Wreath product of Z2 and Z4

Definition
This group is defined as the wreath product of the cyclic group of order two and the cyclic group of order four with the regular group action. In other words, it is the semidirect product $$(\Z_2 \times \Z_2 \times \Z_2 \times \Z_2) \rtimes \Z_4$$, where the acting $$\Z_4$$ acts by cyclic permutation of coordinates.

Other descriptions
The group can be defined using GAP's WreathProduct and GAP:CyclicGroup functions:

WreathProduct(CyclicGroup(2),CyclicGroup(4))