# Direct product of A4 and Z4

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## Definition

This group of order 48 is defined as the external direct product of the alternating group of degree four and the cyclic group of order four. It may be denoted as $A_4 \times \mathbb{Z}_4$ or $A_4 \times C_4$.

## GAP implementation

### Group ID

This finite group has order 48 and has ID 31 among the groups of order 48 in GAP's SmallGroup library. For context, there are groups of order 48. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(48,31)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(48,31);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [48,31]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

### Other descriptions

Description Functions used
DirectProduct(AlternatingGroup(4),CyclicGroup(4)) DirectProduct, AlternatingGroup, CyclicGroup