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A pair of non-conjugate subgroups: a simple proof
$\DeclareMathOperator\SO{SO}$Set
\begin{equation}
\begin{aligned}
\Gamma_1 &=
\left\{
I_{6},
\;
\gamma_1:=
\left(
\begin{smallmatrix}
0&1\\
1&0 \\
&&0&1\\
&&1&0\\
&...
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