Embedding Subgroup in Direct Product in Magma

Question

I am using magma and want to consider a group $H$ as a subgroup of the direct product $H \times G$, written in magma as DirectProduct(H,G).

Right now I am having to use the subgroup generator construction, as in the following example.

It feels like there must be a much simpler way that I am missing!

A further question is how do I simply select an element of $G \setminus H$? An element of $G$ can be chosen by Rep(G), but I am not sure how to ensure this is not in $H$...

Example:

 H := CyclicGroup(8);
 G := DirectProduct(H,CyclicGroup(2))
 H := sub<G | (1,2,3,4,5,6,7,8)>;

Answer 1

> H := CyclicGroup(8);
> G, inj := DirectProduct(H,CyclicGroup(2));
> H := Image(inj[1]);
> H;
Permutation group H acting on a set of cardinality 10
Order = 8 = 2^3
    (1, 2, 3, 4, 5, 6, 7, 8)