= Solution
The <symmetric group> $S_n$ consists of all bijections of $\{1,\ldots,n\}$, with composition as its operation; its <group order> is $n!$.
An $m$-dimensional complex <group representation> of $G$ is a <group homomorphism> $\rho:G\to GL(V)$, where $V$ is an $m$-dimensional complex <vector space>. After a choice of <basis>, each $\rho(g)$ is an invertible $m\times m$ complex <matrix>, with $\rho(gh)=\rho(g)\rho(h)$.
An <irreducible representation> has $V\ne0$ and no invariant <vector subspace> other than $\{0\}$ and $V$. Two <group representations> $\rho$ on $V$ and $\sigma$ on $W$ are <isomorphic representations> if there exists an invertible <linear map> $S:V\to W$ such that $S\rho(g)=\sigma(g)S$ for every $g$. They are inequivalent if no such map exists; in equal-dimensional <matrix> form this says no fixed change of <basis> conjugates every $\rho(g)$ to $\sigma(g)$.
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