The degree of a permutation group is the cardinality of the set upon which it acts.  Since permutation groups act on sets of the form $\left\{1\,2\,\dots \,n\right\}$, the degree is the positive integer $n$. In other words, the degree $n$ is the smallest positive integer such that $G$ is a subgroup of ${\mathbf{S}}_n$ .

The Degree( G ) command returns the degree of a permutation group G.

Some group constructors, such as those for the symmetric and alternating groups take a degree parameter as input, and the Degree command returns this value.

The support of the permutation group $G$ is the set of positive integers displaced by some member of $G$.

The Support( G ) command returns the support of G as a Maple set of positive integers.

The SupportLength( G ) command returns the cardinality of the support of G. Note that this may differ from the degree of G.

The MinSupport( G ) command returns the smallest positive integer that is displaced by the permutation group G. Similarly, the MaxSupport( G ) command returns the largest positive integer displaced by some member of G.

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

$G≔\mathrm{Group}\left(\left\{\mathrm{Perm}\left(\left[\left[1\,2\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[2\,3\,4\right]\right]\right)\right\}\right)$

$G≔⟨\left(1\,2\right)\,\left(2\,3\,4\right)⟩$

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

$4$

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

$\left\{1\,2\,3\,4\right\}$

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

$4$

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

$1$

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

$4$

$G≔\mathrm{DihedralGroup}\left(14\right)$

$G≔{\mathbf{D}}_{14}$

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

$14$

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

$\left\{1\,2\,3\,4\,5\,6\,7\,8\,9\,10\,11\,12\,13\,14\right\}$

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

$14$

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

$1$

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

$14$

$G≔\mathrm{AlternatingGroup}\left(10\right)$

$G≔{\mathbf{A}}_{10}$

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

$10$

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

$\left\{1\,2\,3\,4\,5\,6\,7\,8\,9\,10\right\}$

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

$10$

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

$1$

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

$10$

$G≔\mathrm{Group}\left(\left\{\mathrm{Perm}\left(\left[\left[4\,7\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[2\,4\,7\right]\right]\right)\right\}\right)$

$G≔⟨\left(4\,7\right)\,\left(2\,4\,7\right)⟩$

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

$7$

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

$\left\{2\,4\,7\right\}$

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

$3$

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

$2$

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

$7$

The GroupTheory[Degree], GroupTheory[Support], GroupTheory[SupportLength], GroupTheory[MinSupport] and GroupTheory[MaxSupport] commands were introduced in Maple 17.

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