Faithful irreducible representation of M16

This article describes a particular irreducible linear representation for the following group: M16. The representation is unique up to equivalence of linear representations and is irreducible, at least over its original field of definition in characteristic zero. The representation may also be definable over other characteristics by reducing the matrices modulo that characteristic, though it may behave somewhat differently in these characteristics.
For more on the linear representation theory of the group, see linear representation theory of M16.

We use the group with presentation (here $e$ denotes the identity element):

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

Summary

This is actually a collection of two faithful irreducible two-dimensional representations of the group $M_{16}$, which form a single orbit under the action of the automorphism group, and also form a single orbit under the action of Galois automorphisms in the field of realization.

Item	Value
degree of representation (dimension of space on which it is realized, or order of matrices)	2
Schur index value of representation	1
Kernel of representation	trivial subgroup, i.e., it is a faithful linear representation
Quotient on which it descends to a faithful linear representation	M16
Set of character values	$\{2,-2,0,2i,-2i\}$ where $i$ is a square root of $-1$ Characteristic zero: Ring generated -- $\mathbb {Z} [2i]$, Ideal within ring generated -- $2\mathbb {Z} +2i\mathbb {Z}$, Field generated -- $\mathbb {Q} (i)$
Rings of realization	The representation can be realized precisely over those rings that contain a square root of $-1$.
Fields of realization	The representation can be realized precisely over those fields that contain a square root of $-1$. For a finite field with $q$ elements ($q$ odd), this is equivalent to requiring that $4$ divide $q-1$.
Minimal field of realization	In characteristic zero: $\mathbb {Q} (i)=\mathbb {Q} ({\sqrt {-1}})=\mathbb {Q} [t]/(t^{2}+1)$ In characteristic $p\equiv 1{\pmod {4}}$: $\mathbb {F} _{p}$ In characteristic $p\equiv 1{\pmod {4}}$: $\mathbb {F} _{p^{2}}$
Size of equivalence class under automorphisms	2. An automorphism that flips the two representations is $a\mapsto a^{-1},x\mapsto x$.
Size of equivalence class under Galois automorphisms	characteristic zero or $p\equiv 3{\pmod {4}}$: 2. The automorphism interchanging the square roots of -1 over the prime subfield interchanges the two representations. characteristic $p\equiv 1{\pmod {4}}$: 1. The two representations cannot be interchanged, because the two square roots of -1 cannot be interchanged as they live in the prime subfield.
Smallest size field of realization (characteristic not two)	field:F5