Group cohomology of alternating group:A4

Over the integers
The general description is unclear. The group is not a finite group with periodic cohomology, but there is probably a description that involves a variable power of an abelian group direct summed with another abelian group.

The first few homology groups are given below:

Computation of integral homology
The homology groups for trivial group action with coefficients in $$\mathbb{Z}$$ can be computed in GAP using the GroupHomology function in the HAP package, which can be loaded by the command LoadPackage("hap"); if it is installed but not loaded. The function outputs the orders of cyclic groups for which the homology or cohomology group is the direct product of these (more technically, it outputs the elementary divisors for the homology or cohomology group that we are trying to compute).

Here are computations of the first few homology groups:

Computation of first homology group
gap> GroupHomology(AlternatingGroup(4),1); [ 3 ]

This says that the abelianization (which is also the first homology group for trivial group action on the integers) is cyclic group:Z3, which we already know from the fact that the derived subgroup is V4 in A4.

Computation of second homology group
gap> GroupHomology(AlternatingGroup(4),2); [ 2 ]

This says that the Schur multiplier (the second homology group for trivial group action on the integers) is cyclic group:Z2.

Computation of first few homology groups
gap> List([1,2,3,4,5,6,7,8],i->[i,GroupHomology(AlternatingGroup(4),i)]); [ [ 1, [ 3 ] ], [ 2, [ 2 ] ], [ 3, [ 6 ] ], [ 4, [ ] ], [ 5, [ 2, 6 ] ],   [ 6, [ 2 ] ], [ 7, [ 6 ] ], [ 8, [ 2, 2 ] ] ]

Here's a more comprehensive listing:

gap> List([1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24],i->[i,GroupHomology(AlternatingGroup(4),i)]); [ [ 1, [ 3 ] ], [ 2, [ 2 ] ], [ 3, [ 6 ] ], [ 4, [ ] ], [ 5, [ 2, 6 ] ],   [ 6, [ 2 ] ], [ 7, [ 6 ] ], [ 8, [ 2, 2 ] ], [ 9, [ 2, 6 ] ],   [ 10, [ 2 ] ], [ 11, [ 2, 2, 6 ] ], [ 12, [ 2, 2 ] ], [ 13, [ 2, 6 ] ],   [ 14, [ 2, 2, 2 ] ], [ 15, [ 2, 2, 6 ] ], [ 16, [ 2, 2 ] ],   [ 17, [ 2, 2, 2, 6 ] ], [ 18, [ 2, 2, 2 ] ], [ 19, [ 2, 2, 6 ] ],   [ 20, [ 2, 2, 2, 2 ] ], [ 21, [ 2, 2, 2, 6 ] ], [ 22, [ 2, 2, 2 ] ],   [ 23, [ 2, 2, 2, 2, 6 ] ], [ 24, [ 2, 2, 2, 2 ] ] ]