Linear representation theory of central product of D8 and Z4

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This article gives specific information, namely, linear representation theory, about a particular group, namely: central product of D8 and Z4.
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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
Same as ring generated by character values
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 \ne 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)
number: 9
orbits of irreducible representations over a splitting field under action of automorphism group 2 orbits of size 1 of degree 1 representations, 2 orbits of size 3 of degree 1 representations, 1 orbit of size 2 of degree 2 representations
number: 5