CT_{(n)} is isomorphic to S_N for n > 3, with N = lcm(2, 3, ..., n).
Kohl, A simple group generated by involutions interchanging residue classes of the integers, Math
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A note on the horizontal class transposition group
CT_{(n)} is isomorphic to S_N for n > 3, with N = lcm(2, 3, ..., n).