# Direct product of A5 and Z4

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## Definition

This group is defined as the external direct product of alternating group:A5 and cyclic group:Z4.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 240#Arithmetic functions
Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 240 groups with same order order of direct product is product of orders: the order is $60 \times 4$, where $60 = 5!/2$ is the order of alternating group:A5 and $4$ is the order of cyclic group:Z4.
exponent of a group 60 groups with same order and exponent of a group | groups with same exponent of a group lcm of direct product is lcm of exponents: the exponent is $\operatorname{lcm} \{ 30, 4 \}$.
composition length 3 groups with same order and composition length | groups with same composition length composition length of direct product is sum of composition lengths: The composition length is $1 + 2 = 3$ where 1 is the composition length of alternating group:A5 (because it is simple) and 2 is the composition length of cyclic group:Z4.
chief length 3 groups with same order and chief length | groups with same chief length chief length of direct product is sum of chief lengths: The chief length is thus $1 + 2 = 3$.
minimum size of generating set 2 groups with same order and minimum size of generating set | groups with same minimum size of generating set

## GAP implementation

### Group ID

This finite group has order 240 and has ID 92 among the groups of order 240 in GAP's SmallGroup library. For context, there are 208 groups of order 240. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(240,92)

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

gap> G := SmallGroup(240,92);

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

IdGroup(G) = [240,92]

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(5),CyclicGroup(4)) DirectProduct, AlternatingGroup, CyclicGroup