Projective general linear group:PGL(2,7)

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Definition

This group is defined in the following equivalent ways:

1. It is the projective general linear group of degree two over the field of seven elements, i.e., the group $PGL(2,7)$.
2. It is the automorphism group of projective special linear group:PSL(3,2) which in turn is isomorphic to $PSL(2,7)$ $PGL(3,2)$, $SL(3,2)$, and $GL(3,2)$.
3. It is the outer linear group $OL(3,2)$.

GAP implementation

Group ID

This group has ID $208$ among the groups of order $336$. It can be invoked using GAP's SmallGroup function:

SmallGroup(336,208)

Other definitions

The group can be defined using GAP's ProjectiveGeneralLinearGroup function:

PGL(2,7)

It can also be defined using the AutomorphismGroup function on $PSL(3,2)$:

AutomorphismGroup(PSL(3,2))