| Summary: | The real forms of complex groups (or algebras) are important in physics and mathematics. The Lie group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>L</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mi>C</mi></mrow></mfenced></mrow></semantics></math></inline-formula> is one of these important groups. There are real forms of the classical Lie group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>L</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mi>C</mi></mrow></mfenced></mrow></semantics></math></inline-formula> and the quantum group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi><mi>L</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mi>C</mi></mrow></mfenced></mrow></semantics></math></inline-formula> in the literature. Inspired by this, in our study, we obtain the real forms of the fractional supergroups shown with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>A</mi><mn>3</mn><mi>N</mi></msubsup><mfenced><mrow><mi>S</mi><mi>L</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mi>C</mi></mrow></mfenced></mrow></mfenced></mrow></semantics></math></inline-formula>, for the non-trivial N = 1 and N = 2 cases, that is, the real forms of the fractional supergroups <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>A</mi><mn>3</mn><mn>1</mn></msubsup><mfenced><mrow><mi>S</mi><mi>L</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mi>C</mi></mrow></mfenced></mrow></mfenced></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>A</mi><mn>3</mn><mn>2</mn></msubsup><mfenced><mrow><mi>S</mi><mi>L</mi><mfenced><mrow><mn>2</mn><mo>,</mo><mi>C</mi></mrow></mfenced></mrow></mfenced></mrow></semantics></math></inline-formula>.
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