A group metacyclic if it has a cyclic normal subgroup the quotient by which is also cyclic. Every such group $G$ can be generated by two elements $a$ and $b$, with the subgroup $⟨a⟩$ normal in $G$. The group $G$ is then determined by the action of $⟨b⟩$ on $⟨a⟩$. Since $⟨a⟩$ is normal in $G$, it follows that the conjugate $a^b$ belongs to $⟨a⟩$ so there is a positive integer $k$ for which $a^b=a^{-k}$. Thus, a finite metacyclic group $G$ is completely determined by the orders of $a$ and $b$ and the integer $k$.

The MetacyclicGroup( m, n, k ) command constructs a metacyclic group with generators $a$ and $b$ as described above, such that $a^b=a^{-k}$, and where $a^n=1$ and $b^m=1$.

Note that the generators $a$ and $b$ need not have orders $n$ and $m$, respectively, but that their orders are necessarily divisors of $n$ and $m$.

By default, a permutation group is returned, but you can create a finitely presented group by passing the 'form' = "fpgroup" option.

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

$\mathrm{MetacyclicGroup}\left(6\,8\,5\right)$

$⟨\left(1\,2\,6\,14\,9\,3\right)\left(4\,7\,15\,25\,19\,10\right)\left(5\,8\,16\,26\,20\,11\right)\left(12\,17\,27\,36\,31\,21\right)\left(13\,18\,28\,37\,32\,22\right)\left(23\,29\,38\,44\,41\,33\right)\left(24\,30\,39\,45\,42\,34\right)\left(35\,40\,46\,48\,47\,43\right)\,\left(1\,4\,12\,23\,35\,24\,13\,5\right)\left(2\,7\,17\,29\,40\,30\,18\,8\right)\left(3\,10\,21\,33\,43\,34\,22\,11\right)\left(6\,15\,27\,38\,46\,39\,28\,16\right)\left(9\,19\,31\,41\,47\,42\,32\,20\right)\left(14\,25\,36\,44\,48\,45\,37\,26\right)⟩$

$\mathrm{MetacyclicGroup}\left(6\,8\,5\,'\mathrm{form}'=''permgroup''\right)$

$⟨\left(1\,2\,6\,14\,9\,3\right)\left(4\,7\,15\,25\,19\,10\right)\left(5\,8\,16\,26\,20\,11\right)\left(12\,17\,27\,36\,31\,21\right)\left(13\,18\,28\,37\,32\,22\right)\left(23\,29\,38\,44\,41\,33\right)\left(24\,30\,39\,45\,42\,34\right)\left(35\,40\,46\,48\,47\,43\right)\,\left(1\,4\,12\,23\,35\,24\,13\,5\right)\left(2\,7\,17\,29\,40\,30\,18\,8\right)\left(3\,10\,21\,33\,43\,34\,22\,11\right)\left(6\,15\,27\,38\,46\,39\,28\,16\right)\left(9\,19\,31\,41\,47\,42\,32\,20\right)\left(14\,25\,36\,44\,48\,45\,37\,26\right)⟩$

$\mathrm{MetacyclicGroup}\left(6\,8\,5\,'\mathrm{form}'=''fpgroup''\right)$

$⟨a\,b\mid a^6\,b^8\,b^{-1}ab{a}^5⟩$

$a,b≔\mathrm{op}\left(\mathrm{Generators}\left(\mathrm{MetacyclicGroup}\left(6\,8\,4\right)\right)\right)$

$a,b≔\left(1\,2\,3\right)\left(4\,8\,6\right)\left(5\,9\,7\right)\left(10\,12\,14\right)\left(11\,13\,15\right)\left(16\,20\,18\right)\left(17\,21\,19\right)\left(22\,23\,24\right),\left(1\,4\,10\,16\,22\,17\,11\,5\right)\left(2\,6\,12\,18\,23\,19\,13\,7\right)\left(3\,8\,14\,20\,24\,21\,15\,9\right)$

$\mathrm{PermOrder}\left(a\right)$

$3$

$\mathrm{PermOrder}\left(b\right)$

$8$

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

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