# Linear representation theory of direct product of Z4 and Z2

From Groupprops

## 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) | (the ring of Gaussian integers), equivalently Same as ring generated by character values |

minimal splitting field, i.e., smallest field of realization of irreducible representations (characteristic zero) | , equivalently 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 , and the polynomial should split over the field. For a finite field of size , equivalent to dividing |

minimal splitting field in characteristic | Case : prime field Case : field , 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 , the field of rational numbers | 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 |