# Tabella de parve gruppos

La prime tabella lista le gruppos finite con un ordine minor o equal a 20 excepte isomorphitate.

 Ordine Quantitate de gruppos abelian Quantitate de gruppos non-abelian Quantitate de gruppos in toto 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 1 1 2 1 1 1 3 2 1 1 2 1 1 1 5 1 2 1 2 0 0 0 0 0 1 0 2 0 1 0 3 0 1 0 9 0 3 0 3 1 1 1 2 1 2 1 5 2 2 1 5 1 2 1 14 1 5 1 5

## Abbreviaturas

• ${\displaystyle A_{2}\cong S_{1}\cong \mathbb {Z} _{1}}$ es le gruppo trivial.
• ${\displaystyle A_{n}}$ es le gruppo alternante del grado ${\displaystyle n}$, con ${\displaystyle n!/2}$ permutationes de ${\displaystyle n}$ elementos pro ${\displaystyle n\geq 2}$.
• ${\displaystyle C_{n}}$ = ${\displaystyle \mathbb {Z} _{n}}$ = ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ es le gruppo cyclic del ordine ${\displaystyle n}$.
• ${\displaystyle D_{n}}$ es le gruppo dihedre del ordine ${\displaystyle 2n}$.
• ${\displaystyle \mathrm {Dic} _{n}}$ es le gruppo dicyclic del ordine ${\displaystyle 4n}$.
• ${\displaystyle Q_{8}}$ es le gruppo de quaterniones del ordine ${\displaystyle 8}$.
• ${\displaystyle S_{n}}$ es le gruppo symmetric del grado ${\displaystyle n}$, con ${\displaystyle n!}$ permutationes de ${\displaystyle n}$ elementos.
• ${\displaystyle V_{4}}$ es le gruppo de Klein del ordine ${\displaystyle 4}$.

## Tabella

Ordine Gruppo Subgruppos non-trivial Proprietates Graphico cyclo
1 ${\displaystyle \mathbb {Z} _{1}\cong S_{1}\cong A_{2}}$ abelian, cyclic
2 ${\displaystyle \mathbb {Z} _{2}\cong S_{2}\cong D_{1}}$ abelian, finite, simple, cyclic, le minus grande gruppo non-trivial
3 ${\displaystyle \mathbb {Z} _{3}\cong A_{3}}$ abelian, simple, cyclic
4 ${\displaystyle \mathbb {Z} _{4}\cong \mathrm {Dic} _{1}}$ ${\displaystyle \mathbb {Z} _{2}}$ abelian, cyclic
${\displaystyle V_{4}\cong \mathbb {Z} _{2}^{2}\cong D_{2}}$ ${\displaystyle 3\cdot \mathbb {Z} _{2}}$ abelian, le minus grande gruppo non-cyclic
5 ${\displaystyle \mathbb {Z} _{5}}$ abelian, simple, cyclic
6 ${\displaystyle \mathbb {Z} _{6}\cong \mathbb {Z} _{2}\times \mathbb {Z} _{3}}$ ${\displaystyle \mathbb {Z} _{3}}$, ${\displaystyle \mathbb {Z} _{2}}$ abelian, cyclic
${\displaystyle S_{3}\cong D_{3}}$   gruppo symetric ${\displaystyle S_{3}}$ ${\displaystyle \mathbb {Z} _{3}}$, ${\displaystyle 3\cdot \mathbb {Z} _{2}}$ le minus grande gruppo non-ablian
7 ${\displaystyle \mathbb {Z} _{7}}$ abelian, simple, cyclic
8 ${\displaystyle \mathbb {Z} _{8}}$ ${\displaystyle \mathbb {Z} _{4}}$, ${\displaystyle \mathbb {Z} _{2}}$ abelian, cyclic
${\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{4}}$ ${\displaystyle 2\cdot \mathbb {Z} _{4}}$, ${\displaystyle 3\cdot \mathbb {Z} _{2}}$, ${\displaystyle D_{2}}$ abelian
${\displaystyle \mathbb {Z} _{2}^{3}\cong D_{2}\times \mathbb {Z} _{2}}$ ${\displaystyle 7\cdot \mathbb {Z} _{2}}$, ${\displaystyle 7\cdot D_{2}}$ abelian
${\displaystyle D_{4}}$ ${\displaystyle \mathbb {Z} _{4}}$, ${\displaystyle 2\cdot D_{2}}$, ${\displaystyle 5\cdot \mathbb {Z} _{2}}$ non-abelian
${\displaystyle Q_{8}\cong \mathrm {Dic} _{2}}$ ${\displaystyle 3\cdot \mathbb {Z} _{4}}$, ${\displaystyle \mathbb {Z} _{2}}$ non-abelian; le minus grande gruppo hamiltonian
9 ${\displaystyle \mathbb {Z} _{9}}$ ${\displaystyle \mathbb {Z} _{3}}$ abelian, cyclic
${\displaystyle \mathbb {Z} _{3}^{2}}$ ${\displaystyle 4\cdot \mathbb {Z} _{3}}$ abelian
10 ${\displaystyle \mathbb {Z} _{10}\cong \mathbb {Z} _{2}\times \mathbb {Z} _{5}}$ ${\displaystyle \mathbb {Z} _{5}}$, ${\displaystyle \mathbb {Z} _{2}}$ abelian, cyclic
${\displaystyle D_{5}}$ ${\displaystyle \mathbb {Z} _{5}}$, ${\displaystyle 5\cdot \mathbb {Z} _{2}}$ non-abelian
11 ${\displaystyle \mathbb {Z} _{11}}$ abelian, simple, cyclic
12 ${\displaystyle \mathbb {Z} _{12}\cong \mathbb {Z} _{4}\times \mathbb {Z} _{3}}$ ${\displaystyle \mathbb {Z} _{6}}$, ${\displaystyle \mathbb {Z} _{4}}$, ${\displaystyle \mathbb {Z} _{3}}$, ${\displaystyle \mathbb {Z} _{2}}$ abelian, cyclic
${\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{6}\cong \mathbb {Z} _{2}^{2}\times \mathbb {Z} _{3}\cong D_{2}\times \mathbb {Z} _{3}}$ ${\displaystyle 3\cdot \mathbb {Z} _{6}}$, ${\displaystyle \mathbb {Z} _{3}}$, ${\displaystyle D_{2}}$, ${\displaystyle 3\cdot \mathbb {Z} _{2}}$ abelian
${\displaystyle D_{6}\cong D_{3}\times \mathbb {Z} _{2}}$ ${\displaystyle \mathbb {Z} _{6}}$, ${\displaystyle 2\cdot D_{3}}$, ${\displaystyle 3\cdot D_{2}}$, ${\displaystyle \mathbb {Z} _{3}}$, ${\displaystyle 7\cdot \mathbb {Z} _{2}}$ non-abelian
${\displaystyle A_{4}}$ ${\displaystyle D_{2}}$, ${\displaystyle 4\cdot \mathbb {Z} _{3}}$, ${\displaystyle 3\cdot \mathbb {Z} _{2}}$ non-abelian; nulle subgruppo de ordine 6
${\displaystyle \mathrm {Dic} _{3}}$ ${\displaystyle \mathbb {Z} _{6}}$, ${\displaystyle 3\cdot \mathbb {Z} _{4}}$, ${\displaystyle \mathbb {Z} _{3}}$, ${\displaystyle \mathbb {Z} _{2}}$ non-abelian
13 ${\displaystyle \mathbb {Z} _{13}}$ abelian, simple, cyclic
14 ${\displaystyle \mathbb {Z} _{14}\cong \mathbb {Z} _{2}\times \mathbb {Z} _{7}}$ ${\displaystyle \mathbb {Z} _{7}}$, ${\displaystyle \mathbb {Z} _{2}}$ abelian, cyclic
${\displaystyle D_{7}}$ ${\displaystyle \mathbb {Z} _{7}}$, ${\displaystyle 7\cdot \mathbb {Z} _{2}}$ non-abelian
