# Linear representation theory of groups of order 16

This article gives specific information, namely, linear representation theory, about a family of groups, namely: groups of order 16.

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 16

## 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

### Full listing

The following sets of degrees of irreducible representations works over any splitting field not of characteristic two.

See also nilpotency class and order determine degrees of irreducible representations for groups up to prime-fourth order. This says that for groups of order , the nilpotency class of the group, and the order, together determine the degrees of irreducible representations. In particular, for groups of order 16, there are three cases: the abelian case, where there are 16 of degree one, the class two case, where there are 8 of degree one and 2 of degree two, and the class three case, where there are 4 of degree one and 3 of degree two.

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 16 mod 3, and hence congruent to 1 mod 3.

Group | GAP ID second part | Hall-Senior number | Hall-Senior symbol | Nilpotency class | Degrees as list | Number of irreps of degree 1 (= order of abelianization) | Number of irreps of degree 2 | Total number of irreps (= number of conjugacy classes) |
---|---|---|---|---|---|---|---|---|

cyclic group:Z16 | 1 | 5 | 1 | 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 | 16 | 0 | 16 | |

direct product of Z4 and Z4 | 2 | 3 | 1 | 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 | 16 | 0 | 16 | |

SmallGroup(16,3) | 3 | 9 | 2 | 1,1,1,1,1,1,1,1,2,2 | 8 | 2 | 10 | |

nontrivial semidirect product of Z4 and Z4 | 4 | 10 | 2 | 1,1,1,1,1,1,1,1,2,2 | 8 | 2 | 10 | |

direct product of Z8 and Z2 | 5 | 4 | 1 | 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 | 16 | 0 | 16 | |

M16 | 6 | 11 | 2 | 1,1,1,1,1,1,1,1,2,2 | 8 | 2 | 10 | |

dihedral group:D16 | 7 | 12 | 3 | 1,1,1,1,2,2,2 | 4 | 3 | 7 | |

semidihedral group:SD16 | 8 | 13 | 3 | 1,1,1,1,2,2,2 | 4 | 3 | 7 | |

generalized quaternion group:Q16 | 9 | 14 | 3 | 1,1,1,1,2,2,2 | 4 | 3 | 7 | |

direct product of Z4 and V4 | 10 | 2 | 1 | 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 | 16 | 0 | 16 | |

direct product of D8 and Z2 | 11 | 6 | 2 | 1,1,1,1,1,1,1,1,2,2 | 8 | 2 | 10 | |

direct product of Q8 and Z2 | 12 | 7 | 2 | 1,1,1,1,1,1,1,1,2,2 | 8 | 2 | 10 | |

central product of D8 and Z4 | 13 | 8 | 2 | 1,1,1,1,1,1,1,1,2,2 | 8 | 2 | 10 | |

elementary abelian group:E16 | 14 | 1 | 1 | 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 | 16 | 0 | 16 |

### Grouping by Hall-Senior families

Note that isoclinic groups have same proportions of degrees of irreducible representations, and in particular isoclinic groups of the *same* order have precisely the same degrees of irreducible representations. Thus, all groups in the same Hall-Senior family have the same degrees of irreducible representations. However, the same multiset of degrees of irreducible representations could be attained by more than one Hall-Senior family, though this phenomenon does not occur for order 16.

For the first two Hall-Senior families , there are isoclinic groups of smaller order, hence the degrees of irreducible representations can be computed by first computing the degrees of irreducible representations of those isoclinic groups of smaller order and then scaling up the proportions based on the order. For instance, dihedral group:D8 of order 8 and family has 4 irreps of degree 1 and 1 of degree 2, so the groups in family and of order 16 have irreps of degree 1 and irreps of degree 2.

For more background on the Hall-Senior families business, see Groups of order 16#Families and classification.

Family name | Isomorphism class of inner automorphism group | Isomorphism class of derived subgroup | Number of members | Nilpotency class | Members | Second part of GAP ID of members (sorted ascending) | Hall-Senior numbers of members (sorted ascending) | Number of irreps of degree 1 (= order of abelianization) | Number of irreps of degree 2 | Total number of irreps | Smallest order of group isoclinic to it | Degrees of irreps of smallest order isoclinic stem group |
---|---|---|---|---|---|---|---|---|---|---|---|---|

trivial group | trivial group | 5 | 1 | abelian groups: [SHOW MORE] | 1, 2, 5, 10, 14 | 1--5 | 16 | 0 | 16 | 1 | 1 | |

Klein four-group | cyclic group:Z2 | 6 | 2 | [SHOW MORE] | 3, 4, 6, 11, 12, 13 | 6--11 | 8 | 2 | 10 | 8 | 1 (4 times), 2 (1 time) | |

dihedral group:D8 | cyclic group:Z4 | 3 | 3 | maximal class groups: [SHOW MORE] | 7, 8, 9 | 12--14 | 4 | 3 | 7 | 16 | 1 (4 times), 2 (3 times) |

### Grouping by degrees of irreducible representations

Note that isoclinic groups have same proportions of degrees of irreducible representations, and in particular isoclinic groups of the *same* order have precisely the same degrees of irreducible representations. Thus, all groups in the same Hall-Senior family have the same degrees of irreducible representations.

