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

outopt : option of the form output = "list" or output = "iterator"

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".

The AllHamiltonianGroups command accepts an option of the form output = "list" (the default) or output = "iterator". By default, a sequence of the Hamiltonian groups of order n is returned. If you pass the option output = "iterator" to AllHamiltonianGroups, then an iterator object is returned instead.

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

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

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

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

$0$

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

$3$

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

$G≔⟨\left(1\,2\,3\,4\right)\left(5\,6\,8\,7\right)\,\left(1\,5\,3\,8\right)\left(2\,7\,4\,6\right)\,\left(9\,10\right)\,\left(11\,12\,13\right)\,\left(14\,15\,16\,17\,18\,19\,20\,21\,22\right)⟩$

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

$\mathrm{true}$

$\mathrm{AllHamiltonianGroups}\left(432\,'\mathrm{form}'=''fpgroup''\right)$

$⟨\mathrm{\_x3}\,\mathrm{\_x4}\,\mathrm{\_x5}\,\mathrm{\_x6}\,\mathrm{\_x7}\,\mathrm{\_x8}\mid {\mathrm{\_x5}}^2\,{\mathrm{\_x6}}^3\,{\mathrm{\_x7}}^3\,{\mathrm{\_x8}}^3\,{\mathrm{\_x3}}^4\,{\mathrm{\_x3}}^2{\mathrm{\_x4}}^2\,\mathrm{\_x3}\mathrm{\_x4}{\mathrm{\_x3}}^{-1}\mathrm{\_x4}\,{\mathrm{\_x5}}^{-1}{\mathrm{\_x3}}^{-1}\mathrm{\_x5}\mathrm{\_x3}\,{\mathrm{\_x5}}^{-1}{\mathrm{\_x4}}^{-1}\mathrm{\_x5}\mathrm{\_x4}\,{\mathrm{\_x6}}^{-1}{\mathrm{\_x3}}^{-1}\mathrm{\_x6}\mathrm{\_x3}\,{\mathrm{\_x6}}^{-1}{\mathrm{\_x4}}^{-1}\mathrm{\_x6}\mathrm{\_x4}\,{\mathrm{\_x6}}^{-1}{\mathrm{\_x5}}^{-1}\mathrm{\_x6}\mathrm{\_x5}\,{\mathrm{\_x7}}^{-1}{\mathrm{\_x3}}^{-1}\mathrm{\_x7}\mathrm{\_x3}\,{\mathrm{\_x7}}^{-1}{\mathrm{\_x4}}^{-1}\mathrm{\_x7}\mathrm{\_x4}\,{\mathrm{\_x7}}^{-1}{\mathrm{\_x5}}^{-1}\mathrm{\_x7}\mathrm{\_x5}\,{\mathrm{\_x7}}^{-1}{\mathrm{\_x6}}^{-1}\mathrm{\_x7}\mathrm{\_x6}\,{\mathrm{\_x8}}^{-1}{\mathrm{\_x3}}^{-1}\mathrm{\_x8}\mathrm{\_x3}\,{\mathrm{\_x8}}^{-1}{\mathrm{\_x4}}^{-1}\mathrm{\_x8}\mathrm{\_x4}\,{\mathrm{\_x8}}^{-1}{\mathrm{\_x5}}^{-1}\mathrm{\_x8}\mathrm{\_x5}\,{\mathrm{\_x8}}^{-1}{\mathrm{\_x6}}^{-1}\mathrm{\_x8}\mathrm{\_x6}\,{\mathrm{\_x8}}^{-1}{\mathrm{\_x7}}^{-1}\mathrm{\_x8}\mathrm{\_x7}⟩,⟨\mathrm{\_x12}\,\mathrm{\_x13}\,\mathrm{\_x14}\,\mathrm{\_x15}\,\mathrm{\_x16}\mid {\mathrm{\_x14}}^2\,{\mathrm{\_x15}}^3\,{\mathrm{\_x12}}^4\,{\mathrm{\_x12}}^2{\mathrm{\_x13}}^2\,\mathrm{\_x12}\mathrm{\_x13}{\mathrm{\_x12}}^{-1}\mathrm{\_x13}\,{\mathrm{\_x14}}^{-1}{\mathrm{\_x12}}^{-1}\mathrm{\_x14}\mathrm{\_x12}\,{\mathrm{\_x14}}^{-1}{\mathrm{\_x13}}^{-1}\mathrm{\_x14}\mathrm{\_x13}\,{\mathrm{\_x15}}^{-1}{\mathrm{\_x12}}^{-1}\mathrm{\_x15}\mathrm{\_x12}\,{\mathrm{\_x15}}^{-1}{\mathrm{\_x13}}^{-1}\mathrm{\_x15}\mathrm{\_x13}\,{\mathrm{\_x15}}^{-1}{\mathrm{\_x14}}^{-1}\mathrm{\_x15}\mathrm{\_x14}\,{\mathrm{\_x16}}^{-1}{\mathrm{\_x12}}^{-1}\mathrm{\_x16}\mathrm{\_x12}\,{\mathrm{\_x16}}^{-1}{\mathrm{\_x13}}^{-1}\mathrm{\_x16}\mathrm{\_x13}\,{\mathrm{\_x16}}^{-1}{\mathrm{\_x14}}^{-1}\mathrm{\_x16}\mathrm{\_x14}\,{\mathrm{\_x16}}^{-1}{\mathrm{\_x15}}^{-1}\mathrm{\_x16}\mathrm{\_x15}\,{\mathrm{\_x16}}^9⟩,⟨\mathrm{\_x20}\,\mathrm{\_x21}\,\mathrm{\_x22}\,\mathrm{\_x23}\mid {\mathrm{\_x22}}^2\,{\mathrm{\_x20}}^4\,{\mathrm{\_x20}}^2{\mathrm{\_x21}}^2\,\mathrm{\_x20}\mathrm{\_x21}{\mathrm{\_x20}}^{-1}\mathrm{\_x21}\,{\mathrm{\_x22}}^{-1}{\mathrm{\_x20}}^{-1}\mathrm{\_x22}\mathrm{\_x20}\,{\mathrm{\_x22}}^{-1}{\mathrm{\_x21}}^{-1}\mathrm{\_x22}\mathrm{\_x21}\,{\mathrm{\_x23}}^{-1}{\mathrm{\_x20}}^{-1}\mathrm{\_x23}\mathrm{\_x20}\,{\mathrm{\_x23}}^{-1}{\mathrm{\_x21}}^{-1}\mathrm{\_x23}\mathrm{\_x21}\,{\mathrm{\_x23}}^{-1}{\mathrm{\_x22}}^{-1}\mathrm{\_x23}\mathrm{\_x22}\,{\mathrm{\_x23}}^{27}⟩$

$\mathrm{it}≔\mathrm{AllHamiltonianGroups}\left(194400000\,'\mathrm{output}'=''iterator''\right)$

$\mathrm{it}≔\mathrm{\⟨Hamiltonian\; Groups\; Iterator\; for\; Order\; 194400000\⟩}$

$\mathrm{nops}\left(\left[\mathrm{seq}\right]\left(\mathrm{it}\right)\right)$

$49$

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.