Linear representation theory of direct product of Z4 and Z2

This article gives specific information, namely, linear representation theory, about a particular group, namely: direct product of Z4 and Z2.
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Summary

Item	Summary
degrees of irreducible representations over a splitting field	1,1,1,1,1,1,1,1 (1 occurs 8 times) maximum: 1, lcm: 1, number: 8, sum of squares: 8
Schur index values of irreducible representations over a splitting field	1,1,1,1,1,1,1,1 (1 occurs 8 times)
smallest ring of realization of irreducible representations (characteristic zero)	$\mathbb {Z} [i]$ (the ring of Gaussian integers), equivalently $\mathbb {Z} [t]/(t^{2}+1)$ Same as ring generated by character values
minimal splitting field, i.e., smallest field of realization of irreducible representations (characteristic zero)	$\mathbb {Q} (i)$, equivalently $\mathbb {Q} [x]/(x^{2}+1)$ Same as field generated by character values, also the unique smallest sufficiently large field. All these coincide because it's an abelian group.
condition for a field to be a splitting field	Characteristic not $2$, and the polynomial $t^{2}+1$ should split over the field. For a finite field of size $q$, equivalent to $4$ dividing $q-1$
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, field with five elements
degrees of irreducible representations over a non-splitting field, such as the field of real numbers $\mathbb {R}$, the field of rational numbers $\mathbb {Q}$	1,1,1,1,2,2 maximum: 2, lcm: 2, number: 6
orbit structure of irreducible representations over a splitting field under the automorphism group	orbits of sizes 1,1,2,2,2 number: 5