formopt : option of the form form = "permgroup" or form = "fpgroup"

A group is Hamiltonian if it is non-Abelian, and if every subgroup is normal. Every Hamiltonian group has the quaternion group as a direct factor, so the order of every finite Hamiltonian group is a multiple of $8$.

For a positive integer n, the NumHamiltonianGroups( n ) command returns the number of Hamiltonian groups of order n. (This is $0$ if n is not a multiple of $8$.)

The HamiltonianGroup( n, k ) command returns the k-th Hamiltonian group of order n. An exception is raised if n is not a multiple of $8$.

The AllHamiltonianGroups( n ) command returns an expression sequence of all the Hamiltonian groups of order n, where n is a positive integer. Note that NULL is returned if n is not a multiple of $8$.

The HamiltonianGroup and AllHamiltonianGroups commands accept an option of the form form = F, where F may be either of the strings "permgroup" (the default), or "fpgroup".

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

$\mathrm{seq}\left(\mathrm{NumHamiltonianGroups}\left(2^i\right)\,i=1..20\right)$

$0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1$

$\mathrm{NumHamiltonianGroups}\left(25\right)$

$0$

$\mathrm{NumHamiltonianGroups}\left(432\right)$

$3$

$G≔\mathrm{HamiltonianGroup}\left(432\,2\right)$

$G≔\mathrm{<\; a\; permutation\; group\; on\; 22\; letters\; with\; 5\; generators\; >}$

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

$\mathrm{true}$

$\mathrm{AllHamiltonianGroups}\left(432\&comma;'\mathrm{form}'=''fpgroup''\right)$

$⟨i\&comma;j\&comma;\mathrm{\_g}\&comma;\mathrm{\_a1}\&comma;\mathrm{\_a2}\&comma;\mathrm{\_a3}\mid {\mathrm{\_g}}^2\&comma;{\mathrm{\_a1}}^3\&comma;{\mathrm{\_a2}}^3\&comma;{\mathrm{\_a3}}^3\&comma;i^4\&comma;i^2{j}^2\&comma;ij{i}^{-1}j\&comma;{\mathrm{\_a1}}^{-1}{\mathrm{\_g}}^{-1}\mathrm{\_a1}\mathrm{\_g}\&comma;{\mathrm{\_a1}}^{-1}{i}^{-1}\mathrm{\_a1}i\&comma;{\mathrm{\_a1}}^{-1}{j}^{-1}\mathrm{\_a1}j\&comma;{\mathrm{\_a2}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a2}\mathrm{\_a1}\&comma;{\mathrm{\_a2}}^{-1}{\mathrm{\_g}}^{-1}\mathrm{\_a2}\mathrm{\_g}\&comma;{\mathrm{\_a2}}^{-1}{i}^{-1}\mathrm{\_a2}i\&comma;{\mathrm{\_a2}}^{-1}{j}^{-1}\mathrm{\_a2}j\&comma;{\mathrm{\_a3}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a3}\mathrm{\_a1}\&comma;{\mathrm{\_a3}}^{-1}{\mathrm{\_a2}}^{-1}\mathrm{\_a3}\mathrm{\_a2}\&comma;{\mathrm{\_a3}}^{-1}{\mathrm{\_g}}^{-1}\mathrm{\_a3}\mathrm{\_g}\&comma;{\mathrm{\_a3}}^{-1}{i}^{-1}\mathrm{\_a3}i\&comma;{\mathrm{\_a3}}^{-1}{j}^{-1}\mathrm{\_a3}j\&comma;i^{-1}{\mathrm{\_g}}^{-1}i\mathrm{\_g}\&comma;j^{-1}{\mathrm{\_g}}^{-1}j\mathrm{\_g}⟩,⟨i\&comma;j\&comma;\mathrm{\_g0}\&comma;\mathrm{\_a1}\&comma;\mathrm{\_a2}\mid {\mathrm{\_g0}}^2\&comma;{\mathrm{\_a1}}^3\&comma;i^4\&comma;i^2{j}^2\&comma;ij{i}^{-1}j\&comma;{\mathrm{\_a1}}^{-1}{\mathrm{\_g0}}^{-1}\mathrm{\_a1}\mathrm{\_g0}\&comma;{\mathrm{\_a1}}^{-1}{i}^{-1}\mathrm{\_a1}i\&comma;{\mathrm{\_a1}}^{-1}{j}^{-1}\mathrm{\_a1}j\&comma;{\mathrm{\_a2}}^{-1}{\mathrm{\_a1}}^{-1}\mathrm{\_a2}\mathrm{\_a1}\&comma;{\mathrm{\_a2}}^{-1}{\mathrm{\_g0}}^{-1}\mathrm{\_a2}\mathrm{\_g0}\&comma;{\mathrm{\_a2}}^{-1}{i}^{-1}\mathrm{\_a2}i\&comma;{\mathrm{\_a2}}^{-1}{j}^{-1}\mathrm{\_a2}j\&comma;i^{-1}{\mathrm{\_g0}}^{-1}i\mathrm{\_g0}\&comma;j^{-1}{\mathrm{\_g0}}^{-1}j\mathrm{\_g0}\&comma;{\mathrm{\_a2}}^9⟩,⟨i\&comma;j\&comma;\mathrm{\_g1}\&comma;\mathrm{\_a1}\mid {\mathrm{\_g1}}^2\&comma;i^4\&comma;i^2{j}^2\&comma;ij{i}^{-1}j\&comma;{\mathrm{\_a1}}^{-1}{\mathrm{\_g1}}^{-1}\mathrm{\_a1}\mathrm{\_g1}\&comma;{\mathrm{\_a1}}^{-1}{i}^{-1}\mathrm{\_a1}i\&comma;{\mathrm{\_a1}}^{-1}{j}^{-1}\mathrm{\_a1}j\&comma;i^{-1}{\mathrm{\_g1}}^{-1}i\mathrm{\_g1}\&comma;j^{-1}{\mathrm{\_g1}}^{-1}j\mathrm{\_g1}\&comma;{\mathrm{\_a1}}^{27}⟩$

The GroupTheory[HamiltonianGroup], GroupTheory[NumHamiltonianGroups] and GroupTheory[AllHamiltonianGroups] commands were introduced in Maple 2019.

For more information on Maple 2019 changes, see Updates in Maple 2019.