# Linear representation theory of maximal unipotent subgroup of symplectic group of degree four over a finite field

From Groupprops

This article gives specific information, namely, linear representation theory, about a family of groups, namely: maximal unipotent subgroup of symplectic group of degree four. This article restricts attention to the case where the underlying ring is a finite field.

View linear representation theory of group families | View other specific information about maximal unipotent subgroup of symplectic group of degree four | View other specific information about group families for rings of the type finite field

Let be a prime power with underlying prime (we may have ). This page covers the linear representation theory of the maximal unipotent subgroup of the symplectic group of degree four (in our naming convention, this stands for the group of symplectic matrices) over the finite field of size . This group can also be described as the -Sylow subgroup of .

## Summary

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

number of conjugacy classes (equals number of irreducible representations over a splitting field) | Case even (i.e., a power of 2): Case odd: See number of irreducible representations equals number of conjugacy classes, element structure of maximal unipotent subgroup of symplectic group of degree four over a finite field |

degrees of irreducible representations over a splitting field (such as or ) | Case even (i.e., a power of 2): 1 (occurs times), (occurs times), (occurs times). NOTE: For , , so we have to count both types of occurences, and we get 1 (occurs times), 2 (occurs 2 times)Case odd: 1 (occurs times), (occurs times) |

sum of squares of degrees of irreducible representations | (equals order of the group) see sum of squares of degrees of irreducible representations equals order of group |

lcm of degrees of irreducible representations | |

condition for a field (characteristic not equal to ) to be a splitting field | The polynomial should split completely. For a finite field of size , this is equivalent to . |

field generated by character values, which in this case also coincides with the unique minimal splitting field (characteristic zero) | Field where is a primitive root of unity. This is a degree extension of the rationals. |

unique minimal splitting field (characteristic ) | The field of size where is the order of mod . |

## Related information

- Linear representation theory of maximal unipotent subgroup of symplectic group of degree six over a finite field
- Linear representation theory of maximal unipotent subgroup of symplectic group over a finite field

## Particular cases

Field size (a prime power) | Field characteristic | Maximal unipotent subgroup of | Order of the group (equals ) | Second part of GAP ID | Number of irreducible representations equls number of conjugacy classes (equals if , equals otherwise) | Degrees of irreducible representations (check against summary) | Information on linear representation theory | |
---|---|---|---|---|---|---|---|---|

2 | 2 | 1 | direct product of D8 and Z2 | 16 | 11 | 10 | 1 (occurs 8 times), 2 (occurs 2 times) | linear representation theory of direct product of D8 and Z2 |

3 | 3 | 1 | wreath product of Z3 and Z3 | 81 | 7 | 17 | 1 (occurs 9 times), 3 (occurs 8 times) | linear representation theory of wreath product of Z3 and Z3 |

4 | 2 | 2 | free product of class two of two Klein four-groups | 256 | 8935 | 58 | 1 (occurs 16 times), 2 (occurs 36 times), 4 (occurs 6 times) | linear representation theory of free product of class two of two Klein four-groups |

5 | 5 | 1 | maximal unipotent subgroup of symplectic group:Sp(4,5) | 625 | 7 | 49 | 1 (occurs 25 times), 5 (occurs 24 times) | linear representation theory of maximal unipotent subgroup of symplectic group:Sp(4,5) |