Establish a finite presentation for using two generators.
We know that is generated by an element of order 2 and an element of order 3. Let and . Evidently, these generators satisfy the relations . To show that these relations provide a presentation for , it suffices to show that the group has order at most 24. To that end, we will compute the reduced words in this group.
- There is one word of length 0: 1.
- There are two words of length 1: and . Both are reduced by definition.
- There are at most four words of length 2: , , , and . Since , it is not reduced. Thus 3 words of length 2 might be reduced.
- There are at most six words of length 3: , , , , , and . Since and , these words are not reduced. The remaining 4 words might be reduced.
- There are at most eight words of length 4: , , , , , , , and . Since , , , , , and , these words are not reduced. The remaining two words might be reduced.
- There are at most four words of length 5: , , , and . None of these are reduced.
Thus any group presented by has cardinality at most 12, and we have the presentation .