Dihedral group:D18

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The group is defined as the dihedral group of order math>18</math> and degree 9. In other words, it has the presentation:

\langle a,x \mid a^9 = x^2 = e, xax = a^{-1} \rangle.

GAP implementation

Group ID

This finite group has order 18 and has ID 1 among the groups of order 18 in GAP's SmallGroup library. For context, there are groups of order 18. It can thus be defined using GAP's SmallGroup function as:


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

gap> G := SmallGroup(18,1);

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

IdGroup(G) = [18,1]

or just do:


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

Other descriptions

The group can be constructed using GAP's DihedralGroup function: