In group theory, a metacyclic group is an extension of a cyclic group by a cyclic group. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a group extension is a general means of describing a group in terms of a particular Normal subgroup and Quotient group. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an That is, it is a group G for which there is a short exact sequence

Equivalently, a metacyclic group is a group G having a cyclic normal subgroup N, such that the quotient G/N is also cyclic. In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G

Properties

Metacyclic groups are both supersolvable and metabelian. In Mathematics, a group is supersolvable (or supersoluble) if it has an invariant Normal series where all the factors are cyclic groups In Mathematics, a metabelian group is a group whose Commutator subgroup is abelian.

An interesting property of metacyclic groups is that every complex irreducible representation of every metacyclic group can be easily constructed from a diagram. In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of

Examples

• Any cyclic group is metacyclic. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an
• The direct product or semidirect product of two cyclic groups is metacyclic. In Mathematics, one can often define a direct product of objectsalready known giving a new one In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can These include the dihedral groups, the quasidihedral groups, and the dicyclic groups. In Mathematics, a dihedral group is the group of symmetries of a Regular polygon, including both rotations and reflections In Mathematics, the quasi-dihedral groups and semi-dihedral groups are non-abelian groups of order a power of 2 In Group theory, a dicyclic group is a member of a class of groups Dic n ( n > 1 a Non-abelian group of order 4 n
• Every finite group of squarefree order is metacyclic. In Mathematics, a finite group is a group which has finitely many elements In Mathematics, a square-free, or quadratfrei, Integer is one divisible by no perfect square, except 1
• More generally every Z-group is metacyclic. In Mathematics, especially in the area of Algebra known as Group theory, the term Z-group refers to a number of distinct types of groups: A Z-group is a group whose Sylow subgroups are cyclic.

References

• A. L. Shmel'kin (2001), “Metacyclic group”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104 

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