Cyclic group:Z28

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Definition

This group, denoted $C_{28},\mathbb {Z} _{28}$ or $\mathbb {Z} /28\mathbb {Z}$, is defined in the following equivalent ways:

1. It is a cyclic group of order $28$.
2. It is the direct product of the cyclic group of order four and the cyclic group of order seven.

Verbal definition

It can also be viewed as:

• The quotient group of the group of integers by the subgroup of multiples of 28.
• The group of orientation-preserving symmetries (rotational symmetries) of the regular 28-gon.

Properties

Property	Satisfied?	Explanation	Comment
Abelian group	Yes	Cyclic implies abelian
Nilpotent group	Yes	Abelian implies nilpotent
Leinster group	Yes

GAP implementation

Group ID

This finite group has order 28 and has ID 2 among the groups of order 28 in GAP's SmallGroup library. For context, there are groups of order 28. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(28,2)

For instance, we can use the following assignment in GAP to create the group and name it $G$:

gap> G := SmallGroup(28,2);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [28,2]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

Other descriptions

The group can be defined using GAP's CyclicGroup function:

CyclicGroup(28)