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Question about the presentation of finite groups
How to determine this presentation of $D_{12}$, the group of symmetries of a regular decagon,
$\langle a,b| a^6=b^2=1, b^{ -1}ab=a^3 \rangle$, with $a$ exchanging two points, and $b$, $b^{ -1}ab$ rotating one edge by $1/3$ of the way around the decagon.
I’ve been trying to find a related group, and as far as I can tell this is the only presentation that satisfies the criteria, but I could be wrong.
A:
You might try to compute the Cayley-Hamilton theorem for this group, in order to find the diagonal form of the generators (take the characteristic polynomial) and then use this form of the group to write down a presentation in terms of the generators $a$ and $b$.
For instance, in the case of cyclic groups, the Cayley-Hamilton theorem says that the characteristic polynomial of
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