# Supercharacter theories for quaternion group

From Groupprops

This article gives specific information, namely, supercharacter theories, about a particular group, namely: quaternion group.

View supercharacter theories for particular groups | View other specific information about quaternion group

This page lists the various possible supercharacter theories for the quaternion group of order eight. It builds on a thorough understanding of element structure of quaternion group, subgroup structure of quaternion group, and linear representation theory of quaternion group.

We use the standard notation for the elements of the quaternion group: the identity element is denoted 1, the non-identity element of the center is denoted -1, and the other elements are .

## Character table

Below is the character table of the quaternion group.

This character table works over characteristic zero and over any other characteristic not equal to two once we reduce the entries mod the characteristic:

Representation/Conjugacy class | (identity; size 1) | (size 1) | (size 2) | (size 2) | (size 2) |
---|---|---|---|---|---|

Trivial representation | 1 | 1 | 1 | 1 | 1 |

-kernel | 1 | 1 | 1 | -1 | -1 |

-kernel | 1 | 1 | -1 | 1 | -1 |

-kernel | 1 | 1 | -1 | -1 | 1 |

2-dimensional | 2 | -2 | 0 | 0 | 0 |

## Supercharacter theories

### Summary

Quick description of supercharacter theory | Number of such supercharacter theories under automorphism group action | Number of blocks of conjugacy classes = number of blocks of irreducible representations | Block sizes for conjugacy classses (in number of conjugacy class terms) (should add up to 5, the total number of conjugacy classes) | Block sizes for conjugacy classes (in number of elements terms) (should add up to 8, the order of the group) | Block sizes for irreducible representations (in number of representations terms) (should add up to 5, the total number of conjugacy classes) | Block sizes for irreducible representations (in sum of squares of degrees terms) (should add up to 8, the order of the group) |
---|---|---|---|---|---|---|

ordinary character theory | 1 | 5 | 1,1,1,1,1 | 1,1,2,2,2 | 1,1,1,1,1 | 1,1,1,1,4 |

all non-identity elements form one block | 1 | 2 | 1,4 | 1,7 | 1,4 | 1,7 |

supercharacter theory corresponding to normal series with middle group center of quaternion group (this is also the supercharacter theory obtained by taking the orbits of the automorphism group action) | 1 | 3 | 1,1,3 | 1,1,6 | 1,3,1 | 1,3,4 |

supercharacter theory corresponding to normal series with middle group one of the cyclic maximal subgroups of quaternion group | 3 | 3 | 1,2,2 | 1,3,4 | 1,1,3 | 1,1,6 |

superchararacter theory corresponding to normal series that goes through center of quaternion group and one of the cyclic maximal subgroups of quaternion group | 3 | 4 | 1,1,1,2 | 1,1,2,4 | 1,1,2,1 | 1,1,2,4 |

Total (5 rows) | 9 (number of supercharacter theories) | -- | -- | -- | -- | -- |