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As we know, quaternion group $Q_8$ and dihedral group $D_8$ have the same character table, but they are not isomorphic.I have proven that if the order of a group $G$ is not divided by $8$,and $G$ has the same character table as $D_{2n}$,then $G$ is isomorphic to $D_{2n}$. My question is what groups have the same character table with $D_{2n}$. I want to know all of them.

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The only ways you can express a real number as the sum of two roots of unity are as $w+w^{-1}$ or as $w+(-w) = 0$. We have the equality $w+w^{-1}=1+(-1)$ when $w$ is a primitive $4$th root of $1$, which is why we cannot always distinguish between elements of orders $2$ and $4$ from their characters in $2$-dimensional representations.

But for $n \ge 3$ and $n \ne 4$, if you choose a faithful character $\chi$ of $D_{2n}$, then there will be an element $g$ with $\chi(g) = w+w^{-1}$ with $w$ a primitive $n$-th root of $1$, so $g$ must have order $n$. It has two conjugates, so the other one must be $g^{-1}$ and hence there exists $h \in G$ with $h^{-1}gh=g^{-1}$. Since $h^2 \in \langle g \rangle$, $h^2=1$ or (when $n$ is even) $h^2=g^{n/2}$. So the group is either $D_{2n}$ or (when $n$ is even) the generalized quaternion group $Q_{2n}$, and these do indeed have the same character table.

This argument does not work for $n=4$, but you can do that case separately.

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  • $\begingroup$ Thank you for your answer.Your corollary is that if the order of a group G is divided by 8,and G has the same character table with D2n,then G is iomorphic to the generalized quaternion group Q2n or D2n.But actually there are two groups of order 16 have the same character table with D16,but they are not iomorphic to D2n by the book "representations and characters of groups"(Gorden James.Martin Liebeck) in page 304. $\endgroup$
    – user94782
    Commented Sep 15, 2013 at 1:37
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    $\begingroup$ @user94782: There are three groups of order 16 with 7 conjugacy classes: D16, Q16, and QD16. The character tables (without power maps) are isomorphic for D16 and Q16, but not with QD16 since QD16 has non-real character values. This is also what page 304 of James–Liebeck says. $\endgroup$ Commented Sep 15, 2013 at 4:39

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