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:

$G=M_{16}:=⟨a,x\mid a^8=x^2=e,xa{x}^{-1}=a^5⟩$

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 $q\equiv 1(\mathrm{mod}4)$
smallest splitting field in characteristic $p\ne 0,2$	Case $p\equiv 1(\mathrm{mod}4)$: prime field ${\mathbb{F}}_p$ Case $p\equiv 3(\mathrm{mod}4)$: 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)