# Linear representation theory of groups of order 72

From Groupprops

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 72.

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## 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

Number of irreps of degree 1 | Number of irreps of degree 2 | Number of irreps of degree 3 | Number of irreps of degree 4 | Number of irreps of degree 6 | Number of irreps of degree 8 | Total number of irreps = number of conjugacy classes | Number of groups with these degrees of irreps | Nilpotency class(es) attained | Derived lengths attained | Description of groups | List of GAP IDs (second part) |
---|---|---|---|---|---|---|---|---|---|---|---|

72 | 0 | 0 | 0 | 0 | 0 | 72 | 6 | 1 | 1 | abelian groups | 2, 9, 14, 18, 36, 50 |

36 | 9 | 0 | 0 | 0 | 0 | 45 | 4 | 2 | 2 | ? | 10, 11, 37, 38 |

24 | 12 | 0 | 0 | 0 | 0 | 36 | 4 | not nilpotent | 2 | ? | 12, 27, 29, 48 |

18 | 0 | 6 | 0 | 0 | 0 | 24 | 2 | not nilpotent | 2 | ? | 16, 47 |

12 | 15 | 0 | 0 | 0 | 0 | 27 | 3 | not nilpotent | 2 | ? | 26, 28, 30 |

9 | 9 | 3 | 0 | 0 | 0 | 21 | 2 | not nilpotent | 3 | ? | 3, 25 |

8 | 0 | 0 | 0 | 0 | 1 | 9 | 1 | not nilpotent | 2 | ? | 39 |

8 | 0 | 0 | 4 | 0 | 0 | 12 | 2 | not nilpotent | 2 | ? | 19, 45 |

8 | 16 | 0 | 0 | 0 | 0 | 24 | 8 | not nilpotent | 2 | ? | 1, 5, 7, 13, 17, 32, 34, 49 |

8 | 8 | 0 | 2 | 0 | 0 | 18 | 3 | not nilpotent | 2 | ? | 20, 21, 46 |

6 | 3 | 6 | 0 | 0 | 0 | 15 | 1 | not nilpotent | 3 | ? | 42 |

6 | 3 | 2 | 0 | 1 | 0 | 12 | 1 | not nilpotent | 2 | ? | 44 |

4 | 17 | 0 | 0 | 0 | 0 | 21 | 6 | not nilpotent | 2 | ? | 4, 6, 8, 31, 33, 35 |

4 | 9 | 0 | 2 | 0 | 0 | 15 | 3 | not nilpotent | 2 | ? | 22, 23, 24 |

4 | 1 | 0 | 0 | 0 | 1 | 6 | 1 | not nilpotent | 3 | ? | 41 |

4 | 1 | 0 | 4 | 0 | 0 | 9 | 1 | not nilpotent | 3 | ? | 40 |

2 | 4 | 2 | 0 | 1 | 0 | 9 | 2 | not nilpotent | 3 | ? | 15, 43 |