# Linear representation theory of groups of order 96

From Groupprops

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 96.

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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

There are 29 possibilities for the degrees of irreducible representations for the 231 groups of order 96.

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 | Total number of irreducible representations = number of conjugacy classes | Number of groups with these irreps | Nilpotency class(es) attained | Derived lengths attained | Description of groups | List of GAP IDs (second part) |
---|---|---|---|---|---|---|---|---|---|---|

96 | 0 | 0 | 0 | 0 | 96 | 7 | 1 | 1 | the abelian groups | 2, 46, 59, 161, 176, 220, 231 |

48 | 12 | 0 | 0 | 0 | 60 | 15 | 2 | 2 | ? | [SHOW MORE] |

48 | 0 | 0 | 3 | 0 | 51 | 2 | 2 | 2 | ? | 224, 225 |

32 | 16 | 0 | 0 | 0 | 48 | 12 | not nilpotent | 2 | ? | [SHOW MORE] |

24 | 18 | 0 | 0 | 0 | 42 | 19 | 2, 3 | 2 | ? | [SHOW MORE] |

24 | 6 | 0 | 3 | 0 | 33 | 5 | 3 | 2 | ? | 49, 50, 51, 183, 184 |

24 | 0 | 8 | 0 | 0 | 32 | 3 | not nilpotent | 2 | ? | 73, 196, 228 |

16 | 20 | 0 | 0 | 0 | 36 | 25 | not nilpotent | 2 | ? | [SHOW MORE] |

16 | 12 | 0 | 2 | 0 | 30 | 17 | not nilpotent | 2 | ? | [SHOW MORE] |

16 | 8 | 0 | 3 | 0 | 27 | 4 | not nilpotent | 2 | ? | 211, 214, 216, 217 |

12 | 21 | 0 | 0 | 0 | 33 | 3 | 4 | 2 | ? | 61, 62, 63 |

12 | 12 | 4 | 0 | 0 | 28 | 4 | not nilpotent | 3 | ? | 69, 74, 198, 200 |

12 | 3 | 4 | 0 | 1 | 20 | 2 | not nilpotent | 2 | ? | 197, 199 |

12 | 0 | 4 | 3 | 0 | 19 | 2 | not nilpotent | 3 | ? | 201, 202 |

8 | 22 | 0 | 0 | 0 | 30 | 19 | not nilpotent | 2 | ? | [SHOW MORE] |

8 | 14 | 0 | 2 | 0 | 24 | 38 | not nilpotent | 2 | ? | [SHOW MORE] |

8 | 10 | 0 | 3 | 0 | 21 | 18 | not nilpotent | 2 | ? | [SHOW MORE] |

8 | 6 | 0 | 4 | 0 | 18 | 4 | not nilpotent | 2 | ? | 118, 121, 122, 125 |

8 | 4 | 8 | 0 | 0 | 20 | 4 | not nilpotent | 3 | ? | 65, 186, 194, 226 |

6 | 0 | 10 | 0 | 0 | 16 | 2 | not nilpotent | 2 | ? | 68, 229 |

6 | 0 | 2 | 0 | 2 | 10 | 3 | not nilpotent | 2 | ? | 70, 71, 72 |

4 | 23 | 0 | 0 | 0 | 27 | 3 | not nilpotent | 2 | ? | 6, 7, 8 |

4 | 11 | 0 | 3 | 0 | 18 | 4 | not nilpotent | 2 | ? | 33, 34, 35, 36 |

4 | 6 | 4 | 2 | 0 | 16 | 5 | not nilpotent | 4 | ? | 66, 67, 188, 189, 192 |

4 | 5 | 4 | 0 | 1 | 14 | 3 | not nilpotent | 3 | ? | 185, 187, 195 |

4 | 2 | 4 | 3 | 0 | 13 | 3 | not nilpotent | 4 | ? | 190, 191, 193 |

3 | 3 | 5 | 0 | 1 | 12 | 2 | not nilpotent | 3 | ? | 3, 203 |

3 | 0 | 5 | 3 | 0 | 11 | 1 | not nilpotent | 3 | ? | 204 |

2 | 1 | 6 | 0 | 1 | 10 | 2 | not nilpotent | 3 | ? | 64, 227 |