Important: The group package has been deprecated. Use the superseding command GroupTheory[TransitiveGroup] instead.

The function looks up in tables the information specified in opts for the group or groups specified by deg and num.  The information is returned as a sequence, of which the ith element is the information specified by the ith element of opts.

The first two forms of this function use both the degree and number, and hence return information about a specific conjugacy class of transitive groups in the symmetric group on $\{1,...,\mathrm{deg}\}$.

The numbering of the classes used is described in "On Transitive Permutation Groups" by J.H. Conway, A. Hulpke, and J. McKay, LMS J. Comput. Math. 1 (1998), 1-8.

For instance, $\left[9\,15\right]$ is the 15-th class of degree 9 groups in the paper aforementioned. An alternative valid input for this class is the string $''9T15''$.

The available options are

'generators'     : set of group generators for a representative of the class,

'names'          : set of group names, using the notation of group[transnames]

'order'          : group order

'parity'         : parity of the group, given by 1 for even or -1 for odd

'SnConjugates'      : Array in which the ith element is the number of group elements with a cycle type given by the partition part = combinat[decodepart](deg, i)

'SnConjugates(part)': number of group elements with a cycle type given by the partition part

The third form of this function specifies only a group degree. All of the above options can be used here, and for each option an Array is returned, of which the ith element corresponds to the option information for the ith group of degree deg.  In addition, the option 'number' can be used which returns an integer corresponding to the number of classes of degree deg.

The command with(group,transgroup) allows the use of the abbreviated form of this command.

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

$\mathrm{transgroup}\left(''5T3''\,'\mathrm{names}'\right)$

$\left\{''5:4''\,''F(5)''\right\}$

$\mathrm{transgroup}\left(\left[5\,3\right]\,'\mathrm{names}'\,'\mathrm{parity}'\right)$

$\left\{''5:4''\,''F(5)''\right\},\mathrm{-1}$

$\mathrm{transgroup}\left(''6T12''\,'\mathrm{order}'\,'\mathrm{parity}'\,'\mathrm{generators}'\right)$

$60,1,\left\{\left[\left[1\,2\,3\,4\,6\right]\right]\,\left[\left[5\,6\right]\,\left[1\,4\right]\right]\right\}$

$\mathrm{transgroup}\left(10\,'\mathrm{number}'\right)$

$45$

$\mathrm{transgroup}\left(5\,'\mathrm{number}'\,'\mathrm{order}'\right)$

$5,\left[\begin{array}{ccccc}5& 10& 20& 60& 120\end{array}\right]$

$\mathrm{transgroup}\left(\left[4\,3\right]\,'\mathrm{SnConjugates}'\right)$

$\left[\begin{array}{ccccc}1& 2& 3& 0& 2\end{array}\right]$

$\mathrm{transgroup}\left(\left[4\,3\right]\,'\mathrm{SnConjugates}\left(\left[2\,2\right]\right)'\right)$

$3$

$\mathrm{transgroup}\left(6\,'\mathrm{SnConjugates}\left(\left[2\,2\,2\right]\right)'\right)$

$\left[\begin{array}{cccccccccccccccc}1& 3& 4& 0& 3& 1& 0& 6& 6& 0& 7& 0& 6& 10& 0& 15\end{array}\right]$