M 22 has three rank 3 permutation representations : one on the 77 hexads with point stabilizer 2 4 : A 6, and two rank 3 actions on 176 heptads that are conjugate under an outer automorphism and have point stabilizer A 7.
22.
This permutation representation preserves a Steiner system S ( 5, 6, 12 ) of 132 special hexads, such that each pentad is contained in exactly 1 special hexad, and the hexads are the supports of the weight 6 codewords of the extended ternary Golay code.
23.
This permutation representation preserves a Steiner system S ( 5, 6, 12 ) of 132 special hexads, such that each pentad is contained in exactly 1 special hexad, and the hexads are the supports of the weight 6 codewords of the extended ternary Golay code.
24.
This permutation representation preserves a Steiner system S ( 5, 6, 12 ) of 132 special hexads, such that each pentad is contained in exactly 1 special hexad, and the hexads are the supports of the weight 6 codewords of the extended ternary Golay code.
25.
The enneagram shows the Law of Seven and the Law of Three in a single symbol with the three forces joined in a triangle at the 3, 6 and 9 points, and in addition six inner lines ( hexad ) following the 1, 4, 2, 8, 5, 7, 1, . . . sequence.
26.
Sets of higher cardinalities are called tetrachords ( or tetrads ), pentachords ( or pentads ), hexachords ( or hexads ), heptachords ( heptads or, sometimes, mixing Latin and Greek roots, " septachords " e . g ., ), octachords ( octads ), nonachords ( nonads ), decachords ( decads ), undecachords, and, finally, the dodecachord.
27.
Allen Forte in " The Structure of Atonal Music " redefines the term " hexachord " to mean what other theorists ( notably Howard Hanson in his " Harmonic Materials of Modern Music : Resources of the Tempered Scale " ) mean by the term " hexad ", a six-note pitch collection which is not necessarily a contiguous segment of a scale or a tone row.