15 ${\displaystyle \mathbb {Z} _{15}\cong \mathbb {Z} _{3}\times \mathbb {Z} _{5}}$ ${\displaystyle \mathbb {Z} _{5}}$, ${\displaystyle \mathbb {Z} _{3}}$ abelian, cyclic
16 ${\displaystyle \mathbb {Z} _{16}}$ ${\displaystyle \mathbb {Z} _{8}}$, ${\displaystyle \mathbb {Z} _{4}}$, ${\displaystyle \mathbb {Z} _{2}}$ abelian, cyclic
${\displaystyle \mathbb {Z} _{2}^{4}}$ ${\displaystyle 15\cdot \mathbb {Z} _{2}}$, ${\displaystyle 35\cdot D_{2}}$, ${\displaystyle 15\cdot \mathbb {Z} _{2}^{3}}$ abelian
${\displaystyle \mathbb {Z} _{4}\times \mathbb {Z} _{2}^{2}}$ ${\displaystyle 7\cdot \mathbb {Z} _{2}}$, ${\displaystyle 4\cdot \mathbb {Z} _{4}}$, ${\displaystyle 7\cdot D_{2}}$, ${\displaystyle \mathbb {Z} _{2}^{3}}$, ${\displaystyle 6\cdot \mathbb {Z} _{4}\times \mathbb {Z} _{2}}$ abelian
${\displaystyle \mathbb {Z} _{8}\times \mathbb {Z} _{2}}$ ${\displaystyle 3\cdot \mathbb {Z} _{2}}$, ${\displaystyle 2\cdot \mathbb {Z} _{4}}$, ${\displaystyle D_{2}}$, ${\displaystyle 2\cdot \mathbb {Z} _{8}}$, ${\displaystyle \mathbb {Z} _{4}\times \mathbb {Z} _{2}}$ abelian
${\displaystyle \mathbb {Z} _{4}^{2}}$ ${\displaystyle 3\cdot \mathbb {Z} _{2}}$, ${\displaystyle 6\cdot \mathbb {Z} _{4}}$, ${\displaystyle D_{2}}$,${\displaystyle 3\cdot \mathbb {Z} _{4}\times \mathbb {Z} _{2}}$ abelian
${\displaystyle D_{8}}$ ${\displaystyle \mathbb {Z} _{8}}$, ${\displaystyle 2\cdot D_{4}}$, ${\displaystyle 4\cdot D_{2}}$, ${\displaystyle \mathbb {Z} _{4}}$, ${\displaystyle 9\cdot \mathbb {Z} _{2}}$ non-abelian
${\displaystyle D_{4}\times \mathbb {Z} _{2}}$ ${\displaystyle 4\cdot D_{4}}$, ${\displaystyle \mathbb {Z} _{4}\times \mathbb {Z} _{2}}$, ${\displaystyle 2\cdot \mathbb {Z} _{2}^{3}}$, ${\displaystyle 13\cdot \mathbb {Z} _{2}^{2}}$, ${\displaystyle 2\cdot \mathbb {Z} _{4}}$, ${\displaystyle 11\cdot \mathbb {Z} _{2}}$ non-abelian
${\displaystyle Q_{16}\cong \mathrm {Dic_{4}} }$ ${\displaystyle \mathbb {Z} _{8}}$, ${\displaystyle 2\cdot Q_{8}}$, ${\displaystyle 5\cdot \mathbb {Z} _{4}}$, ${\displaystyle \mathbb {Z} _{2}}$ non-abelian
${\displaystyle Q_{8}\times \mathbb {Z} _{2}}$ ${\displaystyle 3\cdot \mathbb {Z} _{2}\times \mathbb {Z} _{4}}$, ${\displaystyle 4\cdot Q_{8}}$, ${\displaystyle 6\cdot \mathbb {Z} _{4}}$, ${\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}}$, ${\displaystyle 3\cdot \mathbb {Z} _{2}}$ non-abelian, gruppo hamiltonian
gruppo quasi-dihedre ${\displaystyle \mathbb {Z} _{8}}$, ${\displaystyle Q_{8}}$, ${\displaystyle D_{4}}$, ${\displaystyle 3\cdot \mathbb {Z} _{4}}$, ${\displaystyle 2\cdot \mathbb {Z} _{2}\times \mathbb {Z} _{2}}$, ${\displaystyle 5\cdot \mathbb {Z} _{2}}$ non-abelian
