Linear representation theory of M16

This article gives specific information, namely, linear representation theory, about a particular group, namely: M16.
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This article discusses the linear representation theory of the group M16, a group of order 16 given by the presentation:

$G=M_{16}:=\langle a,x\mid a^{8}=x^{2}=e,xax^{-1}=a^{5}\rangle$

Summary

Item	Value
degrees of irreducible representations over a splitting field (such as ${\overline {\mathbb {Q} }}$ or $\mathbb {C}$)	1,1,1,1,1,1,1,1,2,2 (1 occurs 8 times, 2 occurs 2 times) maximum: 2, lcm: 2, number: 10, sum of squares: 16
Schur index values of irreducible representations	1 (all of them)
smallest ring of realization (characteristic zero)	$\mathbb {Z} [i]=\mathbb {Z} [{\sqrt {-1}}]=\mathbb {Z} [t]/(t^{2}+1)$ -- ring of Gaussian integers
ring generated by character values (characteristic zero)	$\mathbb {Z} [2i]=\mathbb {Z} [{\sqrt {-4}}]=\mathbb {Z} [t]/(t^{2}+4)$
minimal splitting field, i.e., smallest field of realization (characteristic zero)	$\mathbb {Q} (i)=\mathbb {Q} ({\sqrt {-1}})=\mathbb {Q} [t]/(t^{2}+1)$ Same as field generated by character values, because all Schur index values are 1.
condition for a field to be a splitting field	The characteristic should not be equal to 2, and the polynomial $t^{2}+1$ should split. For a finite field of size $q$, this is equivalent to saying that $q\equiv 1{\pmod {4}}$
minimal splitting field in characteristic $p\neq 0,2$	Case $p\equiv 1{\pmod {4}}$: prime field $\mathbb {F} _{p}$ Case $p\equiv 3{\pmod {4}}$: Field $\mathbb {F} _{p^{2}}$, quadratic extension of prime field
smallest size splitting field	Field:F5, i.e., the field with five elements.
degrees of irreducible representations over the rational numbers	1,1,1,1,2,2,4 (1 occurs 4 times, 2 occurs 2 times, 4 occurs 1 time) number: 7
orbits of irreducible representations over a splitting field under action of automorphism group	2 orbits of size 1 of degree 1 representations, 3 orbits of size 2 of degree 1 representations, 1 orbit of size 2 of degree 2 representations. number: 6

Representations

Summary information

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 $\mathbb {Z}$ or $\mathbb {Z} [i]$-representation over bad characteristics?
trivial	1	--	any	remains the same	whole group	trivial group	1	1	no bad characteristics
sign, kernel a non-cyclic maximal subgroup	1	--	any	remains the same	direct product of Z4 and Z2 in M16 -- $\langle a^{2},x\rangle$	cyclic group:Z2	1	1	no bad characteristics, but reduced mod 2, this representation becomes the trivial representation
sign, kernel a cyclic maximal subgroup	2	--	any	remains the same	Z8 in M16 -- either $\langle a\rangle$ or $\langle ax\rangle$	cyclic group:Z2	1	1	no bad characteristics, but reduced mod 2, this representation becomes the trivial representation
representation with kernel $\langle a^{2}x\rangle$	2	--	must contain a primitive fourth root of unity, or equivalently, $t^{2}+1$ must split	remains the same	non-central Z4 in M16	cyclic group:Z4	1	1	no bad characteristics, but in characteristic 2, becomes the same as the trivial representation.
representation with kernel $\langle a^{2},x\rangle$	2	--	must contain a primitive fourth root of unity, or equivalently, $t^{2}+1$ must split	remains the same	V4 in M16	cyclic group:Z4	1	1	no bad characteristics, but in characteristic 2, becomes the same as the trivial representation.
faithful irreducible representation of M16	2	2	must contain a primitive fourth root of unity, or equivalently, $t^{2}+1$ must split	remains the same	trivial subgroup	M16	2	1	in characteristic 2, acquires a kernel of order eight, and we get an indecomposable but not irreducible representation of the quotient group.
not over a splitting field: two-dimensional representation with kernel $\langle a^{2}x\rangle$	1	2	must not contain a primitive fourth root of unity, or equivalently, $t^{2}+1$ does not split	splits into the one-dimensional representations with kernel $\langle a^{2}x\rangle$	non-central Z4 in M16	cyclic group:Z4	2	1	indecomposable but not irreducible.
not over a splitting field: two-dimensional representation with kernel $\langle a^{2}x\rangle$	1	2	must not contain a primitive fourth root of unity, or equivalently, $t^{2}+1$ does not split	splits into the one-dimensional representations with kernel $\langle a^{2}x\rangle$	Klein four-group	cyclic group:Z4	2	1	indecomposable but not irreducible.
not over a splitting field: four-dimensional representation	1	2	must not contain a primitive fourth root of unity, or equivalently, $t^{2}+1$ does not split	splits into the two faithful irreducible representations of degree two	trivial subgroup	M16	4	1

