# Direct product of S3 and S3

## Definition

This group is defined as the external direct product of two copies of the symmetric group of degree three.

It can alternatively be defined as the subgroup of the symmetric group on $\{ 1,2,3,4,5,6 \}$ generated by the elements $(1,2,3), (1,2), (4,5,6), (4,5)$.

## Arithmetic functions

Function Value Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 36 groups with same order direct product of two groups of order $6$ each
exponent of a group 6 groups with same order and exponent of a group | groups with same exponent of a group direct product of two groups of exponent $6$ each
Frattini length 1 groups with same order and Frattini length | groups with same Frattini length direct product of two Frattini-free groups
derived length 2 groups with same order and derived length | groups with same derived length
minimum size of generating set 2 groups with same order and minimum size of generating set | groups with same minimum size of generating set In the permutation notation, it is generated by $(1,2,3)(4,5)$ and $(1,2)(4,5,6)$

## GAP implementation

### Group ID

This finite group has order 36 and has ID 10 among the groups of order 36 in GAP's SmallGroup library. For context, there are groups of order 36. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(36,10)

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

gap> G := SmallGroup(36,10);

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

IdGroup(G) = [36,10]

or just do:

IdGroup(G)

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

### Other descriptions

The group can be constructed using GAP's DirectProduct and SymmetricGroup functions:

DirectProduct(SymmetricGroup(3),SymmetricGroup(3))