Dihedral group:D36

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Definition

This group, usually denoted $D_{36}$ (though denoted $D_{18}$ in an alternate convention) is defined in the following equivalent ways:

• It is the dihedral group of order 36. In other words, it is the dihedral group of degree 18, i.e., the group of symmetries of a regular 18-gon.

The usual presentation is:

$⟨a,x\mid a^{18}=x^2=e,xax=a^{-1}⟩$.

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 12#Arithmetic functions

Function	Value	Explanation
order	36
exponent	18
nilpotency class	--	not a nilpotent group.
minimum size of generating set	2

GAP implementation

Group ID

This finite group has order 36 and has ID 4 among the groups of order 36 in GAP's SmallGroup library. For context, there are groups of order 36. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(36,4)

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

gap> G := SmallGroup(36,4);

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

IdGroup(G) = [36,4]

or just do:

IdGroup(G)

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

Other definitions

Description	Functions used
DihedralGroup(36)	DihedralGroup