M-gruppo (gruppo non-abelian, non-hamiltonian, modular) ${\displaystyle 2\cdot \mathbb {Z} _{8}}$, ${\displaystyle \mathbb {Z} _{4}\times \mathbb {Z} _{2}}$, ${\displaystyle 2\cdot \mathbb {Z} _{4}}$, ${\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}}$, ${\displaystyle 3\cdot \mathbb {Z} _{2}}$ non-abelian
producto semidirecte ${\displaystyle \mathbb {Z} _{4}\rtimes \mathbb {Z} _{4}}$ ${\displaystyle 3\cdot \mathbb {Z} _{2}\times \mathbb {Z} _{4}}$, ${\displaystyle 6\cdot \mathbb {Z} _{4}}$, ${\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}}$, ${\displaystyle 3\cdot \mathbb {Z} _{2}}$ non-abelian
le gruppo create per matrices de Pauli ${\displaystyle 3\cdot \mathbb {Z} _{2}\times \mathbb {Z} _{4}}$, ${\displaystyle 3\cdot D_{4}}$, ${\displaystyle Q_{8}}$, ${\displaystyle 4\cdot \mathbb {Z} _{4}}$, ${\displaystyle 3\cdot \mathbb {Z} _{2}\times \mathbb {Z} _{2}}$, ${\displaystyle 7\cdot \mathbb {Z} _{2}}$ non-abelian
${\displaystyle G_{4,4}=V_{4}\rtimes \mathbb {Z} _{4}}$ ${\displaystyle 2\cdot \mathbb {Z} _{2}\times \mathbb {Z} _{4}}$, ${\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{2}\times \mathbb {Z} _{2}}$, ${\displaystyle 4\cdot \mathbb {Z} _{4}}$, ${\displaystyle 7\cdot \mathbb {Z} _{2}\times \mathbb {Z} _{2}}$, ${\displaystyle 7\cdot \mathbb {Z} _{2}}$ non-abelian
17 ${\displaystyle \mathbb {Z} _{17}}$ abelian, simple, cyclic
18 ${\displaystyle \mathbb {Z} _{18}\cong \mathbb {Z} _{9}\times \mathbb {Z} _{2}}$ ${\displaystyle \mathbb {Z} _{9},\mathbb {Z} _{6},\mathbb {Z} _{3},\mathbb {Z} _{2}}$ abelian, cyclic
${\displaystyle \mathbb {Z} _{6}\times \mathbb {Z} _{3}}$ ${\displaystyle \mathbb {Z} _{6},\mathbb {Z} _{3},\mathbb {Z} _{2}}$ abelian
${\displaystyle D_{9}}$ non-abelian
${\displaystyle S_{3}\times \mathbb {Z} _{3}}$ non-abelian
${\displaystyle (\mathbb {Z} _{3}\times \mathbb {Z} _{3})\rtimes _{\alpha }\mathbb {Z} _{2}}$ con ${\displaystyle \alpha (1)={\begin{pmatrix}2&0\\0&2\end{pmatrix}}}$ non-abelian
19 ${\displaystyle \mathbb {Z} _{19}}$ abelian, simple, cyclic
20 ${\displaystyle \mathbb {Z} _{20}\cong \mathbb {Z} _{5}\times \mathbb {Z} _{4}}$ ${\displaystyle \mathbb {Z} _{10},\mathbb {Z} _{5},\mathbb {Z} _{4},\mathbb {Z} _{2}}$ abelian, cyclic
${\displaystyle \mathbb {Z} _{10}\times \mathbb {Z} _{2}\cong \mathbb {Z} _{5}\times \mathbb {Z} _{2}\times \mathbb {Z} _{2}}$ ${\displaystyle \mathbb {Z} _{5},\mathbb {Z} _{2}}$ abelian
${\displaystyle Q_{20}\cong \mathrm {Dic} _{5}}$ non-abelian
${\displaystyle \mathbb {Z} _{5}\rtimes \mathbb {Z} _{4}\cong }$ gruppo affine ${\displaystyle \mathrm {AGL} _{1}}$(5) non-abelian
${\displaystyle D_{10}\cong D_{5}\times \mathbb {Z} _{2}}$ ${\displaystyle \mathbb {Z} _{10},D_{5},\mathbb {Z} _{5},5\cdot V_{4},6\cdot \mathbb {Z} _{2}}$ non-abelian