The sporadic second Janko group $J_2$ is generated by an element of order two and one of order three and hence is a quotient of the modular group $PSL_2(\mathbb{Z}) = C_2 \ast C_3$.

This Janko group has a 100-dimensional permutation representation and hence there is an index 100 subgroup $G$ of the modular group such that the fundamental domain $\mathbb{H}/G$ for the action of $G$ on the upper-half plane by Moebius transformations consists of 100 triangles in the Dedekind tessellation.

Four years ago i tried to depict this fundamental domain in the Farey symbols of sporadic groups-post using Chris Kurth’s **kfarey** package in Sage, but the result was rather disappointing.

Now, the kfarey-package has been greatly extended by Hartmut Monien of Bonn University and is integrated in the latest version of Sage, SAGE 5.0, released a few weeks ago.

Using the Farey symbol sage-documentation it is easy to repeat the calculations from four years ago and, this time, we do obtain this rather nice picture of the fundamental domain

But, there’s a lot more one can do with the new package. By combining the **.fractions()** with the **.pairings()** info it is now possible to get the corresponding Farey code which consists of 34 edges, starting off with

Perhaps surprisingly (?) $G$ turns out to be a genus zero modular subgroup. Naturally, i couldn’t resist drawing the fundamental domain for the 12-dimensional permutations representation of the Mathieu group $M_{12}$ and compare it with that of last time.

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