For order 16 (and more generally for fixed order with prime and ), the degrees of irreducible representations uniquely determine the Hall-Senior family, i.e., different Hall-Senior families have different degrees of irreducible representations. This breaks down for and in particular for 32, where there are some cases of multiple Hall-Senior families having the same degrees of irreducible representations.

Number of irreps of degree 1 | Number of irreps of degree 2 | Total number of irreps | Total number of groups | Nilpotency class(es) attained by these | List of groups | List of GAP IDs (ascending order) | List of Hall-Senior numbers (ascending order) | List of Hall-Senior families (equivalence classes under isoclinism) | Order of inner automorphism group = index of center (bounds square of degree of irreducible representation) | Minimum possible index of abelian normal subgroup (degree of irreducible representation divides index of abelian normal subgroup) |
---|---|---|---|---|---|---|---|---|---|---|

16 | 0 | 16 | 5 | 1 | [SHOW MORE] | 1, 2, 5, 10, 14 | 1--5 | (abelian groups) | 1 | 1 |

8 | 2 | 10 | 6 | 2 | [SHOW MORE] | 3, 4, 6, 11, 12, 13 | 6--11 | 4 | 2 | |

4 | 3 | 7 | 3 | 3 | [SHOW MORE] | 7, 8, 9 | 12--14 | (maximal class groups) | 8 | 2 |

## Splitting field

### Characteristic zero case

Note that for the abelian cases, the smallest field of realization of representations is the same as the field generated by the character values, because the irreducible representations are all one-dimensional and can be identified with their character values.

Key shorthands in this table:

- stands for
- stands for
- is the same as
- is the same as

### Grouping by minimal splitting field

Note that since minimal splitting field need not be unique, some groups have multiple minimal splitting fields. All the minimal splitting fields for direct product of Q8 and Z2 and generalized quaternion group:Q16 are not mentioned.

### Grouping by field generated by character values

### Rationals and reals: properties

This table can be completely reconstructed based on the above/previous tables, but is included for additional clarity.

Group | GAP ID 2nd part | Hall-Senior number | nilpotency class | rational-representation group (all representations realized over rationals)? | rational group (all characters take rational values)? | real-representation group (all representations realized over reals)? | ambivalent group (all characters take real values)? |
---|---|---|---|---|---|---|---|

cyclic group:Z16 | 1 | 5 | 1 | No | No | No | No |

direct product of Z4 and Z4 | 2 | 3 | 1 | No | No | No | No |

SmallGroup(16,3) | 3 | 9 | 2 | No | No | No | No |

nontrivial semidirect product of Z4 and Z4 | 4 | 10 | 2 | No | No | No | No |

direct product of Z8 and Z2 | 5 | 4 | 1 | No | No | No | No |

M16 | 6 | 11 | 2 | No | No | No | No |

dihedral group:D16 | 7 | 12 | 3 | No | No | Yes | Yes |

semidihedral group:SD16 | 8 | 13 | 3 | No | No | No | No |

generalized quaternion group | 9 | 14 | 3 | No | No | No | Yes |

direct product of Z4 and V4 | 10 | 2 | 1 | No | No | No | No |

direct product of D8 and Z2 | 11 | 6 | 2 | Yes | Yes | Yes | Yes |

direct product of Q8 and Z2 | 12 | 7 | 2 | No | Yes | No | Yes |

central product of D8 and Z4 | 13 | 8 | 2 | No | No | No | No |

elementary abelian group:E16 | 14 | 1 | 1 | Yes | Yes | Yes | Yes |

### General case

**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]

## Relation with other orders

### Divisors of the order

Divisor | Quotient value | Information on linear representation theory | Relationship (subgroup perspective) | Relationship (quotient perspective) |
---|---|---|---|---|

2 | 8 | linear representation theory of cyclic group:Z2 | ||

4 | 4 | linear representation theory of groups of order 4 (linear representation theory of cyclic group:Z4, linear representation theory of Klein four-group) | ||

8 | 2 | linear representation theory of groups of order 8 |

### Multiples of the order

Multiplier (other factor) | Multiple | Information on linear representation theory | Relationship (subgroup perspective) | Relationship (quotient perspective) |
---|---|---|---|---|

2 | 32 | linear representation theory of groups of order 32 | ||

3 | 48 | linear representation theory of groups of order 48 | ||

4 | 64 | linear representation theory of groups of order 64 | ||

5 | 80 | linear representation theory of groups of order 80 | ||

6 | 96 | linear representation theory of groups of order 96 |