The Janko groups $J_1$, $J_2$, $J_3$ and $J_4$ are four of the sporadic simple groups.  The first Janko group $J_1$ was the first sporadic simple group to be discovered (in 1965, by Zvonimir Janko) since the Mathieu groups were discovered in the nineteenth century.  Janko predicted the existence of the remaining Janko groups, which were later proved to exist by others.

Three of the Janko groups (all except $J_2$) are pariahs: they do not occur as subquotients of the Monster. The group $J_2$ is also called the Hall-Janko group or the Hall-Janko-Wales group; its existence was proven by Marshall Hall Jr. and David Wales in 1968. The third Janko group $J_3$ was shown to exist in 1969 by Graham Higman and John McKay. Simon Norton proved that the fourth Janko group $J_4$ exists in 1980.

The JankoGroup( n ) command returns a permutation group, or a finitely presented group, isomorphic to the Janko group $J_n$, for n = 1, 2, 3, 4.

For n = 1, 2, 3, you may specify either form = "permgroup" (the default) or form = "fpgroup". The fourth Janko group $J_4$ is not available as a permutation group in the current version of Maple. Thus, for n=4, form="fpgroup" is the default.

In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

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

$G≔\mathrm{JankoGroup}\left(2\right)$

$G≔J_2$

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

$100$

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

$604800$

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

$\mathrm{true}$

$G≔\mathrm{JankoGroup}\left(4\,'\mathrm{form}'=''fpgroup''\right)$

$G≔J_4$

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

$86775571046077562880$

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

$\left\{\left[t\,t\right]\,\left[x\,x\right]\,\left[y\,y\,y\right]\,\left[\frac{1}{t}\,\frac{1}{x}\,t\,x\right]\,\left[\frac{1}{t}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,y\,\frac{1}{x}\,y\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,t\,y\,x\,y\,x\,\frac{1}{y}\,x\,\frac{1}{y}\,x\,y\,x\,y\,x\,y\right]\,\left[x\,y\,x\,y\,x\,\frac{1}{y}\,x\,y\,x\,y\,x\,\frac{1}{y}\,x\,y\,x\,y\,x\,\frac{1}{y}\,x\,y\,x\,\frac{1}{y}\,x\,\frac{1}{y}\,x\,y\,x\,\frac{1}{y}\,x\,\frac{1}{y}\,x\,y\,x\,\frac{1}{y}\,x\,\frac{1}{y}\right]\,\left[\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,x\,y\,x\,y\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,x\,y\,x\,y\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,x\,y\,x\,y\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,x\,y\,x\,y\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,x\,y\,x\,y\right]\,\left[x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\right]\,\left[\frac{1}{x}\,\frac{1}{y}\,x\,y\,\frac{1}{x}\,\frac{1}{y}\,x\,y\,\frac{1}{x}\,\frac{1}{y}\,x\,y\,\frac{1}{x}\,\frac{1}{y}\,x\,y\,\frac{1}{x}\,\frac{1}{y}\,x\,y\,\frac{1}{x}\,\frac{1}{y}\,x\,y\,\frac{1}{x}\,\frac{1}{y}\,x\,y\,\frac{1}{x}\,\frac{1}{y}\,x\,y\,\frac{1}{x}\,\frac{1}{y}\,x\,y\,\frac{1}{x}\,\frac{1}{y}\,x\,y\,\frac{1}{x}\,\frac{1}{y}\,x\,y\,\frac{1}{x}\,\frac{1}{y}\,x\,y\right]\,\left[y\,\frac{1}{x}\,y\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,y\,\frac{1}{x}\,\frac{1}{y}\,t\,y\,x\,\frac{1}{y}\,x\,y\,x\,\frac{1}{y}\,x\,y\,\frac{1}{x}\,y\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,y\,\frac{1}{x}\,\frac{1}{y}\,t\,y\,x\,\frac{1}{y}\,x\,y\,x\,\frac{1}{y}\,x\,y\,\frac{1}{x}\,y\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,y\,\frac{1}{x}\,\frac{1}{y}\,t\,y\,x\,\frac{1}{y}\,x\,y\,x\,\frac{1}{y}\,x\right]\,\left[x\,y\,x\,y\,x\,\frac{1}{y}\,x\,y\,x\,\frac{1}{y}\,x\,y\,x\,\frac{1}{y}\,x\,y\,x\,y\,x\,\frac{1}{y}\,x\,y\,x\,\frac{1}{y}\,x\,y\,x\,\frac{1}{y}\,x\,y\,x\,y\,x\,\frac{1}{y}\,x\,y\,x\,\frac{1}{y}\,x\,y\,x\,\frac{1}{y}\,x\,y\,x\,y\,x\,\frac{1}{y}\,x\,y\,x\,\frac{1}{y}\,x\,y\,x\,\frac{1}{y}\right]\,\left[y\,x\,y\,x\,y\,x\,y\,y\,x\,y\,x\,y\,x\,y\,y\,x\,y\,x\,y\,x\,y\,t\,\frac{1}{y}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,t\,x\,y\,x\,y\,x\,y\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,y\,y\,x\,y\,x\,y\,x\,y\,y\,x\,y\,x\,y\,x\,y\,y\,x\,y\,x\,y\,x\,y\,t\,\frac{1}{y}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,\frac{1}{y}\,\frac{1}{x}\,t\,x\,y\,x\,y\,x\,y\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,x\,y\,y\right]\right\}$

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

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