# Linear representation theory of groups of order 128

From Groupprops

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 128.

View linear representation theory of group families | View linear representation theory of groups of a particular order |View other specific information about groups of order 128

This article describes the linear representation theory of groups of order 128. There are a total of 2328 groups of order 128, so we do not present information group by group but rather present key summary information.

To understand these in a broader context, see linear representation theory of groups of prime-seventh order |linear representation theory of groups of order 2^n

## Degrees of irreducible representations

FACTS TO CHECK AGAINST FOR DEGREES OF IRREDUCIBLE REPRESENTATIONS OVER SPLITTING FIELD:Divisibility facts: degree of irreducible representation divides group order | degree of irreducible representation divides index of abelian normal subgroupSize bounds: order of inner automorphism group bounds square of degree of irreducible representation| degree of irreducible representation is bounded by index of abelian subgroup| maximum degree of irreducible representation of group is less than or equal to product of maximum degree of irreducible representation of subgroup and index of subgroupCumulative facts: sum of squares of degrees of irreducible representations equals order of group | number of irreducible representations equals number of conjugacy classes | number of one-dimensional representations equals order of abelianization

### Grouping by degrees of irreducible representations

Note that since number of conjugacy classes in group of prime power order is congruent to order of group modulo prime-square minus one, all the values for the total number of irreducible representations, which equals the number of conjugacy classes, is congruent to 128 mod 3, and hence congruent to 2 mod 3.

Number of irreps of degree 1 | Number of irreps of degree 2 | Number of irreps of degree 4 | Number of irreps of degree 8 | Total number of irreps (= number of conjugacy classes) | Total number of groups | Nilpotency class(es) attained by these | Derived length(s) attained by these | Description of groups |
---|---|---|---|---|---|---|---|---|

128 | 0 | 0 | 0 | 128 | 15 | 1 | 1 | abelian groups |

64 | 16 | 0 | 0 | 80 | 60 | 2 | 2 | |

64 | 0 | 4 | 0 | 68 | 21 | 2 | 2 | |

64 | 0 | 0 | 1 | 65 | 2 | 2 | 2 | extraspecial groups |

32 | 24 | 0 | 0 | 56 | 158 | 2,3 | 2 | |

32 | 0 | 6 | 0 | 38 | 57 | 2 | 2 | |

32 | 16 | 2 | 0 | 50 | 80 | 2 | 2 | |

32 | 8 | 4 | 0 | 44 | 204 | 2,3 | 2 | |

32 | 8 | 0 | 1 | 41 | 6 | 3 | 2 | |

16 | 28 | 0 | 0 | 44 | 149 | 2,3,4 | 2 | |

16 | 20 | 2 | 0 | 38 | 361 | 2,3 | 2 | |

16 | 16 | 3 | 0 | 35 | 50 | 3 | 2 | |

16 | 12 | 4 | 0 | 32 | 538 | 2,3,4 | 2 | |

16 | 12 | 0 | 1 | 29 | 25 | 3 | 2 | |

16 | 8 | 5 | 0 | 29 | 88 | 3 | 2 | |

16 | 8 | 1 | 1 | 26 | 2 | 3 | 2 | |

16 | 4 | 6 | 0 | 26 | 134 | 2,3,4 | 2 | |

16 | 4 | 2 | 1 | 21 | 11 | 4 | 2 | |

8 | 30 | 0 | 0 | 38 | 32 | 3,4,5 | 2 | |

8 | 22 | 2 | 0 | 32 | 52 | 3 | 2 | |

8 | 18 | 3 | 0 | 29 | 54 | 3,4 | 2 | |

8 | 14 | 0 | 1 | 23 | 10 | 3 | 2 | |

8 | 14 | 4 | 0 | 26 | 108 | 3,4,5 | 2 | |

8 | 10 | 5 | 0 | 23 | 65 | 3,4 | 2 | |

8 | 10 | 1 | 1 | 20 | 10 | 4 | 2 | |

8 | 6 | 6 | 0 | 20 | 15 | 3,4 | 2,3 | |

8 | 6 | 2 | 1 | 17 | 5 | 4 | 3 | |

8 | 2 | 7 | 0 | 17 | 9 | 5 | 2,3 | |

8 | 2 | 3 | 1 | 14 | 4 | 5 | 2 | |

4 | 31 | 0 | 0 | 35 | 3 | 6 | 2 | the maximal class groups |