Projective semilinear group of degree two
It can be described as an external semidirect product of the projective general linear group of degree two over by the Galois group of over its prime subfield , where the latter acts on the former by applying the Galois automorphism to all the matrix entries in any representing matrix:
For a prime power , we denote by the group , where is the (unique up to isomorphism) field of size .
Over finite field
We consider the case where is the (unique up to isomorphism) field of size , with , so is the field characteristic and is the order of the Galois group .
|order||--||order of semidirect product is product of orders: the order of is and the order of is .|
|(field size)||(underlying prime, field characteristic)||(size of Galois group)||Order of (= )|
|7||7||1||projective general linear group:PGL(2,7)||336|
|9||3||2||automorphism group of alternating group:A6||1440|
|11||11||1||projective general linear group:PGL(2,11)||1320|