# Linear representation theory of direct product of Z4 and Z2

## Contents

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 $p \ne 0,2$ 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, field with five elements
degrees of irreducible representations over a non-splitting field, such as the field of real numbers $\R$, 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