# Direct product of A6 and Z2

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## Definition

This group is defined as the external direct product of alternating group:A6 and cyclic group:Z2.

## Arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 720 groups with same order order of direct product is product of orders: the direct factors have orders $360$ and $2$, so the order is $360 \times 2 = 720$
exponent of a group 60 groups with same order and exponent of a group | groups with same exponent of a group exponent of direct product is lcm of exponents: the direct factors have exponents $360$ and $2$ respectively.

## GAP implementation

### Group ID

This finite group has order 720 and has ID 766 among the groups of order 720 in GAP's SmallGroup library. For context, there are 840 groups of order 720. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(720,766)

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

gap> G := SmallGroup(720,766);

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

IdGroup(G) = [720,766]

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(6),CyclicGroup(2)) DirectProduct, AlternatingGroup, CyclicGroup