Group action invariants
| Degree $n$: | $33$ | |
| Transitive number $t$: | $5$ | |
| Group: | $S_3\times D_{11}$ | |
| Parity: | $-1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $1$ | |
| Generators: | (1,16,32,14,30,11,27,8,24,4,19)(2,18,33,13,28,10,25,7,22,6,20,3,17,31,15,29,12,26,9,23,5,21), (1,6,2,4,3,5)(7,33,8,31,9,32)(10,28,11,29,12,30)(13,25,14,26,15,27)(16,23,17,24,18,22)(19,21,20) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $12$: $D_{6}$ $22$: $D_{11}$ $44$: $D_{22}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 11: $D_{11}$
Low degree siblings
There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| Cycle Type | Size | Order | Representative |
| $ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $11$ | $2$ | $( 4,32)( 5,33)( 6,31)( 7,29)( 8,30)( 9,28)(10,26)(11,27)(12,25)(13,23)(14,24) (15,22)(16,19)(17,20)(18,21)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 2, 3)( 5, 6)( 7, 9)(10,12)(13,15)(17,18)(20,21)(22,23)(25,26)(28,29)(31,33)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $33$ | $2$ | $( 2, 3)( 4,32)( 5,31)( 6,33)( 7,28)( 8,30)( 9,29)(10,25)(11,27)(12,26)(13,22) (14,24)(15,23)(16,19)(17,21)(18,20)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)$ |
| $ 6, 6, 6, 6, 6, 3 $ | $22$ | $6$ | $( 1, 2, 3)( 4,33, 6,32, 5,31)( 7,30, 9,29, 8,28)(10,27,12,26,11,25) (13,24,15,23,14,22)(16,20,18,19,17,21)$ |
| $ 11, 11, 11 $ | $2$ | $11$ | $( 1, 4, 8,11,14,16,19,24,27,30,32)( 2, 5, 9,12,15,17,20,22,25,28,33) ( 3, 6, 7,10,13,18,21,23,26,29,31)$ |
| $ 22, 11 $ | $6$ | $22$ | $( 1, 4, 8,11,14,16,19,24,27,30,32)( 2, 6, 9,10,15,18,20,23,25,29,33, 3, 5, 7, 12,13,17,21,22,26,28,31)$ |
| $ 33 $ | $4$ | $33$ | $( 1, 5, 7,11,15,18,19,22,26,30,33, 3, 4, 9,10,14,17,21,24,25,29,32, 2, 6, 8, 12,13,16,20,23,27,28,31)$ |
| $ 33 $ | $4$ | $33$ | $( 1, 7,15,19,26,33, 4,10,17,24,29, 2, 8,13,20,27,31, 5,11,18,22,30, 3, 9,14, 21,25,32, 6,12,16,23,28)$ |
| $ 22, 11 $ | $6$ | $22$ | $( 1, 7,14,21,27,31, 4,10,16,23,30, 3, 8,13,19,26,32, 6,11,18,24,29) ( 2, 9,15,20,25,33, 5,12,17,22,28)$ |
| $ 11, 11, 11 $ | $2$ | $11$ | $( 1, 8,14,19,27,32, 4,11,16,24,30)( 2, 9,15,20,25,33, 5,12,17,22,28) ( 3, 7,13,21,26,31, 6,10,18,23,29)$ |
| $ 33 $ | $4$ | $33$ | $( 1,10,20,30, 6,15,24,31, 9,16,26, 2,11,21,28, 4,13,22,32, 7,17,27, 3,12,19, 29, 5,14,23,33, 8,18,25)$ |
| $ 22, 11 $ | $6$ | $22$ | $( 1,10,19,29, 4,13,24,31, 8,18,27, 3,11,21,30, 6,14,23,32, 7,16,26) ( 2,12,20,28, 5,15,22,33, 9,17,25)$ |
| $ 11, 11, 11 $ | $2$ | $11$ | $( 1,11,19,30, 4,14,24,32, 8,16,27)( 2,12,20,28, 5,15,22,33, 9,17,25) ( 3,10,21,29, 6,13,23,31, 7,18,26)$ |
| $ 33 $ | $4$ | $33$ | $( 1,13,25, 4,18,28, 8,21,33,11,23, 2,14,26, 5,16,29, 9,19,31,12,24, 3,15,27, 6,17,30, 7,20,32,10,22)$ |
| $ 22, 11 $ | $6$ | $22$ | $( 1,13,27, 6,16,29, 8,21,32,10,24, 3,14,26, 4,18,30, 7,19,31,11,23) ( 2,15,25, 5,17,28, 9,20,33,12,22)$ |
| $ 11, 11, 11 $ | $2$ | $11$ | $( 1,14,27, 4,16,30, 8,19,32,11,24)( 2,15,25, 5,17,28, 9,20,33,12,22) ( 3,13,26, 6,18,29, 7,21,31,10,23)$ |
| $ 11, 11, 11 $ | $2$ | $11$ | $( 1,16,32,14,30,11,27, 8,24, 4,19)( 2,17,33,15,28,12,25, 9,22, 5,20) ( 3,18,31,13,29,10,26, 7,23, 6,21)$ |
| $ 22, 11 $ | $6$ | $22$ | $( 1,16,32,14,30,11,27, 8,24, 4,19)( 2,18,33,13,28,10,25, 7,22, 6,20, 3,17,31, 15,29,12,26, 9,23, 5,21)$ |
| $ 33 $ | $4$ | $33$ | $( 1,17,31,14,28,10,27, 9,23, 4,20, 3,16,33,13,30,12,26, 8,22, 6,19, 2,18,32, 15,29,11,25, 7,24, 5,21)$ |
Group invariants
| Order: | $132=2^{2} \cdot 3 \cdot 11$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | [132, 5] |
| Character table: not available. |