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:
“Love’s boat has been shattered against the life of everyday. You and I are quits, and it’s useless to draw up a list of mutual hurts, sorrows, and pains.”
—Vladimir Mayakovsky (1893–1930)
“Hey, you dress up our town very nicely. You don’t look out the Chamber of Commerce is going to list you in their publicity with the local attractions.”
—Robert M. Fresco, and Jack Arnold. Dr. Matt Hastings (John Agar)
“In the pink light
the small red sun goes rolling, rolling,
round and round and round at the same height
in perpetual sunset,”
—Elizabeth Bishop (1911–1979)
“And seniors grow tomorrow
From the juniors today,
And even swimming groups can fade,
Games mistresses turn grey.”
—Philip Larkin (1922–1986)