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{{group family-specific information|
group family = projective general linear group of degree two|
information type = linear representation theory|
connective = of}}
 
This article describes the linear representation theory of the [[general linear group of degree two]] over a [[finite field]]. The order (size) of the field is <math>q</math>, and the characteristic prime is <math>p</math>. <math>q</math> is a power of <math>p</math>.
See also the linear representation theory for: [[linear representation theory of special linear group of degree twoover a finite field|special linear group]], [[linear representation theory of projective special linear group of degree twoover a finite field|projective special linear group]], and [[linear representation theory of general linear group of degree twoover a finite field|general linear group]]. ==Summary==
{| class="sortable" border="1"
! Item !! Value
|-
| [[degrees of irreducible representations]] over a [[splitting field]] || Case <math>q</math> odd: 1 (2 times), <math>q</math> (2 times), <math>q + 1</math> (<math>(q - 3)/2</math> times), <math>q - 1</math> (<math>(q - 1)/2</math> times)<br> Case <math>q</math> even:
|-
| number of irreducible representations || Case <math>q</math> odd: <math>q + 2</math>, case <math>q</math> even: <math>q + 1</math><br>See [[number of irreducible representations equals number of conjugacy classes]], [[element structure of projective general linear group of degree two over a finite field#Conjugacy class structure]]
|-
| [[maximum degree of irreducible representation]] || <math>q + 1</math>
|-
| [[lcm of degrees of irreducible representations]] || <math>q(q + 1)(q - 1) = q^3 - q</math>
|-
| sum of squares of degrees of irreducible representations || <math>q(q + 1)(q - 1) = q^3 - q</math>, equal to the group order; see [[sum of squares of degrees of irreducible representations equals group order]]
|}
==Particular cases==
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