# Direct product of A5 and V4

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## Contents

## Definition

This group is defined as the external direct product of alternating group:A5 and Klein four-group (which in turn is the direct product of two copies of cyclic group:Z2). In symbols, it is denoted where stands for the alternating group of degree five and stands for the Klein four-group.

## 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 , where is the order of alternating group:A5 and is the order of the Klein four-group. |

## GAP implementation

### Group ID

This finite group has order 240 and has ID 190 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(240,190)`

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

`gap> G := SmallGroup(240,190);`

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

`IdGroup(G) = [240,190]`

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),ElementaryAbelianGroup(4)) |
DirectProduct, AlternatingGroup, ElementaryAbelianGroup|ElementaryAbelianGroup |