Classification of groups of order four times a prime congruent to 3 modulo 4
From Groupprops
Statement
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 ![]() |
---|---|---|---|---|---|
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 ![]() |
4 | Yes | Klein four-group | Yes | Yes |