A group $G$ is characteristically simple if it is non-trivial and has no proper, non-trivial characteristic subgroups. Clearly, every simple group is characteristically simple.

The IsCharacteristicallySimple( G ) command returns true if the group G is characteristically simple, and returns false otherwise.

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

$\mathrm{IsCharacteristicallySimple}\left(\mathrm{CyclicGroup}\left(37\right)\right)$

$\mathrm{true}$

$\mathrm{IsCharacteristicallySimple}\left(\mathrm{CyclicGroup}\left(300\right)\right)$

$\mathrm{false}$

$\mathrm{IsCharacteristicallySimple}\left(\mathrm{Alt}\left(8\right)\right)$

$\mathrm{true}$

$\mathrm{IsCharacteristicallySimple}\left(\mathrm{Symm}\left(8\right)\right)$

$\mathrm{false}$

$\mathrm{IsCharacteristicallySimple}\left(\mathrm{SmallGroup}\left(8\,5\right)\right)$

$\mathrm{true}$

$\mathrm{IsCharacteristicallySimple}\left(\mathrm{SmallGroup}\left(8\,2\right)\right)$

$\mathrm{false}$

$\mathrm{IsCharacteristicallySimple}\left(\mathrm{DirectProduct}\left(\mathrm{Alt}\left(5\right)\,\mathrm{Alt}\left(6\right)\right)\right)$

$\mathrm{false}$

$\mathrm{IsCharacteristicallySimple}\left(\mathrm{DirectProduct}\left(\mathrm{Alt}\left(6\right)\,\mathrm{Alt}\left(6\right)\right)\right)$

$\mathrm{true}$