# Conjugacy Class of Dihedral Group of Order 12

I am given the dihedral group of order 12: $$D_{12}=$$, where $$a$$ is a rotation of a hexagon by 60 degrees, and $$b$$ is a reflection across a diagonal of two vertices.

I am looking to find the conjugacy class $$cl_{D_{12}}(b)$$. I have, $$cl_{D_{12}}(b)$$ = $$\{b, aba^{-1}, a^2ba^{-2}, a^3ba^{-3}\}$$.

Have I found this conjugacy class correctly?

Thanks.

• I think you mean $a^6 = b^2 = 1$, not $3$. – rogerl Nov 25 '18 at 1:07

Note that $$a^3 = a^{-3}$$, so that $$a^3ba^{-3} = a^3ba^3 = a^3a^{15}b = a^{18}b = b.$$
It might be easier if you think of the elements of $$D_{12}$$ as $$1, a, a^2, a^3, a^4, a^5, b, ba, ba^2, ba^3, ba^4, ba^5,$$ and remember that always $$a^kb = ba^{6-k}$$.
As a hint, $$D_{12}$$ has two conjugacy classes of size $$1$$, two of size $$2$$, and $$2$$ of size $$3$$.
By hand, we have $$b,aba^{-1},a^2ba-{2},a^3ba^{-3},a^4ba{-4},a^5ba^{-5},bab(ba)^{-1},ba^2b(ba^2)^{-1},ba^3b(ba^3)^{-1},ba^4b(ba^4)^{-1},ba^5b(ba^5)^{-1}$$.
But these simplify to $$\{b,ba^4,ba^2\}$$, using the relations.