abelian group:E27||5||1|
Second cohomology groups and extensions
Schur multiplier and Schur covering groups
The Schur multiplier is defined as second cohomology group for trivial group action, , and also as the second homology group . A corresponding Schur covering group of is a group that arises as a stem extension with base normal subgroup the Schur multiplier and the quotient group is .
|Group||GAP ID 2nd part||Nilpotency class||(= Schur multiplier)||Order of||Possibilities for Schur covering groups||Cohomology group information||Orders of Schur covering groups|
|cyclic group:Z27||1||1||trivial group||1||cyclic group:Z27||--||27|
|direct product of Z9 and Z3||2||1||cyclic group:Z3||3||SmallGroup(27,3), nontrivial semidirect product of Z9 and Z9, semidirect product of Z27 and Z3||second cohomology group for trivial group action of direct product of Z9 and Z3 on Z3||81|
|prime-cube order group:U(3,3)||3||2||elementary abelian group:E9||9||SmallGroup(243,3)||second cohomology group for trivial group action of U(3,3) on E9||243|
|elementary abelian group:E27||5||1||elementary abelian group:E27||27||lots of them||second cohomology group for trivial group action of E27 on E27||729|