List of Small Non-abelian Groups
| Order | Group | Subgroups | Properties | Cycle Graph |
|---|---|---|---|---|
| 6 | S3 = Dih3 | Z3, Z2 (3) | the smallest non-abelian group | |
| 8 | Dih4 | Z4, Z22 (2), Z2 (5) | ||
| quaternion group, Q8 = Dic2 | Z4 (3), Z2 | the smallest Hamiltonian group; smallest group demonstrating that all subgroups may be normal without the group being abelian; the smallest group G demonstrating that for a normal subgroup H the quotient group G/H need not be isomorphic to a subgroup of G | ||
| 10 | Dih5 | Z5, Z2 (5) | ||
| 12 | Dih6 = Dih3 × Z2 | Z6, Dih3 (2), Z22 (3), Z3, Z2 (7) | ||
| A4 | Z22, Z3 (4), Z2 (3) | smallest group demonstrating that a group need not have a subgroup of every order that divides the group's order: no subgroup of order 6 (See Lagrange's theorem and the Sylow theorems.) | ||
| Dic3 = Z3 Z4 | Z2, Z3, Z4 (3), Z6 | |||
| 14 | Dih7 | Z7, Z2 (7) | ||
| 16 | Dih8 | Z8, Dih4 (2), Z22 (4), Z4, Z2 (9) | ||
| Dih4 × Z2 | Dih4 (2), Z4 × Z2, Z23 (2), Z22 (11), Z4 (2), Z2 (11) | |||
| generalized quaternion group, Q16 = Dic4 | ||||
| Q8 × Z2 | Hamiltonian | |||
| The order 16 quasidihedral group | ||||
| The order 16 modular group | ||||
| Z4 Z4 | ||||
| The group generated by the Pauli matrices | ||||
| G4,4 = Z22 Z4 |
Read more about this topic: List Of Small Groups
Famous quotes containing the words list of, list, small and/or groups:
“Religious literature has eminent examples, and if we run over our private list of poets, critics, philanthropists and philosophers, we shall find them infected with this dropsy and elephantiasis, which we ought to have tapped.”
—Ralph Waldo Emerson (1803–1882)
“Religious literature has eminent examples, and if we run over our private list of poets, critics, philanthropists and philosophers, we shall find them infected with this dropsy and elephantiasis, which we ought to have tapped.”
—Ralph Waldo Emerson (1803–1882)
“When parents fail to set appropriate limits, children may feel more vulnerable at night: the aggressive urges that have not been “tamed” by day may be terrifying to a small child alone in the dark.”
—Cathy Rindner Tempelsman (20th century)
“... until both employers’ and workers’ groups assume responsibility for chastising their own recalcitrant children, they can vainly bay the moon about “ignorant” and “unfair” public criticism. Moreover, their failure to impose voluntarily upon their own groups codes of decency and honor will result in more and more necessity for government control.”
—Mary Barnett Gilson (1877–?)