Tables of all commutation symmetries belonging to subgroups of Sr with r £ 6

B. Fiedler
Eichelbaumstr. 13, D-04249 Leipzig
email: bfiedler@fiemath.de

April, 1999

The following Tables 1, 2, 3, 4 present a complete list of all commutation symmetries e: C ® C× that can be defined over arbitrary subgroups C of S6. The calculation of this list is described in Section ..

We list only representatives C of classes of conjugate subgroups in our Tables. Every row of a Table contains information relating to such a representative C Í S6. The first column of a Table gives the order |C| of C, the third column contains a set of generators of C. If C is a conjugate group of a subgroup of a Sr with r < 6, then we have chosen such a representative C that possesses the fixed points r+1, ¼, 6. Thus all commutation symmetries belonging to subgroups of a Sr with r < 6 can be found in our Tables, too. (Look at the generators.)

The second column of a Table contains the orders of the images e(C) of commutation symmetries e that are possible on C. An entry '663321' means that there are two groups e(C) of order 6, two groups e(C) of order 3, one group e(C) of order 2 and one group e(C) of order 1 up to the equivalence of commutation symmetries defined in Section ..

The colums 4, 5, ... give the values that a homomorphism e: C ® S1 takes on the generators of C. The sequence of these columns corresponds to the sequence of the orders |e(C)| in column 2. If an entry '663321' is given in the second column, then the columns 4 and 5 contain the e-values of groups e(C) of order 6. The e-values of the groups e(C) of order 3 are arranged in columns 6 and 7 and the e-values of the groups groups e(C) of order 2 and 1 are given in columns 8 and 9, respectively.

If for two such homomorphisms e, e¢: C ® S1 an isomorphism f: e(C) ® e(C) with f ¹ id  exists such that there holds true e¢ = f°e, then we write a sign ' @ ' in the Table. A ' @ ' in the k-th columns means that the homomorphism e¢ in the k-th column can be obtained from the homomorphism e in the (k - 1)-th column by an isomorphism f.

 

Table 1: All commutation symmetries of solvable subgroups of the S6.
|C| |e(C)| generators of C values of e
2 21 {2,1,3,4,5,6} -1 1
2 21 {2,1,4,3,5,6} -1 1
2 21 {2,1,4,3,6,5} -1 1
3 331 {2,3,1,4,5,6} e[(-2 i)/ 3] p @ e[2 i/ 3] p 1
3 331 {2,3,1,5,6,4} e[(-2 i)/ 3] p @ e[2 i/ 3] p 1
4 4421 {3,4,2,1,5,6} -i @ i -1 1
4 221 {1,2,4,3,5,6} -1 1 1
{2,1,3,4,5,6} -1 -1 1
4 21 {2,1,4,3,5,6} -1 1
{3,4,1,2,5,6} -1 1
4 4421 {2,1,5,6,4,3} -i @ i -1 1
4 2221 {1,2,3,4,6,5} -1 1 -1 1
{2,1,4,3,5,6} -1 -1 1 1
4 221 {1,2,4,3,6,5} -1 1 1
{2,1,5,6,3,4} -1 -1 1
4 21 {1,2,4,3,6,5} -1 1
{2,1,3,4,6,5} -1 1
5 55551 {2,3,4,5,1,6} e[(-2 i)/ 5] p @ e[4 i/ 5] p @ e[(-4 i)/ 5] p @ e[2 i/ 5] p 1
6 21 {2,3,1,4,5,6} 1 1
{1,3,2,4,5,6} -1 1
6 663321 {2,3,1,5,4,6} e[(-i)/ 3] p @ ei/3 p e[(-2 i)/ 3] p @ e[2 i/ 3] p -1 1
6 21 {1,2,4,5,3,6} 1 1
{2,1,3,5,4,6} -1 1
6 663321 {4,3,6,5,2,1} e[(-i)/ 3] p @ ei/3 p e[(-2 i)/ 3] p @ e[2 i/ 3] p -1 1
6 21 {3,5,4,1,6,2} 1 1
{1,2,4,3,6,5} -1 1
6 21 {4,6,2,5,1,3} 1 1
{2,1,4,3,6,5} -1 1
8 2221 {3,4,2,1,5,6} 1 -1 -1 1
{1,2,4,3,5,6} -1 1 -1 1
8 4444 {3,4,2,1,5,6} -i @ i -i @ i -1 -1 1 1
2221 {3,4,2,1,6,5} i @ -i -i @ i 1 -1 -1 1
8 2221 {2,1,5,6,4,3} 1 -1 -1 1
{1,2,3,4,6,5} -1 1 -1 1

 

