The QuaternionGroup( n ) calling sequence constructs a generalized quaternion group of order $2^n$, where $3\le n$ is an integer.

The argument n is optional, and is taken to be equal to $3$ by default, so the calling sequence QuaternionGroup() returns a quaternion group of order $8$.

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 generalized quaternion group is constructed as a permutation group by default.

But, 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.

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

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

$Q$

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

$Q$

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

$Q$

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

$8$

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

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

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

$8,4$

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

$\mathrm{false}$

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

$\mathrm{ctQ}≔\left[\begin{array}{cccccc}C& \mathrm{1a}& \mathrm{2a}& \mathrm{4a}& \mathrm{4b}& \mathrm{4c}\\ \mathrm{|C|}& 1& 1& 2& 2& 2\\ \mathrm{X1}& 1& 1& 1& 1& 1\\ \mathrm{X2}& 1& 1& \mathrm{-1}& \mathrm{-1}& 1\\ \mathrm{X3}& 1& 1& \mathrm{-1}& 1& \mathrm{-1}\\ \mathrm{X4}& 1& 1& 1& \mathrm{-1}& \mathrm{-1}\\ \mathrm{X5}& 2& \mathrm{-2}& 0& 0& 0\end{array}\right]$

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

$\mathrm{ctD}≔\left[\begin{array}{cccccc}C& \mathrm{1a}& \mathrm{2a}& \mathrm{2b}& \mathrm{2c}& \mathrm{4a}\\ \mathrm{|C|}& 1& 1& 2& 2& 2\\ \mathrm{X1}& 1& 1& 1& 1& 1\\ \mathrm{X2}& 1& 1& \mathrm{-1}& \mathrm{-1}& 1\\ \mathrm{X3}& 1& 1& \mathrm{-1}& 1& \mathrm{-1}\\ \mathrm{X4}& 1& 1& 1& \mathrm{-1}& \mathrm{-1}\\ \mathrm{X5}& 2& \mathrm{-2}& 0& 0& 0\end{array}\right]$

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

$\mathrm{true}$

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

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

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

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

$\mathrm{true}$

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

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

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

$\left\{\left(1\,3\right)\left(2\,4\right)\left(5\,8\right)\left(6\,7\right)\right\}$

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

$\mathrm{true}$

$\mathrm{IsHamiltonian}\left(\mathrm{QuaternionGroup}\left(5\right)\right)$

$\mathrm{false}$

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

$\mathrm{false}$

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

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