The order class profile of a finite group G is the sequence of orders of elements of G, including their multiplicities.

The OrderClassProfile( G ) command computes the order class profile of a finite group G. By default, this is returned as a list of pairs of the form [ order, multiplicity ]. The sorted list of element orders can be returned by using the 'output' = "list" option. To produce, instead, a MultiSet, use the 'output' = "multiset" option.

The OrderClassPolynomial( G, x ) command returns a polynomial encoding of the order class data of the finite group G. It is a univariate polynomial in the indeterminate x for which the coefficient of x^k is equal to the number of elements of order k in G.

The order class number of a finite group $G$ is the number of order classes of elements of $G$.

The OrderClassNumber( G ) command returns the order class number of the finite group G.

The order rank of a finite group $G$ is the number of distinct order class lengths of $G$ greater than $1$.

The OrderRank( G ) command returns the order rank of the finite group G.

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

$G≔\mathrm{Alt}\left(4\right)$

$G≔{\mathbf{A}}_4$

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

$\left[\left[1\,1\right]\,\left[2\,3\right]\,\left[3\,8\right]\right]$

$\mathrm{OrderClassProfile}\left(G\,'\mathrm{output}'=''list''\right)$

$\left[1\,2\,2\,2\,3\,3\,3\,3\,3\,3\,3\,3\right]$

$\mathrm{OrderClassProfile}\left(G\,'\mathrm{output}'=''multiset''\right)$

$\left\{\left[1\,1\right]\,\left[2\,3\right]\,\left[3\,8\right]\right\}$

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

$3$

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

$2$

$\mathrm{OrderClassPolynomial}\left(\mathrm{Symm}\left(6\right)\,'x'\right)$

$240x^6+144x^5+180x^4+80x^3+75x^2+x$

The GroupTheory[OrderClassProfile] command was introduced in Maple 2019.

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

The GroupTheory[OrderClassPolynomial] and GroupTheory[OrderClassNumber] commands were introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.

The GroupTheory[OrderRank] command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.