Wreath product of Z2 and A4

Definition
This group is defined as the defining ingredient::external wreath product of defining ingredient::cyclic group:Z2 by defining ingredient::alternating group:A4, where the permutation action of the latter is taken as the natural action on a set of size four.

More explicitly, it is the external semidirect product with base elementary abelian group:E16 (the direct product of four copies of cyclic group:Z2) by alternating group:A4:

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

where the latter acts on the former by coordinate permutations given by the natural action of $$A_4$$ on a set of size four.