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This paper explores the genetic code's algebraic structures associated with the four mRNA (or DNA) bases A, G, C, and U. We have obtained quotient group structures of codons by considering the transition and substitution mutation. In these quotient group structures, cosets (codon members) explain intriguing interactions between the algebraic properties of codons and the physico-chemical properties (polarity, hydrophilicity, and hydrophobicity) of amino acids. Considering the evolutionary impacts of base locations in a codon, the base's hydrogen bond number, and the base's chemical form distinctions, we have generated a distance-based amino acids matrix. This matrix exhibits a fascinating association between distance measurements and amino acids' physico-chemical aspects. Also, we have obtained multiple amino acid graphs relating to this distance-giving matrix, which explores the evolutionary organization of amino acids.
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