Classification of groups of order four times a prime

We split the classification of groups of order $4p$, $p$ prime into three cases:

• Classification of groups of order 8 ($p=2$, which is more generally classified as a group of prime-cubed order.)
• Classification of groups of order four times a prime congruent to 1 modulo 4
• Classification of groups of order four times a prime congruent to 3 modulo 4

Summary of results

• The groups of order 8 are dihedral group:D8, quaternion group, cyclic group:Z8, elementary abelian group:E8 and Direct product of Z4 and Z2.

• Otherwise, in both the $p\equiv 1,3mod4$ cases, there are the cyclic, dihedral and dicyclic groups of order $4p$ as well as the direct product of the cyclic group of order $2p$ and cyclic group:Z2.

• • If $p\equiv 1mod4$ there is an additional group - the 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 $p$)

See also

• Classification of groups of order a power of two times a prime