# Classification of groups of order four times a prime congruent to 1 modulo 4

From Groupprops

## Statement

Suppose is an odd prime that is congruent to 1 modulo 4, i.e., 4 divides .

Then, there are five isomorphism classes of groups of order , as detailed below:

Group | Second part of GAP ID | Abelian? | Isomorphism class of 2-Sylow subgroup | Is the 2-Sylow subgroup normal? | Is the -Sylow subgroup normal? |
---|---|---|---|---|---|

dicyclic group of order | 1 | No | cyclic group:Z4 | No | Yes |

cyclic group of order | 2 | Yes | cyclic group:Z4 | Yes | Yes |

semidirect product of cyclic group of order by unique cyclic subgroup of order 4 in its automorphism group (which is the multiplicative group mod ) | 3 | No | cyclic group:Z4 | No | Yes |

dihedral group of order | 4 | No | Klein four-group | No | Yes |

direct product of cyclic group of order and cyclic group:Z2 (also, direct product of group of prime order and Klein four-group | 5 | Yes | Klein four-group | Yes | Yes |