groups of small order

Below is a list of all possible groups per order up to isomorphism.

Groups of prime order:

• All groups of prime order are isomorphic to a cyclic group of that order.

Groups of prime square order:

• All groups of order $p^2$, where $p$ is a prime, are isomorphic to one of the following:

  • $C_{p^2}$(Abelian): cyclic group of order $p^2$.

  • $C_p\times C_p$(Abelian): elementary abelian group of order $p^2$.

Groups of order 1:

• trivial group (i.e. $\mathrm{\{}e\mathrm{\}}$).

Groups of order 6:

• $C_6$(Abelian): cyclic group of order 6.

• $S_3$(non-Abelian): symmetric group where $n=3$.

Groups of order 8:

• $C_8$(Abelian): cyclic group of order 8.

• $C_4\times C_2$(Abelian): direct product of two groups of a cyclic group of order 4 and a cyclic group of order 2.

• $C_2\times C_2\times C_2$(Abelian): direct product of three groups of a cyclic group of order 2.

• $D_4$(non-Abelian): octic group; dihedral group of degree 4.

• $Q_8$(non-Abelian): quaternion group.

Groups of order 10:

• $C_{10}$(Abelian): cyclic group of order 10.

• $D_5$(non-Abelian): dihedral group of degree 5.

Groups of order 12:

• $C_{12}$(Abelian): cyclic group of order 12.

• $C_2\times C_6$(Abelian).

• $A_4$(non-Abelian): alternating group of degree 4.

• $D_6$(non-Abelian): dihedral group of degree 6.

• $\mathrm{Dic}(C_6)$(non-Abelian): dicyclic group of order 12.This is a generalized quaternion group $Q_{12}$.

Groups of order 14:

• $C_{14}$(Abelian): cyclic group of order 14.

• $D_7$(non-Abelian): dihedral group of degree 7.

Groups of order 15:

• $C_{15}$(Abelian): cyclic group of order 15.