Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$.
Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class transposition
$\tau_{r_1(m_1),r_2(m_2)}$ be the permutation of $\mathbb{Z}$ which interchanges
$r_1+km_1$ and $r_2+km_2$ for every $k \in \mathbb{Z}$ and which fixes everything else.

Given a constant $C \in \mathbb{N}$, what is the largest possible order of a *finite*
group generated by 3 class transpositions interchanging residue classes with moduli
less than or equal to $C$?

The question asks for asymptotic behavior as well as for exact values for small $C$.

Remarks:

Groups generated by 3 class transpositions can either be finite or infinite.
Among the 52394 unordered triples of pairwise distinct class transpositions
interchanging residue classes with moduli less than or equal to $C = 6$,
21948 generate finite groups and 30446 generate infinite groups.

Largest finite groups for $C = 3, \dots, 6$ are as follows:

$C = 3$: $|<\tau_{(0(2),1(2)}, \tau_{0(3),1(3)}, \tau_{(1(3),2(3)}>| = 120 = 2^3 \cdot 3 \cdot 5$.

$C = 4$: $|<\tau_{(1(3),2(3)}, \tau_{(0(2),1(4)}, \tau_{2(4),3(4)}>| =
2^{15} \cdot 3^8 \cdot 5^3 \cdot 7^2 \cdot 11$.

$C = 5$: $|<\tau_{0(3),2(3)}, \tau_{1(2),0(4)}, \tau_{2(5),3(5)}>| =$
$2^{95} \cdot 3^{47} \cdot 5^{20} \cdot 7^{14} \cdot 11^7 \cdot 13^6 \cdot 17^3
\cdot 19^2 \cdot 23^2 \cdot 29^2 \cdot 31^2$.

$C = 6$: $|<\tau_{1(5),4(5)}, \tau_{0(3),1(6)}, \tau_{3(4),0(6)}>| =$
$2^{200} \cdot 3^{103} \cdot 5^{48} \cdot 7^{28} \cdot 11^{16} \cdot 13^{13}
\cdot 17^8 \cdot 19^6 \cdot 23^6 \cdot 29$.

The mentioned groups and their orders have been determined with the GAP package RCWA,
cf. <http://www.gap-system.org/Packages/rcwa.html>.