The quaternion group is one of the two non-abelian groups of order $8$, (the other being the dihedral group of degree $4$). It is notable because it is an example of a Hamiltonian group - every one of its subgroups is normal - and it appears as a subgroup of every non-Abelian Hamiltonian group.

The QuaternionGroup() command returns a quaternion group, either as a permutation group (the default), or as a finitely presented group.

You can pass the option 'form' = "fpgroup" or 'form' = "permgroup" to cause the QuaternionGroup command to return a group of the indicated class.

In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

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

$\mathrm{QuaternionGroup}\left(\right)$

$\mathrm{GroupTheory}:-\mathrm{QuaternionGroup}\left(\mathrm{form}\=''permgroup''\right)$

$\mathrm{QuaternionGroup}\left('\mathrm{form}'=''permgroup''\right)$

$\mathrm{GroupTheory}:-\mathrm{QuaternionGroup}\left(\mathrm{form}\=''permgroup''\right)$

$\mathrm{QuaternionGroup}\left('\mathrm{form}'=''fpgroup''\right)$

$\mathrm{GroupTheory}:-\mathrm{QuaternionGroup}\left(\mathrm{form}\=''fpgroup''\right)$

$\mathrm{GroupOrder}\left(\mathrm{QuaternionGroup}\left(\right)\right)$

$8$

$\mathrm{SearchSmallGroups}\left('\mathrm{order}'=8\,'\mathrm{abelian}'=\mathrm{false}\right)$

$\left[8\,3\right],\left[8\,4\right]$

$\mathrm{IdentifySmallGroup}\left(\mathrm{QuaternionGroup}\left(\right)\right)$

$8,4$

$\mathrm{AreIsomorphic}\left(\mathrm{QuaternionGroup}\left(\right)\,\mathrm{DihedralGroup}\left(4\right)\right)$

$\mathrm{false}$

$\mathrm{ctQ}≔\mathrm{CharacterTable}\left(\mathrm{QuaternionGroup}\left(\right)\right)$

$\mathrm{GroupTheory}:-\mathrm{CharacterTable}\left(\mathbf{module}\left(\right)...\mathbf{end\; module}\right)$

$\mathrm{ctD}≔\mathrm{CharacterTable}\left(\mathrm{DihedralGroup}\left(4\right)\right)$

$\mathrm{GroupTheory}:-\mathrm{CharacterTable}\left(\mathbf{module}\left(\right)...\mathbf{end\; module}\right)$

$\mathrm{EqualEntries}\left(\mathrm{GetMatrix}\left(\mathrm{ctQ}\right)\,\mathrm{GetMatrix}\left(\mathrm{ctD}\right)\right)$

$\mathrm{true}$

$\mathrm{GraphTheory}:-\mathrm{DrawGraph}\left(\mathrm{CayleyGraph}\left(\mathrm{QuaternionGroup}\left(\right)\right)\right)$

$\mathrm{DrawCayleyTable}\left(\mathrm{QuaternionGroup}\left(\right)\right)$

$\mathrm{DrawSubgroupLattice}\left(\mathrm{QuaternionGroup}\left(\right)\right)$

$\mathrm{andmap}\left(\mathrm{IsNormal}\,\mathrm{convert}\left(\mathrm{SubgroupLattice}\left(\mathrm{QuaternionGroup}\left(\right)\right)\,'\mathrm{list}'\right)\,\mathrm{QuaternionGroup}\left(\right)\right)$

$\mathrm{true}$

$\mathrm{map}\left(\mathrm{GroupOrder}\,\left[\mathrm{Centre}\,\mathrm{DerivedSubgroup}\,\mathrm{FrattiniSubgroup}\right]\left(\mathrm{QuaternionGroup}\left(\right)\right)\right)$

$\left[2\,2\,2\right]$

$\mathrm{map}\left(\mathrm{op}@\mathrm{Generators}\,\left\{\mathrm{Centre}\,\mathrm{DerivedSubgroup}\,\mathrm{FrattiniSubgroup}\right\}\left(\mathrm{QuaternionGroup}\left(\right)\right)\right)$

$\left\{\mathbf{module}\left(\right)...\mathbf{end\; module}\right\}$

$\mathrm{IsPerfectOrderClassesGroup}\left(\mathrm{QuaternionGroup}\left(4\right)\right)$

$\mathrm{false}$

The GroupTheory[QuaternionGroup] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.

The GroupTheory[QuaternionGroup] command was updated in Maple 2021.