# Linear representation theory of groups of order 27

From Groupprops

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 27.

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 27

To understand these in a broader context, see linear representation theory of groups of prime-cube order |linear representation theory of groups of order 3^n

Group | GAP ID 2nd part | Linear representation theory page |
---|---|---|

cyclic group:Z27 | 1 | linear representation theory of cyclic group:Z27 |

direct product of Z9 and Z3 | 2 | linear representation of direct product of Z9 and Z3 |

prime-cube order group:U(3,3) | 3 | linear representation theory of prime-cube order group:U(3,3) |

M27 (semidirect product of Z9 and Z3) | 4 | linear representation theory of M27 |

elementary abelian group:E27 | 5 | linear representation theory of elementary abelian group:E27 |

## 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

Group | GAP ID second part | Degrees as list | Number of irreps of degree 1 (= order of abelianization) | Number of irreps of degree | Total number of irreps (= number of conjugacy classes) |
---|---|---|---|---|---|

cyclic group:Z27 | 1 | 1 (27 times) | 27 | 0 | 27 |

direct product of Z9 and Z3 | 2 | 1 (27 times) | 27 | 0 | 27 |

prime-cube order group:U(3,3) | 3 | 1 (9 times), 3 (2 times) | 9 | 2 | 11 |

M27 | 4 | 1 (9 times), 3 (2 times) | 9 | 2 | 11 |

elementary abelian group:E27 | 5 | 1 (27 times) | 27 | 0 | 27 |