# Linear representation theory of semidihedral group:SD16

This article gives specific information, namely, linear representation theory, about a particular group, namely: semidihedral group:SD16.

View linear representation theory of particular groups | View other specific information about semidihedral group:SD16

## Summary

We shall use the semidihedral group of order 16 with the following presentation:

.

Item | Value |
---|---|

degrees of irreducible representations over a splitting field (e.g., or ) | 1,1,1,1,2,2,2 (1 occurs 4 times, 2 occurs 3 times) maximum: 2, lcm: 2, number: 7, sum of squares: 16 |

Schur index values of irreducible representations over a splitting field | 1,1,1,1,1,1,1 |

smallest ring of realization (characteristic zero) | = = Same as ring generated by character values. |

minimal splitting field, i.e., smallest field of realization (characteristic zero) | = = Same as field generated by character values, because all Schur index values are 1. See minimal splitting field need not be cyclotomic |

condition for a field to be a splitting field | The characteristic should not be equal to 2, and the polynomial must split. For a finite field of size , this is equivalent to saying that or . |

minimal splitting field (characteristic ) | Case or : prime field Case or : Field , quadratic extension of prime field. |

smallest size splitting field | Field:F3. |

degrees of irreducible representations over the rational numbers | 1,1,1,1,2,4 |

## Irreducible representations

### Summary information

Below is summary information on irreducible representations that are absolutely irreducible, i.e., they remain irreducible in any bigger field, and in particular are irreducible in a splitting field. We assume that the characteristic of the field is not 2, except in the last column, where we consider what happens in characteristic 2.

Name of representation type | Number of representations of this type | Degree | Schur index | Criterion for field | Kernel (a normal subgroup of semidihedral group:SD16 -- see subgroup structure of semidihedral group:SD16) | Quotient by kernel (on which it descends to a faithful representation) | Characteristic 2 |
---|---|---|---|---|---|---|---|

trivial | 1 | 1 | 1 | any | whole group | trivial group | works |

sign representation with kernel | 1 | 1 | 1 | any | Z8 in SD16: | cyclic group:Z2 | works, same as trivial |

sign representation with kernel a maximal dihedral subgroup | 1 | 1 | 1 | any | D8 in SD16: | cyclic group:Z2 | works, same as trivial |

sign representation with kernel a maximal quaternion subgroup | 1 | 1 | 1 | any | Q8 in SD16: | cyclic group:Z2 | works, same as trivial |

two-dimensional irreducible, not faithful | 1 | 2 | 1 | any | center of semidihedral group:SD16: | dihedral group:D8 | indecomposable but not irreducible |

two-dimensional faithful irreducible | 2 | 2 | 1 | The polynomial must split, i.e., must have a square root | trivial subgroup, i.e., it is a faithful linear representation | semidihedral group:SD16 | ? |

Below are representations that are irreducible over a non-splitting field, but split over a splitting field.

Name of representation type | Number of representations of this type | Degree | Criterion for field | What happens over a splitting field? | Kernel | Quotient by kernel (on which it descends to a faithful representation) |
---|---|---|---|---|---|---|

four-dimensional faithful irreducible | 1 | 4 | The polynomial must not split, i.e., must not have a square root |
splits into the two two-dimensional faithful irreducibles. | trivial subgroup, i.e., it is a faithful linear representation | semidihedral group:SD16 |

### Trivial representation

The trivial representation or principal representation (whose character is called the trivial character or principal character) sends all elements of the group to the matrix :

Element | Matrix | Characteristic polynomial | Minimal polynomial | Trace, character value |
---|---|---|---|---|

1 | ||||

1 | ||||

1 | ||||

1 | ||||

1 | ||||

1 | ||||

1 | ||||

1 | ||||

1 | ||||

1 | ||||

1 | ||||

1 | ||||

1 | ||||

1 | ||||

1 | ||||

1 |

### Sign representation with kernel

This representation is a one-dimensional representation sending everything in the cyclic subgroup (see Z8 in SD16) to and everything outside it to .

To keep the description short, we club together the cosets rather than having one row per element:

Elements | Matrix | Characteristic polynomial | Minimal polynomial | Trace, character value |
---|---|---|---|---|

1 | ||||

-1 |

### Sign representation with kernel

There is a sign representation with kernel which is dihedral group:D8 (see D8 in SD16). Everything inside the subgroup goes to and everything outside the subgroup goes to .

To keep the descriptions short, we club together the cosets rather than having one row per element:

Elements | Matrix | Characteristic polynomial | Minimal polynomial | Trace, character value |
---|---|---|---|---|

1 | ||||

-1 |

### Sign representation with kernel

There is a sign representation with kernel which is quaternion group (see Q8 in SD16). Everything inside the subgroup goes to and everything outside the subgroup goes to .

To keep the descriptions short, we club together the cosets rather than having one row per element:

Elements | Matrix | Characteristic polynomial | Minimal polynomial | Trace, character value |
---|---|---|---|---|

1 | ||||

-1 |

### Two-dimensional irreducible unfaithful representation

This representation has kernel equal to -- center of semidihedral group:SD16. It descends to a faithful irreducible two-dimensional representation of the quotient group, which is isomorphic to dihedral group:D8.

