List of Small Groups - List of Small Non-abelian Groups

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:

    “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)

    “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)

    “Doth it not show vilely in me to desire small beer?”
    —William Shakespeare (1564–1616)

    “And seniors grow tomorrow
    From the juniors today,
    And even swimming groups can fade,
    Games mistresses turn grey.”
    —Philip Larkin (1922–1986)