# Linear representation theory of groups of order 60

From Groupprops

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 60.

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 60

## 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 5 | Total number of irreps = number of conjugacy classes | Number of groups with these irreps | Nilpotency class(es) attained | Derived length(s) attained | Description of groups | List of groups | List of GAP IDs (second part) |
---|---|---|---|---|---|---|---|---|---|---|---|

60 | 0 | 0 | 0 | 0 | 60 | 2 | 1 | 1 | all the abelian groups | cyclic group:Z60, direct product of V4 and Z15 | 4, 13 |

20 | 10 | 0 | 0 | 0 | 30 | 2 | not nilpotent | 2 | ? | ? | 1, 11 |

15 | 0 | 5 | 0 | 0 | 20 | 1 | not nilpotent | 2 | ? | ? | 9 |

12 | 12 | 0 | 0 | 0 | 24 | 2 | not nilpotent | 2 | ? | ? | 2, 10 |

12 | 0 | 0 | 3 | 0 | 15 | 1 | not nilpotent | 2 | ? | ? | 6 |

4 | 14 | 0 | 0 | 0 | 18 | 2 | not nilpotent | 2 | ? | ? | 3, 12 |

4 | 6 | 0 | 2 | 0 | 12 | 1 | not nilpotent | 2 | ? | ? | 8 |

4 | 2 | 0 | 3 | 0 | 11 | 1 | not nilpotent | 2 | ? | ? | 7 |

1 | 0 | 2 | 1 | 1 | 5 | 1 | not nilpotent | not solvable | The unique simple non-abelian group of this order | alternating group:A5 | 5 |