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Particular cases
{| class="sortable" border="1"
! <math>q</math><br> (field size) !! <math>p</math> <br> (underlying prime, field characteristic) !! <math>r = \log_pq</math> !! general affine group <math>GL(2,q)</math> !! [[order]] of the group (= <math>q^2(q^2 - 1)(q^2 - q)</math>)!! number of conjugacy classes (= <math>q^2 + q - 1</math>) !! element structure page
|-
| 2 || 2 || 1 || [[symmetric group:S4]] || 24 || 5 || [[element structure of symmetric group:S4]]
|-
| 3 || 3 || 1 || [[general affine group:GA(2,3)]] || 432 || 11 || [[element structure of general affine group:GA(2,3)]]
|-
| 4 || 2 || 2 || [[general affine group:GA(2,4)]] || 2880 || 19 ||
|-
| 5 || 5 || 1 || [[general affine group:GA(2,5)]] || 12000 || 29 ||
|}
| <math>A</matH> is the identity, <math>v \ne 0</math> || <math>\{ 1,1 \}</math> || <math>(x - 1)^2</math> || <math>x - 1</math> || <math>q^2 - 1</math> || 1 || <matH>q^2 - 1</math> || Yes || Yes
|-
| <math>A</math> is diagonalizable over <math>\mathbb{F}_q</math> with equal diagonal entries not equal to 1, hence a scalar. The value of <matH>v</math> does not affect the conjugacy class. || <math>\{a,a \}</math> where <math>a \in \mathbb{F}_q^\ast \setminus \{ 1 \}</math> || <math>(x - a)^2</math> where <math>a \in \mathbb{F}_q^\ast\setminus \{ 1 \}</math> || <math>x - a</math> where <math>a \in \mathbb{F}_q^\ast\setminus \{ 1 \}</math> || <math>q^2</math> || <math>q - 2</math> || <math>q^2(q - 2)</math> || Yes || Yes
|-
| <math>A</math> is diagonalizable over <math>\mathbb{F}_{q^2}</math>, not over <math>\mathbb{F}_q</math>. Must necessarily have no repeated eigenvalues. The value of <math>v</matH> does not affect the conjugacy class. || Pair of conjugate elements of <math>\mathbb{F}_{q^2}</math> || <math>x^2 - ax + b</math>, irreducible || Same as characteristic polynomial || <math>q^3(q - 1)</math> || <math>q(q - 1)/2 = (q^2 - q)/2</math> || <math>q^4(q-1)^2/2</math> || Yes || No
| <math>A</math> diagonalizable over <math>\mathbb{F}_q</math> with ''distinct'' diagonal entries, one of which is 1, <math>v</math> is not in the image of <math>A - 1</matH> || <math>1,\mu</math>, <math>\mu \in \mathbb{F}_q^\ast \setminus \{ 1 \}</math> || <math>x^2 - (\mu + 1)x + \mu</math> || Same as characteristic polynomial || <math>q(q + 1)(q^2 - q)</math> || <math>q - 2</math> || <math>q^2(q+1)(q - 1)(q-2)</math> || Yes || Yes
|-
| <math>A</math> diagonalizable over <math>\mathbb{F}_q</math> with ''distinct'' diagonal entries, neither of which is 1 || <math>\lambda, \mu</math> (interchangeable) distinct elements of <math>\mathbb{F}_q^\ast</math> , neither equal to 1 || <math>x^2 - (\lambda + \mu)x + \lambda \mu</math> || Same as characteristic polynomial || <math>q^3(q+1)</math>|| <math>(q - 2)(q - 3)/2 </math> || <math>q^3(q+1)(q-2)(q-3)/2</math> || Yes || Yes
|-
! Total !! NA !! NA !! NA !! NA !! <math>q^2 + q - 1</math> !! <math>q^2(q^2 - 1)(q^2 - q)</math> !! !!
|}
<section end="conjugacy class structure"/>
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