Linear representation theory of dihedral group:D10

This article gives specific information, namely, linear representation theory, about a particular group, namely: dihedral group:D10.
View linear representation theory of particular groups | View other specific information about dihedral group:D10

We shall use the dihedral group with the following presentation (here, $e$ is used to denote the identity element):

$\langle a,x\mid a^{5}=x^{2}=e,xax^{-1}=a^{-1}\rangle$.

Character table

FACTS TO CHECK AGAINST (for characters of irreducible linear representations over a splitting field):
Orthogonality relations: Character orthogonality theorem | Column orthogonality theorem
Separation results (basically says rows independent, columns independent): Splitting implies characters form a basis for space of class functions|Character determines representation in characteristic zero
Numerical facts: Characters are cyclotomic integers | Size-degree-weighted characters are algebraic integers
Character value facts: Irreducible character of degree greater than one takes value zero on some conjugacy class| Conjugacy class of more than average size has character value zero for some irreducible character | Zero-or-scalar lemma

This character table works over characteristic zero:

Representation/Conj class	$\{e\}$ (size 1)	$\{x,ax,a^{2}x,a^{3}x,a^{4}x\}$ (size 5)	$\{a,a^{-1}\}$ (size 2)	$\{a^{2},a^{-2}\}$ (size 2)
Trivial representation	$1$	$1$	$1$	$1$
Non-trivial one-dimensional	$1$	$-1$	$1$	$1$
Faithful irreducible representation	$2$	$0$	${\frac {-1-{\sqrt {5}}}{2}}$	${\frac {-1+{\sqrt {5}}}{2}}$
Faithful irreducible representation	$2$	$0$	${\frac {-1+{\sqrt {5}}}{2}}$	${\frac {-1-{\sqrt {5}}}{2}}$

See also

Linear representation theory of dihedral groups