Wreath product of Z2 and A5

Definition
This group is defined as the external wreath product of cyclic group:Z2 by alternating group:A5, where the permutation action of the latter is taken to be its natural action on a set of size five.

More explicitly, it is the external semidirect product of elementary abelian group:E32 by alternating group:A5:

$$(\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2) \rtimes A_5$$

where $$A_5$$ acts on the group $$\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$$ (which is elementary abelian group:E32) by permuting the coordinates based on its natural action on a set of size five.