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:
“Feminism is an entire world view or gestalt, not just a laundry list of women’s issues.”
—Charlotte Bunch (b. 1944)
“Thirty—the promise of a decade of loneliness, a thinning list of single men to know, a thinning brief-case of enthusiasm, thinning hair.”
—F. Scott Fitzgerald (1896–1940)
“If Montaigne is a man in the prime of life sitting in his study on a warm morning and putting down the sum of his experience in his rich, sinewy prose, then Pascal is that same man lying awake in the small hours of the night when death seems very close and every thought is heightened by the apprehension that it may be his last.”
—Cyril Connolly (1903–1974)
“... 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–?)