Lagrange theorem demonstrations in GAP with cosets and cardinalities?

Question

The goal of this question is to introduce basic group theory concepts with GAP: examples about alternating groups, cyclic groups and demonstrate the Lagrange theorem $[G:H]=\frac{\#G}{\# H}$.

Example.

I was surprised to find out that the number of elements in Klein four group, alternating group $A_4$ is 12, not 4,

which is equivalent to Elements(AlternatingGroup(4)) in GAP.

I would be extremely delighted to find GAP commands such as

• command to print cosets ( CosetTable(AlternatingGroup(4)) firing error) and

• command to find subgroups of A4.

which I want to use for more accessible demonstration for the Lagrange theorem in group theory. So

How to demonstrate Lagrange theorem with different introductory groups in GAP?