# 4.6: Classification Groups

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

### Classification of finite groups

The following table gives the number of distinct groups of finite order. Here $$C_n$$ stands for a cyclic group of order $$n$$. Cyclic groups are in red.

$$S_4$$

 Group order Abelian Non-abelian 1 {e} 2 $$C_2=D_1$$ 3 $$C_3$$ 4 $$C_4$$ or  $$C_2 \times C_2=D_2$$ 5 $$C_5$$ 6 $$C_6$$ $$S_3=D_3$$ 7 $$C_7$$ 8 $$C_8$$ or    $$C_2 \times C_4$$ or $$C_2 \times C_2 \times C_2$$ $$D_4$$  or $$Q_8$$ 9 $$C_9$$ or  $$C_3 \times C_3$$ 10 $$C_{10}$$ $$D_5$$ 11 12 13 14 15 16 17

4.6: Classification Groups is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.