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

Below is the character table over a splitting field. Here $i$ denotes a square root of $-1$ in the field.

Representation/conjugacy class and size	$\{e\}$ (size 1)	$\{a^{4}\}$ (size 1)	$\{a^{2}\}$ (size 1)	$\{a^{6}\}$ (size 1)	$\{a,a^{5}\}$ (size 2)	$\{a^{3},a^{7}\}$ (size 2)	$\{ax,a^{5}x\}$ (size 2)	$\{a^{3}x,a^{7}x\}$ (size 2)	$\{x,a^{4}x\}$ (size 2)	$\{a^{2}x,a^{6}x\}$ (size 2)
trivial	1	1	1	1	1	1	1	1	1	1
$\langle a^{2},x\rangle$-kernel	1	1	1	1	-1	-1	-1	-1	1	1
$\langle a\rangle$-kernel	1	1	1	1	1	1	-1	-1	-1	-1
$\langle ax\rangle$-kernel	1	1	1	1	-1	-1	1	1	-1	-1
$\langle a^{2}x\rangle$-kernel (first)	1	1	-1	-1	$i$	$-i$	$-i$	$i$	-1	1
$\langle a^{2}x\rangle$-kernel (second)	1	1	-1	-1	$-i$	$i$	$i$	$-i$	-1	1
$\langle a^{4},x\rangle$-kernel (first)	1	1	-1	-1	$i$	$-i$	$i$	$-i$	1	-1
$\langle a^{4},x\rangle$-kernel (second)	1	1	-1	-1	$-i$	$i$	$-i$	$i$	1	-1
faithful irreducible representation of M16 (first)	2	-2	$2i$	$-2i$	0	0	0	0	0	0
faithful irreducible representation of M16 (second)	2	-2	$-2i$	$2i$	0	0	0	0	0	0

Below are the size-degree-weighted characters, obtained by multiplying each character value by the size of the conjugacy class and dividing by the degree of the irreducible representation:

Representation/conjugacy class and size	$\{e\}$ (size 1)	$\{a^{4}\}$ (size 1)	$\{a^{2}\}$ (size 1)	$\{a^{6}\}$ (size 1)	$\{a,a^{5}\}$ (size 2)	$\{a^{3},a^{7}\}$ (size 2)	$\{ax,a^{5}x\}$ (size 2)	$\{a^{3}x,a^{7}x\}$ (size 2)	$\{x,a^{4}x\}$ (size 2)	$\{a^{2}x,a^{6}x\}$ (size 2)
trivial	1	1	1	1	2	2	2	2	2	2
$\langle a\rangle$-kernel	1	1	1	1	2	2	-2	-2	-2	-2
$\langle ax\rangle$-kernel	1	1	1	1	-2	-2	2	2	-2	-2
$\langle a^{2},x\rangle$-kernel	1	1	1	1	-2	-2	-2	-2	2	2
$\langle a^{2}x\rangle$-kernel (first)	1	1	-1	-1	$2i$	$-2i$	$-2i$	$2i$	-2	2
$\langle a^{2}x\rangle$-kernel (second)	1	1	-1	-1	$-2i$	$2i$	$2i$	$-2i$	-2	2
$\langle a^{4},x\rangle$-kernel (first)	1	1	-1	-1	$2i$	$-2i$	$2i$	$-2i$	2	-2
$\langle a^{4},x\rangle$-kernel (second)	1	1	-1	-1	$-2i$	$2i$	$-2i$	$2i$	2	-2
faithful irreducible representation of M16	1	-1	$i$	$-i$	0	0	0	0	0	0
faithful irreducible representation of M16 (second)	1	-1	$-i$	$i$	0	0	0	0	0	0