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	$\mathbb {Z} [2i]=\mathbb {Z} [{\sqrt {-4}}]=\mathbb {Z} [t]/(t^{2}+4)$
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 $q\equiv 1{\pmod {4}}$
smallest 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	1,1,1,1,1,1,1,1,4 (1 occurs 8 times, 4 occurs 1 time)