# Projective general linear group:PGL(2,9)

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## Definition

This group is the projective general linear group of degree two over the field of nine elements.

## Arithmetic functions

Function Value Similar groups Explanation
order (number of elements, equivalently, cardinality or size of underlying set) 720 groups with same order As $PGL(2,q)$, $q = 9$: order $q^3 - q = 9^3 - 9 = 720$

## GAP implementation

### Group ID

This finite group has order 720 and has ID 764 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(720,764)

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

gap> G := SmallGroup(720,764);

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

IdGroup(G) = [720,764]

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 defined using GAP's ProjectiveGeneralLinearGroup function:

PGL(2,9)