Metacyclic group
A group G is metacyclic if it has a cyclic normal subgroup N with G/N cyclic — an extension of one cyclic group by another (N ◁ G with N and G/N both cyclic).
Key facts
- Every cyclic group is metacyclic.
- Metacyclic ⟹ supersolvable ⟹ monomial ⟹ solvable.
- Metacyclic ⟹ metabelian.
- Closed under subgroups and quotients.
- Needs at most 2 generators.
Examples
- All cyclic Cₙ:
C₆,C₁₂. - Every dihedral Dₙ — cyclic Cₙ normal, quotient C₂:
D₄,D₆. - Every dicyclic / generalized quaternion group:
Q₈,Q₁₆,Dic₃. - The modular maximal-cyclic and semi-dihedral 2-groups:
M₁₆,SD₁₆. C₅ ⋊ C₄(a semidirect product).- Every group of order p² or pq.
Non-examples
- The elementary abelian group
C₂ × C₂ × C₂— needs 3 generators; no cyclic subgroup with cyclic quotient. - The extraspecial (Heisenberg) group of order p³ and exponent p.