To keep the descriptions short, we club together the cosets rather than having one row per element:

Element | Matrix | Characteristic polynomial | Minimal polynomial | Trace, character value | Determinant |
---|---|---|---|---|---|

2 | 1 | ||||

0 | 1 | ||||

-2 | 1 | ||||

0 | 1 | ||||

0 | -1 | ||||

0 | -1 | ||||

0 | -1 | ||||

0 | -1 |

### Two-dimensional faithful irreducible representations

### Four-dimensional faithful irreducible representation over a non-splitting field

## Character table

FACTS TO CHECK AGAINST (for characters of irreducible linear representations over a splitting field):Orthogonality relations: Character orthogonality theorem | Column orthogonality theoremSeparation results(basically says rows independent, columns independent): Splitting implies characters form a basis for space of class functions|Character determines representation in characteristic zeroNumerical facts: Characters are cyclotomic integers | Size-degree-weighted characters are algebraic integersCharacter value facts: Irreducible character of degree greater than one takes value zero on some conjugacy class| Conjugacy class of more than average size has character value zero for some irreducible character | Zero-or-scalar lemma

Below is the character table over a splitting field, where stands for any chosen square root of :

Representation/conjugacy class representative | (size 1) | (size 1) | (size 2) | (size 2) | (size 2) | (size 4) | (size 4) |
---|---|---|---|---|---|---|---|

-kernel sign | 1 | 1 | 1 | 1 | 1 | -1 | -1 |

-kernel sign | 1 | 1 | 1 | -1 | -1 | 1 | -1 |

-kernel sign | 1 | 1 | 1 | -1 | -1 | -1 | 1 |

two-dimensional unfaithful, kernel is center | 2 | 2 | -2 | 0 | 0 | 0 | 0 |

first faithful irreducible representation | 2 | -2 | 0 | 0 | 0 | ||

second faithful irreducible representation | 2 | -2 | 0 | 0 | 0 |

## GAP implementation

### Degrees of irreducible representations

These can be computed using the CharacterDegrees function:

gap> CharacterDegrees(SmallGroup(16,8)); [ [ 1, 4 ], [ 2, 3 ] ]

### Character table

The character table can be computed using the Irr and CharacterTable functions:

gap> Irr(CharacterTable(SmallGroup(16,8))); [ Character( CharacterTable( <pc group of size 16 with 4 generators> ), [ 1, 1, 1, 1, 1, 1, 1 ] ), Character( CharacterTable( <pc group of size 16 with 4 generators> ), [ 1, -1, 1, 1, 1, -1, -1 ] ), Character( CharacterTable( <pc group of size 16 with 4 generators> ), [ 1, 1, -1, 1, 1, -1, -1 ] ), Character( CharacterTable( <pc group of size 16 with 4 generators> ), [ 1, -1, -1, 1, 1, 1, 1 ] ), Character( CharacterTable( <pc group of size 16 with 4 generators> ), [ 2, 0, 0, -2, 2, 0, 0 ] ), Character( CharacterTable( <pc group of size 16 with 4 generators> ), [ 2, 0, 0, 0, -2, -E(8)-E(8)^3, E(8)+E(8)^3 ] ), Character( CharacterTable( <pc group of size 16 with 4 generators> ), [ 2, 0, 0, 0, -2, E(8)+E(8)^3, -E(8)-E(8)^3 ] ) ]

A nicer tabular display can be achieved using the Display function:

gap> Display(CharacterTable(SmallGroup(16,8))); CT1 2 4 2 2 3 4 3 3 1a 4a 2a 4b 2b 8a 8b X.1 1 1 1 1 1 1 1 X.2 1 -1 1 1 1 -1 -1 X.3 1 1 -1 1 1 -1 -1 X.4 1 -1 -1 1 1 1 1 X.5 2 . . -2 2 . . X.6 2 . . . -2 A -A X.7 2 . . . -2 -A A A = -E(8)-E(8)^3 = -ER(-2) = -i2

### Irreducible representations

The irreducible representations can be computed using the IrreducibleRepresentations function:

gap> IrreducibleRepresentations(SmallGroup(16,8)); [ Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ], [ [ 1 ] ] ] , Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ -1 ] ], [ [ -1 ] ], [ [ 1 ] ], [ [ 1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 0, 1 ], [ 1, 0 ] ], [ [ 1, 0 ], [ 0, -1 ] ], [ [ -1, 0 ], [ 0, -1 ] ], [ [ 1, 0 ], [ 0, 1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 0, -E(8) ], [ -E(8)^3, 0 ] ], [ [ 0, 1 ], [ 1, 0 ] ], [ [ E(4), 0 ], [ 0, -E(4) ] ], [ [ -1, 0 ], [ 0, -1 ] ] ], Pcgs([ f1, f2, f3, f4 ]) -> [ [ [ 0, E(8) ], [ E(8)^3, 0 ] ], [ [ 0, 1 ], [ 1, 0 ] ], [ [ E(4), 0 ], [ 0, -E(4) ] ], [ [ -1, 0 ], [ 0, -1 ] ] ] ]