# Linear representation theory of central product of D8 and Z4

## Contents

This article gives specific information, namely, linear representation theory, about a particular group, namely: central product of D8 and Z4.
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## 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
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