The permutation groups of cyclic codes are widely applicable in determining the weight distribution of codes, decoding theory and various other areas. In this paper, by employing two distinct matrix representations, we can relate cyclic codes with very long lengths and special generator polynomials to those with prime lengths. Consequently, we mainly determine the permutation groups of certain cyclic codes over $\mathbb{F}_{r^α}$ with lengths $hp$, $r^mp^n$ and $pq$ and special generator polynomials where $h$ is a positive integer and $p$, $q$ and $r$ are distinct prime numbers. For length $pq$, we manage to provide the permutation groups of cyclic codes with generator polynomials $Q_{pq}(x)$(the $pq$-th cyclotomic polynomial) or others, which seems to be the first work about permutation groups of cyclic codes with generator polynomials that are factors of $x^{pq}-1$ but not factors of $x^p-1(\text{or }x^q-1)$.

翻译：循环码的置换群在确定码的重量分布、译码理论及其他诸多领域中具有广泛应用。本文通过利用两种不同的矩阵表示，将具有极长码长和特殊生成多项式的循环码与具有素数长度的循环码联系起来。基于此，我们主要确定了 $\mathbb{F}_{r^α}$ 上具有长度 $hp$、$r^mp^n$ 和 $pq$ 及特殊生成多项式的某些循环码的置换群，其中 $h$ 为正整数，$p$、$q$ 和 $r$ 为互异的素数。对于长度为 $pq$ 的循环码，我们成功给出了生成多项式为 $Q_{pq}(x)$（即第 $pq$ 个分圆多项式）或其他多项式的置换群，这似乎是关于生成多项式为 $x^{pq}-1$ 的因子（而非 $x^p-1$ 或 $x^q-1$ 的因子）的循环码置换群的首项研究成果。