Linear representation theory of direct product of Z4 and Z2

From Groupprops
Jump to: navigation, search

Contents

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

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