Order formulas for linear groups of degree two
From Groupprops
This article gives a list of formulas for the orders of the general linear group of degree two and some other related groups, both for a finite field of size and for related rings.
For the related formulas for degrees other than two, as well as for more detailed explanations of the order formulas, see order formulas for linear groups.
For a finite field of size
Formulas
In the formulas below, the field size is . The characteristic of the field is a prime number . is a prime power with underlying prime . We let , so and is a nonnegative integer.
Group  Symbolic notation  Order formula  Order formula (maximally factorized)  Order formula (expanded)  Degree as polynomial in (same as algebraic dimension)  Quick explanation for order 

general linear group of degree two  or  4  [SHOW MORE]  
projective general linear group of degree two  or  3  [SHOW MORE]  
special linear group of degree two  or  3  [SHOW MORE]  
projective special linear group of degree two  or  for odd for even 
for odd for even 
for odd for even 
3  [SHOW MORE] 
general semilinear group of degree two  or  4  [SHOW MORE]  
projective semilinear group of degree two  or  3  [SHOW MORE]  
general affine group of degree two  or or or  6  [SHOW MORE]  
special affine group of degree two  or or or  5  [SHOW MORE] 
Particular cases
The links are to the actual groups, which are not explicitly specified in order to save space.
Field size  Field characteristic  so that  







2  2  1  6  6  6  6  6  6  24 
3  3  1  48  24  24  12  48  24  432 
4  2  2  180  60  60  60  360  120  2880 
5  5  1  480  120  120  60  480  120  12000 
7  7  1  2016  336  336  168  2016  336  38304 
8  2  3  3528  504  504  504  10584  1512  225792 
9  3  2  5760  720  720  360  11520  1440  466560 