$U(24)$: internal direct product and generators of a subgroup of a full symmetric group

Question

I'm being confused by the following question:

Build a permutation representation of U(24). List a representation of each element. Then, construct the group as a subgroup of a full symmetric group created with three generators. To determine these three generators, you will likely need to understand U(24) as an internal direct product.

I think that my permutation representation of $U(24)$ is correct (checked in Sage). However, I'm not sure to see the link between understanding $U(24)$ as an internal direct product and determining the three generators to build the subgroup of a full symmetric group - $S_8$ here.

For now, I have the following two subgroups, which look to build $U(24)$ as an internal direct product:

$$ H = \{ 1, 5, 7, 11 \} \\ K = \{ 1, 13 \} \\ \text{we have: } U(24) = H \times K \text{ such that: } H \cap K = \{ 1 \}, hk = kh \text{ for all } k \in K \text{ and } h \in H $$

Is this the right way to proceed? What is then the link with generators?

Thanks!

Answer 1

We know that $U(24)\cong C_2\times C_2\times C_2$, see

What group is $(\mathbb{Z}/24\mathbb{Z})^{*}$ isomorphic to.

Generators of $U(24)$ in $S_8$ are given by $$ (1234)(5678), (15)(26)(37)(48), $$ see here.