# Linear representation theory of extraspecial groups

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## Contents |

This article gives specific information, namely, linear representation theory, about a family of groups, namely: extraspecial group.

View linear representation theory of group families | View other specific information about extraspecial group

This article describes the linear representation theory of extraspecial groups. An extraspecial group of order , with and a prime number, is a non-abelian group of that order such that is a cyclic subgroup of order . We can deduce from this that the quotient group is an elementary abelian group of order .

For every prime and every fixed , there are two isomorphism classes of extraspecial groups of order , known as the extraspecial group of '+' and '-' type respectively.

## Summary

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

degrees of irreducible representations over a splitting field | 1 occurs times, occurs times |

number of irreducible representations over a splitting field | See also number of irreducible representations equals number of conjugacy classes, element structure of extraspecial groups |

maximum degree of irreducible representation over a splitting field | |

lcm of degrees of irreducible representations over a splitting field | |

field generated by character values (characteristic zero) | where is a primitive root of unity. Note that when , this just becomes , indicating that the group is a rational group. |

minimal splitting field, i.e., minimal field of realization of irreducible representations (characteristic zero) | When is odd, then , same as field generated by character values. (See also odd-order p-group implies every irreducible representation has Schur index one). When , it may be strictly bigger than the field generated by character values. |

## Particular cases

extraspecial group of '+' type | linear representation theory | extraspecial group of '-' type | linear representation theory | degrees of irreducible representations over a splitting field (same for both groups) | number of irreducible representations = | |||
---|---|---|---|---|---|---|---|---|

1 | 2 | 8 | dihedral group:D8 | linear representation theory of dihedral group:D8 | quaternion group | linear representation theory of quaternion group | 1,1,1,1,2 (1 occurs 4 times, 2 occurs 1 time) | 5 |

1 | 3 | 27 | prime-cube order group:U(3,3) | linear representation theory of prime-cube order group:U(3,3) | M27 | linear representation theory of M27 | 1 occurs 9 times, 3 occurs 2 times | 11 |

2 | 2 | 32 | inner holomorph of D8 | linear representation theory of inner holomorph of D8 | central product of D8 and Q8 | linear representation theory of central product of D8 and Q8 | 1 occurs 16 times, 4 occurs 1 time | 17 |

2 | 3 | 243 | 1 occurs 81 times, 9 occurs 2 times | 83 | ||||

3 | 2 | 128 | 1 occurs 64 times, 8 occurs 1 time | 65 |

It is also noteworthy that the extraspecial groups are the *only* groups of their order that have the given degrees of irreducible representations, i.e., any group with those degrees of irreducible representations must be extraspecial.