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The complex fuzzy environment is an innovative tool to handle ambiguous situations in different mathematical problems. In this article, we commence the abstraction of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ρ</mi><mo>,</mo><mi>η</mi><mo>)</mo></mrow></semantics></math></inline-formula>-complex fuzzy sets, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ρ</mi><mo>,</mo><mi>η</mi><mo>)</mo></mrow></semantics></math></inline-formula>-complex fuzzy subgroupoid, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ρ</mi><mo>,</mo><mi>η</mi><mo>)</mo></mrow></semantics></math></inline-formula>-complex fuzzy subgroups and describe important examples of the symmetric group under <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ρ</mi><mo>,</mo><mi>η</mi><mo>)</mo></mrow></semantics></math></inline-formula>-complex fuzzy sets. Additionally, we discuss the conjugacy class of the group with respect to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ρ</mi><mo>,</mo><mi>η</mi><mo>)</mo></mrow></semantics></math></inline-formula>-complex fuzzy normal subgroups. We define <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ρ</mi><mo>,</mo><mi>η</mi><mo>)</mo></mrow></semantics></math></inline-formula>-complex fuzzy cosets and elaborate upon the certain operation of this analog to group theoretic operation. We prove that factors regarding the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ρ</mi><mo>,</mo><mi>η</mi><mo>)</mo></mrow></semantics></math></inline-formula>-complex fuzzy normal subgroup form a group and establish an ordinary homomorphism. Moreover, we create the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>ρ</mi><mo>,</mo><mi>η</mi><mo>)</mo></mrow></semantics></math></inline-formula>-complex fuzzy subgroup of the factor group.