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

Statement

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

Then, there are five isomorphism classes of groups of order $4p$, as detailed below:

Group	Second part of GAP ID	Abelian?	Isomorphism class of 2-Sylow subgroup	Is the 2-Sylow subgroup normal?	Is the $p$-Sylow subgroup normal?
dicyclic group of order $4p$	1	No	cyclic group:Z4	No	Yes
cyclic group of order $4p$	2	Yes	cyclic group:Z4	Yes	Yes
semidirect product of cyclic group of order $p$ by unique cyclic subgroup of order 4 in its automorphism group (which is the multiplicative group mod $p$)	3	No	cyclic group:Z4	No	Yes
dihedral group of order $4p$	4	No	Klein four-group	No	Yes
direct product of cyclic group of order $2p$ and cyclic group:Z2 (also, direct product of group of prime order and Klein four-group	5	Yes	Klein four-group	Yes	Yes

Related classifications

General version

• Classification of groups of order a product of a prime-square and another prime

Similar classifications

• Classification of groups of order four times a prime congruent to 3 modulo 4
• Classification of groups of order 12
• Classification of groups of prime-square order
• Classification of groups of prime-cube order