Cyclic group:Z56

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Definition

This group, denoted $C_{56},\mathbb {Z} _{56}$ or $\mathbb {Z} /56\mathbb {Z}$, is defined in the following equivalent ways:

1. It is a cyclic group of order $56$.
2. It is the direct product of the cyclic group of order eight 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 56.
• The group of orientation-preserving symmetries (rotational symmetries) of the regular 56-gon.

Properties

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

GAP implementation

Group ID

This finite group has order 56 and has ID 2 among the groups of order 56 in GAP's SmallGroup library. For context, there are groups of order 56. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(56,2)

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

gap> G := SmallGroup(56,2);

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

IdGroup(G) = [56,2]

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(56)