Classification of groups of order four times a prime congruent to 3 modulo 4
Suppose is an odd prime that is congruent to 3 modulo 4, i.e., 4 divides . Suppose further that .
Then, there are four 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|
|dihedral group of order||3||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||4||Yes||Klein four-group||Yes||Yes|
The case differs from tthe general case. For that case, see classification of group of order 12.