Element structure of general linear group:GL(2,3)

This article gives specific information, namely, element structure, about a particular group, namely: general linear group:GL(2,3).
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This article describes the element structure of general linear group:GL(2,3) which is the general linear group of degree two over field:F3.

See also element structure of general linear group of degree two.

Conjugacy class structure

Interpretation as general linear group of degree two over field:F3

Compare with element structure of general linear group of degree two#Conjugacy class structure.

Nature of conjugacy class	Eigenvalues	Characteristic polynomial	Minimal polynomial	Size of conjugacy class (generic $q$)	Size of conjugacy class ($q=3$)	Number of such conjugacy classes (generic $q$)	Number of such conjugacy classes ($q=3$)	Total number of elements (generic $q$)	Total number of elements ($q=3$)	Representative matrix (one per conjugacy class)
Diagonalizable over $\mathbb {F} _{3}$ with equal diagonal entries, hence a scalar	$\{a,a\}$ where $a\in \mathbb {F} _{3}^{\ast }$	$(x-a)^{2}$ where $a\in \mathbb {F} _{3}^{\ast }$	$x-a$ where $a\in \mathbb {F} _{3}^{\ast }$	1	1	$q-1$	2	$q-1$	2	${\begin{pmatrix}1&0\\0&1\\\end{pmatrix}}$, ${\begin{pmatrix}-1&0\\0&-1\\\end{pmatrix}}$
Diagonalizable over field:F9, not over $\mathbb {F} _{3}$. Must necessarily have no repeated eigenvalues.	Pair of conjugate elements of $\mathbb {F} _{9}$	$x^{2}+1$, $x^{2}+x-1$, $x^{2}-x-1$	Same as characteristic polynomial	$q(q-1)$	6	$q(q-1)/2$	3	$q^{2}(q-1)^{2}/2$	18	${\begin{pmatrix}0&1\\-1&0\\\end{pmatrix}}$, ${\begin{pmatrix}0&1\\1&-1\\\end{pmatrix}}$, ${\begin{pmatrix}0&1\\1&1\\\end{pmatrix}}$
Not diagonal, has Jordan block of size two	$a$ (multiplicity two) where $a\in \mathbb {F} _{3}^{\ast }$	$(x-a)^{2}$ where $a\in \mathbb {F} _{3}^{\ast }$	Same as characteristic polynomial	$q^{2}-1$	8	$q-1$	2	$(q+1)(q-1)^{2}$	16	${\begin{pmatrix}1&1\\0&1\\\end{pmatrix}}$, ${\begin{pmatrix}-1&1\\0&-1\\\end{pmatrix}}$
Diagonalizable over field:F3 with distinct diagonal entries	$\{1,-1\}$	$x^{2}-1$	Same as characteristic polynomial	$q(q+1)$	12	$(q-1)(q-2)/2$	1	$q(q+1)(q-1)(q-2)/2$	12	${\begin{pmatrix}1&0\\0&-1\\\end{pmatrix}}$
Total	NA	NA	NA	NA	NA	$q^{2}-1$	8	$q^{4}-q^{3}-q^{2}+q$	48	NA