Wreath product of A4 and Z2

Definition
This group is defined as the defining ingredient::external wreath product of defining ingredient::alternating group:A4 and defining ingredient::cyclic group:Z2, where the latter acts via the regular group action. More explicitly it is the defining ingredient::external semidirect product:

$$(A_4 \times A_4) \rtimes \mathbb{Z}_2$$

where the non-identity element of $$\mathbb{Z}_2$$ acts by the coordinate exchange automorphism on $$A_4 \times A_4$$.