# Linear representation theory of 2-Sylow subgroup of symmetric group

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## Contents |

This article gives specific information, namely, linear representation theory, about a family of groups, namely: 2-Sylow subgroup of symmetric group.

View linear representation theory of group families | View other specific information about 2-Sylow subgroup of symmetric group

This article describes the linear representation theory of groups arising as the 2-Sylow subgroup of symmetric group.

All these groups are rational groups, i.e., any two elements of such a group that generate the same cyclic subgroup are conjugate. Hence, all the characters in characteristic zero are rational-valued. In fact, all these groups are rational-representation groups: all the representations in characteristic zero can be realized over the rational numbers.

## Particular cases

In the table below, we list only even values of . This is because for even ,the 2-Sylow subgroup of is isomorphic to the 2-Sylow subgroup of .

2-Sylow subgroup of symmetric group | Order of the group | Number of irreducible representations (= number of conjugacy classes) | Number of irreducible representations of degree one | Number of irreducible representations of degree two | Number of irreducible representations of degree four | Linear representation theory page | |
---|---|---|---|---|---|---|---|

0 | trivial group | 1 | 1 | 1 | 0 | 0 | linear representation theory of trivial group |

2 | cyclic group:Z2 | 2 | 2 | 2 | 0 | 0 | linear representation theory of cyclic group:Z2 |

4 | dihedral group:D8 | 8 | 5 | 4 | 1 | 0 | linear representation theory of dihedral group:D8 |

6 | direct product of D8 and Z2 | 16 | 10 | 8 | 2 | 0 | linear representation theory of direct product of D8 and Z2 |

8 | wreath product of D8 and Z2 | 128 | 20 | 8 | 6 | 6 | linear representation theory of wreath product of D8 and Z2 |