Presentations of mapping class groups in dimension $3$ For any closed oriented surface $M$, its mapping class group $MCG(M)$ can be generated by Dehn twists along certain curves on $M$. A presentation for the group $MCG(M)$ was found in [1] and then simplified in [2].
How about in dimension $3$? The first question is that, unlike in dimension $2$ in which the classification of spaces is easy (genus and number of boundaries), in dimension $3$ the classification is complicated. One either uses

*

*Thurston's geometrization

*Lickorish and Kirby's presentation using links

Neither of them is easy, so I'd expect it much harder to get presentation for the mapping class group.
Question: Nevertheless, is there any presentation known?
In the first view-point, the closest answer I have seen is this, in which Allen Hatcher claims that the MCG of three-manifolds are essentially known [3]. There, the natural map from MCG to $Out(\pi_1(M))$ is considered. While the kernel of this map is understood in the nonprime case [3. section 2], it is not necessarily onto. Even if it's onto, we still do not have a presentation for MCG.
In the second view-point, I have not heard of any result.
Reference

*

*[1] A presentation for the mapping class group of a closed orientable surface-[Hatcher and Thurston]


*[2] A simple presentation for the mapping class group of an orientable surface-[Bronislaw Wajnryb]


*[3] Stabilization for mapping class groups of 3-manifolds-[Allen Hatcher and Nathalie wahl]
 A: Suppose that $M$ is finite volume hyperbolic. Then the mapping class group is finite and there are algorithms to build its multiplication table.  On the other hand, there is definitely no overall pattern to these groups - they are “just” the finite quotients of torsion free lattices in $\mathrm{SL}(2, \mathbb{C})$.
And this is only the tip of the iceberg... the mapping class groups of surfaces appear when thinking about Seifert fibered spaces, outer automorphism groups of free groups show up when dealing with the doubles of handlebodies, and if you connect sum the above together you get more craziness...
A: For certain 3-manifolds (irreducible), if you are willing to take finite index subgroups, the description is relatively easy, and has to do with dehn-twists too.
Let me add the following paper by McCullough which was useful to me when considering this question: https://projecteuclid.org/journals/journal-of-differential-geometry/volume-33/issue-1/Virtually-geometrically-finite-mapping-class-groups-of-3-manifolds/10.4310/jdg/1214446029.full
I think this goes in line with the comments by HJRW which are more detailed, but I thought this might be useful.
