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$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

The following examples illustrate that the class of Lagrangian groups is not subgroupclosed.
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$\mathrm{IsLagrangian}\left(\mathrm{Symm}\left(4\right)\right)$

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$\mathrm{IsLagrangian}\left(\mathrm{Alt}\left(4\right)\right)$

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$\mathrm{IsGCLTGroup}\left(\mathrm{Symm}\left(4\right)\right)$

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$\mathrm{IsGCLTGroup}\left(\mathrm{DihedralGroup}\left(6\right)\right)$

The smallest Lagrangian group that is not a GCLTgroup is the direct product of a cyclic group of order $3$ and the symmetric group of degree $3$.
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$G\u2254\mathrm{PermutationGroup}\left(\mathrm{DirectProduct}\left(\mathrm{CyclicGroup}\left(3\right)\,\mathrm{Symm}\left(3\right)\right)\right)$

${G}{\u2254}\u27e8\left({1}{\,}{2}{\,}{3}\right){\,}\left({4}{\,}{5}\right){\,}\left({4}{\,}{5}{\,}{6}\right)\u27e9$
 (5) 
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$\mathrm{IsLagrangian}\left(G\right)$

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$\mathrm{IsGCLTGroup}\left(G\right)$
