Element structure of general linear group of degree three over a finite field

This article gives specific information, namely, element structure, about a family of groups, namely: general linear group of degree three.
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This article discusses the element structure of the general linear group of degree three over a finite field. The group is $GL(3,q)$ where $q$ is the order (size) of the field. We denote by $p$ the prime number that is the characteristic of the field.

Particular cases

Group	$p$	$q$	Order of the group	Number of conjugacy classes	Element structure page
projective special linear group:PSL(3,2)	2	2	168	6	element structure of projective special linear group:PSL(3,2)
general linear group:GL(3,3)	3	3	11232	24	element structure of general linear group:GL(3,3)
general linear group:GL(3,5)	5	5	1488000	120	element structure of general linear group:GL(3,5)

Conjugacy class structure

There is a total of $(q^3 - 1)(q^3 - q)(q^3 - q^2) = q^3(q - 1)^3(q + 1)(q^2 + q + 1)$ elements, and a total of $q^3 - q = q(q - 1)(q + 1)$ conjugacy classes.

Nature of conjugacy class	Eigenvalues	Characteristic polynomial	Minimal polynomial	Size of conjugacy class	Number of such conjugacy classes	Total number of elements	Semisimple?	Diagonalizable over $\mathbb{F}_q$?
Diagonalizable over $\mathbb{F}_q$ with equal diagonal entries, hence a scalar	$\{a,a,a \}$ where $a \in \mathbb{F}_q^\ast$	$(x - a)^3$	$x - a$	1	$q - 1$	$q - 1$	Yes	Yes
Diagonalizable over $\mathbb{F}_q$ with one eigenvalue having multiplicity two, the other eigenvalue having multiplicity one	$\{ a,a,b \}$ where $a \ne b$, both in $\mathbb{F}_q^\ast$	$(x - a)^2(x - b)$	$(x - a)(x - b)$	$q^2(q^2 + q + 1) = q^4 + q^3 + q^2$	$(q - 1)(q - 2)$	$q^2(q^2 + q + 1)(q - 1)(q - 2) = q^6 - 2q^5 - q^3 + 2q^2$	Yes	Yes
Diagonalizable over $\mathbb{F}_q$ with all distinct diagonal entries	$\{ a,b,c \}$, all distinct elements of $\mathbb{F}_q^\ast$	$(x - a)(x - b)(x - c)$	same as characteristic polynomial	$q^3(q + 1)(q^2 + q + 1)$	$(q-1)(q-2)(q-3)/6 = (q^3 - 6q^2 + 11q - 6)/6$	$q^3(q + 1)(q^2 + q + 1)(q - 1)(q - 2)(q - 3)/6 = (q^9 - 4q^8 + q^7 + 5q^6 + 4q^5 - q^4 - 6q^3)/6$	Yes	Yes
Diagonalizable over $\mathbb{F}_{q^3}$, not over $\mathbb{F}_q$	Distinct Galois conjugate triple of elements in $\mathbb{F}_{q^3}^\ast$. If one of the elements is $a$, the other two are $a^q$ and $a^{q^2}$.	irreducible degree three polynomial over $\mathbb{F}_q$	same as characteristic polynomial	$q^3(q - 1)^2(q + 1)$	$q(q + 1)(q - 1)/3 = (q^3 - q)/3$	$q^4(q - 1)^3(q + 1)^2/3 = (q^9 - q^8 - 2q^7 + 2q^6 + q^5 - q^4)/3$	Yes	No
One eigenvalue is in $\mathbb{F}_q^\ast$, the other two are in $\mathbb{F}_{q^2} \setminus \mathbb{F}_q$	one element of $\mathbb{F}_q^\ast$, pair of Galois conjugates over $\mathbb{F}_q$ in $\mathbb{F}_{q^2}$.	product of linear polynomial and irreducible degree two polynomial over $\mathbb{F}_q$	same as characteristic polynomial	$q^3(q - 1)(q^2 + q + 1) = q^6 - q^3$	$q(q - 1)^2/2 = (q^3 - 2q^2 + q)/2$	$q^4(q - 1)^3(q^2 + q + 1)/2 = (q^9 - 2q^8 + q^7 - q^6 + 2q^5 - q^4)/2$	Yes	No
Has Jordan blocks of sizes 2 and 1 with distinct eigenvalues over $\mathbb{F}_q$	$\{ a,a,b \}$ with $a,b \in \mathbb{F}_q^\ast$, $a \ne b$	$(x - a)^2(x - b)$	same as characteristic polynomial	$q^2(q + 1)(q - 1)(q^2 + q + 1)$	$(q - 1)(q - 2)$	$q^2(q -1)^2(q + 1)(q^2 + q + 1)(q - 2) = q^8 - 2q^7 - q^6 + q^5 + 2q^4 + q^3 - 2q^2$	No	No
Has Jordan blocks of sizes 2 and 1 with equal eigenvalues over $\mathbb{F}_q$	$\{ a,a,a \}$ with $a \in \mathbb{F}_q^\ast$	$(x - a)^3$	$(x - a)^2$	$q(q + 1)(q - 1)^2(q^2 + q + 1) = q^6 - q^4 - q^2 + q$	$q - 1$	$q(q + 1)(q - 1)^3(q^2 + q + 1) = q^7 - q^6 - q^5 + q^3 + q^2 - q$	No	No
Has Jordan block of size 3	$\{ a,a,a \}$ with $a \in \mathbb{F}_q^\ast$	$(x - a)^3$	same as characteristic polynomial	$(q - 1)(q + 1)(q^2 + q + 1) = q^4 + q^3 - q - 1$	$q - 1$	$(q - 1)^2(q + 1)(q^2 + q + 1) = q^5 - q^3 - q^2 + 1$	No	No
Total	NA	NA	NA	NA	$q^3 - q$	$(q^3 - 1)(q^3 - q)(q^3 - q^2) = q^9 - q^8 - q^7 + q^5 + q^4 - q^3$	NA	NA