Finite groups · Group

$C_{6}$ Cyclic group of order 6

A cyclic group with 6 one-dimensional ordinary complex irreducibles.

Compare this group

Reference coverageFinite referenceAbelian

Definition

The cyclic group generated by a with a^6 = 1 and exact order 6.

Definition sources

Order: $6$
Center: $C_{6}$

Reference properties

Property	Value	Evidence
Order	$6$	Sources
Center	$C_{6}$	Sources
Abelian	Yes	Sources
Simple	No	Sources
Ordinary complex irreducibles	1, 1, 1, 1, 1, 1	Sources
GAP SmallGroups ID	SmallGroup(6,2)	Sources

Representation scope

The complete list of ordinary complex irrep dimensions is given above. Labels ρ1, ρ2, … follow the cited GroupNames character-table row order; they are local labels, not dimension-based identities. [sources]

Real, Complex and Quaternionic refer to Frobenius–Schur type (+1, 0, −1). The linked character table provides additional data; this entry does not supply matrices or a modular representation catalogue. [sources]

Complete ordinary complex irrep dimension and reality summary
Irrep label	Complex dimension	Reality type
ρ1	1	Real
ρ2	1	Real
ρ3	1	Complex
ρ4	1	Complex
ρ5	1	Complex
ρ6	1	Complex

Complete dimension and reality summary for ordinary complex irreps. This does not supply explicit matrices, every character value, or modular representations. Labels follow the conventions stated in this record.

Structure and conventions

For a generator a, χₖ(a) = exp(2πik/6); source row labels need not be ordered by k. [sources]

Names and aliases

Z6 · Z/6Z · ℤ6

Aliases identify the stated object under this record’s conventions. Type names and broader search terms are kept separate.

Related objects and maps

$\mathrm{C}_{n}$
Finite cyclic groups
Family

Claims and sources

Each assertion has a stable identifier within this record and a source locator or derivation. Cite the dataset edition when referring to a corrected or historical claim.

Definition

The cyclic group generated by a with a^6 = 1 and exact order 6.