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 $a\neq b$, 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 $a,b\in \mathbb {F} _{q}^{\ast }$, $a\neq b$	$(x-a)^{2}(x-b)$	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