Diatonic PCs

Partly because it is a smaller number and partly because it is a prime
number, there are far few transitive permutation groups of degree 7
than there are of degree 12--in fact, only seven of them. These can be
thought of in terms of 7-equal, or in terms of 7-note scales--the
diatonic scale in particular. Here's the scoop:

C(7) Cyclic group of order 7
(1,2,3,4,5,6,7)
x^7+x^6+3*x^5+5*x^4+5*x^3+3*x^2+x+1

D(7) Dihedral group of order 14
(1,2,3,4,5,6,7) and (1,6)(2,5)(3,4)
x^7+x^6+3*x^5+4*x^4+4*x^3+3*x^2+x+1

F21(7) Frobenius group of order 21
(1,2,3,4,5,6,7) and (1,2,4)(3,6,5)
x^7+x^6+x^5+3*x^4+3*x^3+x^2+x+1

F42(7) Frobenius group of order 42 (affine line group)
(1,2,3,4,5,6,7) and (1,3,2,6,4,5)
x^7+x^6+x^5+2*x^4+2*x^3+x^2+x+1

L(3,2) Group of Fano plane of order 168
(1,2,3,4,5,6,7) and (1,2)(3,6)
x^7+x^6+x^5+2*x^4+2*x^3+x^2+x+1

A7 Alternating group, order 7!/2 = 2520
x^7+x^6+x^5+x^4+x^3+x^2+x+1

S7 Symmetric group, order 7! = 5040
x^7+x^6+x^5+x^4+x^3+x^2+x+1

We have five cyclic triad forms, four dihedreal triad forms, three F21
triad forms, two F42 triad forms and two L(3,2) triad forms; for A7
and S7 all chords with the same number of notes are the same.