# Direct product of A5 and S3

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## Definition

This group can be defined in a number of equivalent ways:

1. It is the direct product of the alternating group of degree five and the symmetric group of degree three. In other words, it is $A_5 \times S_3$.
2. It is the outer linear group $OL(2,4)$, i.e., it is the semidirect product of the general linear group:GL(2,4) by a two-element cyclic group acting via the transpose-inverse map.

## GAP implementation

### Group ID

This group has ID $121$ among the groups of order $360$. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(360,121)

### Other definitions

The group can be defined using the DirectProduct function:

DirectProduct(AlternatingGroup(5),SymmetricGroup(3))