Linear representation theory of dihedral group:D16

This article gives specific information, namely, linear representation theory, about a particular group, namely: dihedral group:D16.
View linear representation theory of particular groups | View other specific information about dihedral group:D16

Summary

We shall use the dihedral group with the following presentation:

$\langle a,x\mid a^{8}=x^{2}=e,xax^{-1}=a^{-1}\rangle$.

Item	Value
degrees of irreducible representations over a splitting field (such as $\mathbb {C}$ or ${\overline {\mathbb {Q} }}$)	1,1,1,1,2,2,2 maximum: 2, lcm: 2, number: 7, sum of squares: 16
Schur index values of irreducible representations over a splitting field	1,1,1,1,1,1,1
smallest ring of realization (characteristic zero)	$\mathbb {Z} [{\sqrt {2}}]$ or $\mathbb {Z} [t]/(t^{2}-2)$
smallest splitting field, i.e., smallest field of realization (characteristic zero)	$\mathbb {Q} ({\sqrt {2}})$ or $\mathbb {Q} [t]/(t^{2}-2)$
condition for a field to be a splitting field	The characteristic should not be equal to 2, and the polynomial $t^{2}-2$ must split. For a finite field of size $q$, this is equivalent to saying that $q\equiv \pm 1{\pmod {8}}$.
smallest size splitting field	Field:F7.
degrees of irreducible representations over the rational numbers	1,1,1,1,2,4

Family contexts

Family name	Parameter values	General discussion of linear representation theory of family
dihedral group	degree $n=8$, order $2n=16$	linear representation theory of dihedral groups

COMPARE AND CONTRAST: View linear representation theory of groups of order 16 to compare and contrast the linear representation theory with other groups of order 16.

Representations

Summary information

Below is summary information on irreducible representations. Note that a particular representation may make sense, and be irreducible, only for certain kinds of fields -- see the "Values not allowed for field characteristic" and "Criterion for field" columns to see the condition the field must satisfy for the representation to be irreducible there.

Name of representation type	Number of representations of this type	Values not allowed for field characteristic	Criterion for field	What happens over a splitting field?	Kernel	Quotient by kernel (on which it descends to a faithful representation)	Degree	Schur index	What happens by reducing the $\mathbb {Z}$-representation over bad characteristics?
trivial	1	--	any	remains the same	whole group	trivial group	1	1	--
sign representation with kernel $\langle a\rangle$	1	--	any	remains the same	Z8 in D16: $\langle a\rangle$	cyclic group:Z2	1	1	There are no bad characteristics, but in characteristic two, it becomes equal to the trivial representation.
sign representation with kernel a maximal dihedral subgroup	2	--	any	remains the same	D8 in D16: $\langle a^{2},x\rangle$ or $\langle a^{2},ax\rangle$	cyclic group:Z2	1	1	There are no bad characteristics, but in characteristic two, it becomes equal to the trivial representation.
two-dimensional irreducible, not faithful	1	2	any	remains the same	center of dihedral group:D16: $\langle a^{4}\rangle$	dihedral group:D8	2	1	The exact form of the new representation depends on the choice of matrices before we go mod 2, but the kernel becomes one of the D8 in D16 subgroups, and we thus get a representation of cyclic group:Z2 in characteristic two that sends the non-identity element to ${\begin{pmatrix}0&1\\1&0\\\end{pmatrix}}$. This has an invariant one-dimensional subspace and is not irreducible.
two-dimensional faithful irreducible	2	2	The polynomial $t^{2}-2$ must split, i.e., $2$ must have a square root	remains the same	trivial subgroup, i.e., it is a faithful linear representation	dihedral group:D16	2	1	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.
four-dimensional faithful irreducible	1	2	The polynomial $t^{2}-2$ must not split, i.e., $2$ must not have a square root	splits into the two two-dimensional faithful irreducibles.	trivial subgroup, i.e., it is a faithful linear representation	dihedral group:D16	4	1	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.

Trivial representation

The trivial representation or principal representation (whose character is called the trivial character or principal character) sends all elements of the group to the $1\times 1$ matrix $(1)$:

Element	Matrix	Characteristic polynomial	Minimal polynomial	Trace, character value
$e$	$(1)$	$t-1$	$t-1$	1
$a$	$(1)$	$t-1$	$t-1$	1
$a^{2}$	$(1)$	$t-1$	$t-1$	1
$a^{3}$	$(1)$	$t-1$	$t-1$	1
$a^{4}$	$(1)$	$t-1$	$t-1$	1
$a^{5}$	$(1)$	$t-1$	$t-1$	1
$a^{6}$	$(1)$	$t-1$	$t-1$	1
$a^{7}$	$(1)$	$t-1$	$t-1$	1
$x$	$(1)$	$t-1$	$t-1$	1
$ax$	$(1)$	$t-1$	$t-1$	1
$a^{2}x$	$(1)$	$t-1$	$t-1$	1
$a^{3}x$	$(1)$	$t-1$	$t-1$	1
$a^{4}x$	$(1)$	$t-1$	$t-1$	1
$a^{5}x$	$(1)$	$t-1$	$t-1$	1
$a^{6}x$	$(1)$	$t-1$	$t-1$	1
$a^{7}x$	$(1)$	$t-1$	$t-1$	1

Sign representation with kernel $\langle a\rangle$

This representation is a one-dimensional representation sending everything in the cyclic subgroup $\langle a\rangle$ (see Z8 in D16) to $(1)$ and everything outside it to $(-1)$.

Element	Matrix	Characteristic polynomial	Minimal polynomial	Trace, character value
$e$	$(1)$	$t-1$	$t-1$	1
$a$	$(1)$	$t-1$	$t-1$	1
$a^{2}$	$(1)$	$t-1$	$t-1$	1
$a^{3}$	$(1)$	$t-1$	$t-1$	1
$a^{4}$	$(1)$	$t-1$	$t-1$	1
$a^{5}$	$(1)$	$t-1$	$t-1$	1
$a^{6}$	$(1)$	$t-1$	$t-1$	1
$a^{7}$	$(1)$	$t-1$	$t-1$	1
$x$	$(-1)$	$t+1$	$t+1$	-1
$ax$	$(-1)$	$t+1$	$t+1$	-1
$a^{2}x$	$(-1)$	$t+1$	$t+1$	-1
$a^{3}x$	$(-1)$	$t+1$	$t+1$	-1
$a^{4}x$	$(-1)$	$t+1$	$t+1$	-1
$a^{5}x$	$(-1)$	$t+1$	$t+1$	-1
$a^{6}x$	$(-1)$	$t+1$	$t+1$	-1
$a^{7}x$	$(-1)$	$t+1$	$t+1$	-1

Sign representations with kernels $\langle a^{2},x\rangle$ with $\langle a^{2},ax\rangle$

These are sign representations with kernels one of the D8 in D16 subgroups. There are two such representations, one for each subgroup.

Two-dimensional irreducible unfaithful representation

This representation has kernel equal to $\langle a^{4}\rangle$ -- center of dihedral group:D16. It descends to a faithful irreducible two-dimensional representation of the quotient group, which is isomorphic to dihedral group:D8.

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Four-dimensional faithful irreducible representations

There are two such representations, and they are related by the group automorphism $a\mapsto a^{3},x\mapsto x$, and also by the Galois automorphism ${\sqrt {2}}\mapsto -{\sqrt {2}}$ for the extension $\mathbb {Q} ({\sqrt {2}})$ over $\mathbb {Q}$.