First agemo subgroup of direct product of Z4 and Z2

Definition
The group $$G$$ is a direct product of Z4 and Z2, which, for convenience, we describe by ordered pairs with the first members from the integers mod 4 (the first direct factor cyclic group:Z4) and the second member from the integers mod 2 (the second direct factor cyclic group:Z2). It has elements:

$$\! \{ (0,0), (1,0), (2,0), (3,0), (0,1), (1,1), (2,1), (3,1) \}$$

The subgroup $$H$$ is given as:

$$\! \{ (0,0), (2,0) \}$$

Cosets
$$H$$ is a normal subgroup of $$G$$, so its left cosets coincide with its right cosets. The four cosets are as follows:

$$\! \{ (0,0), (2,0) \}, \{ (1,0), (3,0) \}, \{ (0,1), (2,1) \}, \{ (1,1), (3,1) \}$$

Dual subgroup
We know that subgroup lattice and quotient lattice of finite abelian group are isomorphic, which means that there must exist a subgroup of $$G$$ that plays the role of a dual subgroup to $$H$$ -- in particular, that is isomorphic to the quotient group $$G/H$$ and its quotient group is isomorphic to $$H$$. The subgroup is first omega subgroup of direct product of Z4 and Z2.

Cohomology interpretation
We can think of $$G$$ as an extension with abelian normal subgroup $$H$$ and quotient group $$G/H$$. Since $$G$$ is abelian, $$H$$ is central, so the action of the quotient group on the normal subgroup is the trivial group action. We can thus study $$G$$ as an extension group arising from a cohomology class for the trivial group action of $$G/H$$ (which is a Klein four-group) on $$H$$ (which is cyclic group:Z2).

For more, see second cohomology group for trivial group action of V4 on Z2.

GAP implementation
The group-subgroup pair can be defined using the DirectProduct, CyclicGroup, and Agemo functions:

G := DirectProduct(CyclicGroup(4),CyclicGroup(2)); H := Agemo(G,2,1);