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Let <inline-formula> <math display="inline"> <semantics> <msub> <mi>S</mi> <mn>3</mn> </msub> </semantics> </math> </inline-formula> be the symmetric group on three elements. Let <i>k</i> be a field and <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>M</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> be the full matrix algebra of <inline-formula> <math display="inline"> <semantics> <mrow> <mn>3</mn> <mo>&#215;</mo> <mn>3</mn> </mrow> </semantics> </math> </inline-formula>-matrices over <i>k</i>. In this paper, the <inline-formula> <math display="inline"> <semantics> <mrow> <mi>k</mi> <msub> <mi>S</mi> <mn>3</mn> </msub> </mrow> </semantics> </math> </inline-formula>-module algebra structures on <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>M</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> are described, and classified up to isomorphism.