# Linear representation theory of groups of order 64

From Groupprops

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 64.

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 64

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

## 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 64 mod 3, and hence congruent to 1 mod 3.

Number of irreps of degree 1 | Number of irreps of degree 2 | Number of irreps of degree 4 | Total number of irreps (= number of conjugacy classes) | Total number of groups | Nilpotency class(es) attained by these | Hall-Senior family/families | List of GAP IDs (second part) |
---|---|---|---|---|---|---|---|

64 | 0 | 0 | 64 | 11 | 1 | , all the abelian groups of order 64 | [SHOW MORE] |

32 | 8 | 0 | 40 | 31 | 2 | [SHOW MORE] | |

32 | 0 | 2 | 34 | 7 | 2 | [SHOW MORE] | |

16 | 12 | 0 | 28 | 60 | 2,3 | (class three), (class two) | [SHOW MORE] |

16 | 8 | 1 | 25 | 15 | 2 | ? | [SHOW MORE] |

16 | 4 | 2 | 22 | 41 | 2,3 | [SHOW MORE] | |

16 | 0 | 3 | 19 | 5 | 2 | ? | [SHOW MORE] |

8 | 14 | 0 | 22 | 33 | 2,3,4 | ?, | [SHOW MORE] |

8 | 10 | 1 | 19 | 31 | 3 | [SHOW MORE] | |

8 | 6 | 2 | 16 | 24 | 3,4 | [SHOW MORE] | |

8 | 2 | 3 | 13 | 6 | 4 | [SHOW MORE] | |

4 | 15 | 0 | 19 | 3 | 5 | maximal class groups | [SHOW MORE] |