Classification of groups of order four times a prime congruent to 3 modulo 4

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Statement

Suppose p is an odd prime that is congruent to 3 modulo 4, i.e., 4 divides p - 3. Suppose further that p > 3.

Then, there are four 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
dihedral group of order 4p 3 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 4 Yes Klein four-group Yes Yes

Related classifications

General version

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