Wreath product of Z6 and Z2

Definition
This group is defined as the defining ingredient::external wreath product with base cyclic group:Z6 and acting group cyclic group:Z2, where the latter acts via its regular group action, i.e., its action as permutations on a set of size tow.

More explicitly, it is the external semidirect product:

$$(\mathbb{Z}_6 \times \mathbb{Z}_6) \rtimes \mathbb{Z}_2$$

where the non-identity element of the acting group $$\mathbb{Z}_2$$ acts by permuting the two copies of $$\mathbb{Z}_6$$, i.e., by a coordinate exchange automorphism.