There are 17 different conjugacy classes of subgroups of D12 × D12 which occur as stabilizers of tone rows. For each class Ũi we present the generator of one representative of this class, the order of this representative, the size of the class, i.e. the number of different groups in this conjugacy class, and the number of D12 × D12-orbits of tone rows with stabilizer type Ũi.
In order to search for tone rows with given stabilizer type, determine the conjugacy class of the stabilizer by choosing a number from {1,…,17}.
| name | generators | size of the group | size of the class | size of orbit |
| U1 | identity | 1 | 1 | 827282 |
| U2 | (TI,S6) | 2 | 6 | 912 |
| U3 | (T6,R) | 2 | 6 | 912 |
| U4 | (T6,S6) | 2 | 1 | 130 |
| U5 | (I,SR) | 2 | 36 | 942 |
| U6 | (TI,R) | 2 | 36 | 5649 |
| U7 | (T4,S4) | 3 | 2 | 11 |
| U8 | (T3,S3) | 4 | 2 | 2 |
| U9 | (TI,S6), (T6,R) | 4 | 36 | 96 |
| U10 | (I,SR), (T6,S6) | 4 | 18 | 12 |
| U11 | (TI,R), (T6,S6) | 4 | 18 | 42 |
| U12 | (I,SR), (T4,S4) | 6 | 24 | 2 |
| U13 | (TI,R), (T4,S4) | 6 | 24 | 15 |
| U14 | (I,SR), (T3,S3) | 8 | 36 | 6 |
| U15 | (TI,R), (T2,S2) | 12 | 12 | 2 |
| U16 | (I,SR), (T,S) | 24 | 12 | 1 |
| U17 | (I,SR), (T,S5) | 24 | 12 | 1 |
S is the cyclic shift S=(1,2,3,4,5,6,7,8,9,10,11,12)
R is the retrograde (consisting of 6 transpositions)
R=(1,12)(2,11)(3,10)(4,9)(5,8)(6,7).
Then SR is a retrograde inversion (consisting of 5 transpositions
and two fixed points) SR=(1)(2,12)(3,11)(4,10)(5,9)(6,8)(7).
T is the transposing operator T=(1,2,3,4,5,6,7,8,9,10,11,12)
I is the inversion at pitch class 1 (consisting of 5 transpositions
and two fixed points) I=(1)(2,12)(3,11)(4,10)(5,9)(6,8)(7).
Then TI is an inversion (consisting of 6 transpositions)
TI=(1,2)(3,12)(4,11)(5,10)(6,9)(7,8).