Wreath product of Z5 and Z2

Definition
This group is defined as the defining ingredient::external wreath product of defining ingredient::cyclic group:Z5 and defining ingredient::cyclic group:Z2, where the latter acts via the regular group action on a set of size two. In other words, it is an defining ingredient::external semidirect product:

$$(\mathbb{Z}_5 \times \mathbb{Z}_5) \rtimes \mathbb{Z}_2$$

where the latter acts on the former by a coordinate exchange automorphism.