# Linear representation theory of central product of D8 and Z4

From Groupprops

## Contents |

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

View linear representation theory of particular groups | View other specific information about central product of D8 and Z4

## Summary

Item | Value |
---|---|

degrees of irreducible representations over a splitting field (such as or ) | 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) | -- ring of Gaussian integers Same as ring generated by character values |

minimal splitting field, i.e., smallest field of realization (characteristic zero) | 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 should split. For a finite field of size , this is equivalent to saying that |

minimal splitting field in characteristic | Case : prime field Case : Field , 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 |