# Cyclic group:Z6

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## Definition

### Verbal definition

The cyclic group of order 6 is defined as the group of order six generated by a single element. Equivalently it can be described as a group with six elements where with the exponent reduced mod 3. It can also be viewed as:

- The quotient group of the group of integers by the subgroup of multiples of 6.
- The multiplicative group comprising the six sixth roots of unity (as a subgroup of the multiplicative group of nonzero complex numbers)
- The group of orientation-preserving symmetries (rotational symmetries) of the regular hexagon.