# Linear representation theory of direct product of Z8 and Z2

From Groupprops

## Contents |

This article gives specific information, namely, linear representation theory, about a particular group, namely: direct product of Z8 and Z2.

View linear representation theory of particular groups | View other specific information about direct product of Z8 and Z2

## Summary

Item | Summary |
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degrees of irreducible representations over a splitting field | 1 (16 times) maximum: 1, lcm: 1, number: 16, sum of squares: 16 |

Schur index values of irreducible representations over a splitting field | 1,1,1,1,1,1,1,1 (1 occurs 8 times) Follows from the group being abelian. |

smallest ring of realization of irreducible representations (characteristic zero) | . Same as . Same as ring generated by character values |

minimal splitting field, i.e., smallest field of realization of irreducible representations (characteristic zero) | . Same as . Same as field generated by character values, also the unique smallest sufficiently large field. All coincide because the group is abelian. |

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 : , quadratic extension of . This is because in all cases. |

smallest size splitting field | field:F9 |

degrees of irreducible representations over a non-splitting field, such as the field of real numbers , the field of rational numbers | ? |

orbits of irreducible representations over a splitting field under the action of the automorphism group | ? |