Element structure of projective general linear group:PGL(2,9)

This article gives specific information, namely, element structure, about a particular group, namely: projective general linear group:PGL(2,9).
View element structure of particular groups | View other specific information about projective general linear group:PGL(2,9)

This article describes the element structure of projective general linear group:PGL(2,9), which is the projective general linear group of degree two over field:F9.

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

Note that this group is not isomorphic to symmetric group:S6, even though it has a subgroup of index two that is isomorphic to alternating group:A6.

Conjugacy class structure

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

There is a total of $11$ conjugacy classes.

Nature of conjugacy class upstairs in $GL_{2}$	Eigenvalues	Characteristic polynomial	Minimal polynomial	Size of conjugacy class	Number of such conjugacy classes	Total number of elements
Diagonalizable over field:F9 with equal diagonal entries, hence a scalar	$\{a,a\}$ where $a\in \mathbb {F} _{9}^{\ast }$	$(x-a)^{2}$	$x-a$	1	1	1
Diagonalizable over $\mathbb {F} _{81}$, not over $\mathbb {F} _{9}$, eigenvalues are negatives of each other.	Pair of mutually negative conjugate elements of $\mathbb {F} _{81}$. All such pairs identified.	$x^{2}-\mu$, $\mu$ a nonzero non-square	Same as characteristic polynomial	36	1	$q(q-1)/2$
Diagonalizable over $\mathbb {F} _{9}$ with mutually negative diagonal entries.	$\{\lambda ,-\lambda \}$, all such pairs identified.	$x^{2}-\lambda ^{2}$, all identified	Same as characteristic polynomial	45	1	45
Diagonalizable over $\mathbb {F} _{81}$, not over $\mathbb {F} _{9}$, eigenvalues are not negatives of each other.	Pair of conjugate elements of $\mathbb {F} _{81}$. Each pair identified with anything obtained by multiplying both elements of it by an element of $\mathbb {F} _{9}$.	$x^{2}-ax+b$, $a\neq 0$, irreducible; with identification.	Same as characteristic polynomial	72	4	288
Not diagonal, has Jordan block of size two	$a\in \mathbb {F} _{9}^{\ast }$ (multiplicity 2). Each conjugacy class has one representative of each type.	$(x-a)^{2}$	Same as characteristic polynomial	80	1	80
Diagonalizable over $\mathbb {F} _{9}$ with distinct diagonal entries whose sum is not zero.	$\lambda ,\mu$ where $\lambda ,\mu \in \mathbb {F} _{9}^{\ast }$ and $\lambda +\mu \neq 0$. The pairs $\{\lambda ,\mu \}$ and $\{a\lambda ,a\mu \}$ are identified.	$x^{2}-(\lambda +\mu )x+\lambda \mu$, again with identification.	Same as characteristic polynomial.	90	3	270
Total	NA	NA	NA	NA	11	720