Table 2: All commutation symmetries of solvable subgroups of the S6 (continuation).
|C| |e(C)| generators of C values of e
8 2221 {2,1,5,6,4,3} 1 -1 -1 1
{1,2,5,6,3,4} -1 1 -1 1
8 2221 {1,2,5,6,4,3} -1 1 -1 1
{2,1,3,4,6,5} -1 -1 1 1
8 2221 {1,2,3,4,6,5} 1 1 -1 1
{1,2,4,3,5,6} 1 -1 -1 1
{2,1,3,4,5,6} -1 -1 -1 1
8 2221 {1,2,3,4,6,5} 1 -1 -1 1
{2,1,4,3,5,6} 1 1 1 1
{3,4,1,2,5,6} -1 1 -1 1
9 33331 {1,2,3,5,6,4} e[(-2 i)/ 3] p @ e[2 i/ 3] p 1 @ 1 1
{2,3,1,4,5,6} e[(-2 i)/ 3] p @ e[2 i/ 3] p e[(-2 i)/ 3] p @ e[2 i/ 3] p 1
10 21 {2,4,1,5,3,6} 1 1
{1,3,2,5,4,6} -1 1
12 331 {1,3,4,2,5,6} e[(-2 i)/ 3] p @ e[2 i/ 3] p 1
{2,3,1,4,5,6} e[2 i/ 3] p @ e[(-2 i)/ 3] p 1
12 2221 {2,1,4,5,3,6} -1 -1 1 1
{1,2,3,5,4,6} -1 1 -1 1
12 2221 {5,3,6,2,4,1} -1 -1 1 1
{1,2,4,3,6,5} -1 1 -1 1
12 331 {3,4,5,6,1,2} e[(-2 i)/ 3] p @ e[2 i/ 3] p 1
{3,4,6,5,2,1} e[(-2 i)/ 3] p @ e[2 i/ 3] p 1
16 2222 {1,2,5,6,4,3} 1 -1 -1 -1 -1 1 1 1
2221 {2,1,5,6,4,3} -1 -1 1 -1 1 1 -1 1
{1,2,3,4,6,5} 1 1 1 -1 -1 -1 -1 1
18 663321 {2,3,1,4,6,5} e[(-i)/ 3] p @ ei/3 p e[(-2 i)/ 3] p @ e[2 i/ 3] p -1 1
{2,3,1,5,4,6} e[(-i)/ 3] p @ ei/3 p e[(-2 i)/ 3] p @ e[2 i/ 3] p -1 1
18 663321 {4,5,6,2,3,1} e[(-i)/ 3] p @ ei/3 p e[(-2 i)/ 3] p @ e[2 i/ 3] p -1 1
{4,5,6,3,1,2} ei/3 p @ e[(-i)/ 3] p e[2 i/ 3] p @ e[(-2 i)/ 3] p -1 1
18 21 {1,2,3,5,6,4} 1 1
{2,3,1,4,5,6} 1 1
{1,3,2,4,6,5} -1 1
20 4421 {2,4,1,5,3,6} 1 @ 1 1 1
{1,4,5,3,2,6} -i @ i -1 1

 

Table 3: All commutation symmetries of solvable subgroups of the S6 (continuation).
|C| |e(C)| generators of C values of e
24 21 {2,3,4,1,5,6} -1 1
{2,4,1,3,5,6} -1 1
24 663321 {3,4,5,6,2,1} e[(-i)/ 3] p @ ei/3 p e[(-2 i)/ 3] p @ e[2 i/ 3] p -1 1
{3,4,6,5,1,2} e[(-i)/ 3] p @ ei/3 p e[(-2 i)/ 3] p @ e[2 i/ 3] p -1 1
24 663321 {1,3,4,2,6,5} e[(-i)/ 3] p @ ei/3 p e[(-2 i)/ 3] p @ e[2 i/ 3] p -1 1
{2,3,1,4,6,5} ei/3 p @ e[(-i)/ 3] p e[2 i/ 3] p @ e[(-2 i)/ 3] p -1 1
24 21 {2,1,4,5,6,3} -1 1
{2,1,4,6,3,5} -1 1
24 21 {2,1,5,6,4,3} -1 1
{3,4,2,1,6,5} -1 1
24 21 {1,2,5,6,4,3} -1 1
{3,4,2,1,5,6} -1 1
36 4421 {4,5,6,1,3,2} -i @ i -1 1
{4,5,6,2,1,3} -i @ i -1 1
36 221 {1,3,2,5,6,4} -1 -1 1
{2,1,3,5,6,4} -1 -1 1
{2,3,1,4,6,5} -1 1 1
36 221 {4,5,6,2,3,1} -1 -1 1
{4,5,6,3,1,2} -1 -1 1
{4,6,5,2,1,3} 1 -1 1
48 2221 {2,1,3,5,6,4} -1 -1 1 1
{2,1,4,5,3,6} -1 -1 1 1
{1,2,4,5,6,3} -1 1 -1 1
48 2221 {3,4,5,6,2,1} -1 1 -1 1
{3,4,6,5,1,2} -1 1 -1 1
{1,2,5,6,4,3} 1 -1 -1 1
72 2221 {1,3,2,5,6,4} -1 1 -1 1
{2,1,3,5,6,4} -1 1 -1 1
{2,3,1,4,6,5} -1 1 -1 1
{4,5,6,2,3,1} -1 -1 1 1

 

Table 4: All commutation symmetries of non-solvable subgroups of the S6.
|C| |e(C)| generators of C values of e
60 1 {2,3,4,5,1,6} 1
{2,3,5,1,4,6} 1
60 1 {1,3,6,5,2,4} 1
{2,3,4,5,1,6} 1
120 21 {2,3,4,5,1,6} 1 1
{2,1,3,4,5,6} -1 1
120 21 {2,3,4,5,1,6} 1 1
{6,3,2,5,4,1} -1 1
360 1 {1,3,4,5,6,2} 1
{1,3,4,6,2,5} 1
{2,3,4,5,1,6} 1
720 21 {2,3,4,5,6,1} -1 1
{2,1,3,4,5,6} -1 1

 

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On 6 Apr 1999, 23:07.