Show commands:
Magma
magma: G := TransitiveGroup(33, 4);
Group action invariants
| Degree $n$: | $33$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $4$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $C_3\times D_{11}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
| $\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,31,29,27,23,21,18,14,11,8,5,3,33,28,26,22,20,17,13,10,7,4,2,32,30,25,24,19,16,15,12,9,6), (1,3,2)(4,32,5,33,6,31)(7,28,8,29,9,30)(10,27,11,25,12,26)(13,24,14,22,15,23)(16,19,17,20,18,21) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $22$: $D_{11}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
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 $ | $11$ | $2$ | $( 4,33)( 5,31)( 6,32)( 7,29)( 8,30)( 9,28)(10,25)(11,26)(12,27)(13,22)(14,23) (15,24)(16,20)(17,21)(18,19)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)$ |
| $ 6, 6, 6, 6, 6, 3 $ | $11$ | $6$ | $( 1, 2, 3)( 4,31, 6,33, 5,32)( 7,30, 9,29, 8,28)(10,26,12,25,11,27) (13,23,15,22,14,24)(16,21,18,20,17,19)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)(31,33,32)$ |
| $ 6, 6, 6, 6, 6, 3 $ | $11$ | $6$ | $( 1, 3, 2)( 4,32, 5,33, 6,31)( 7,28, 8,29, 9,30)(10,27,11,25,12,26) (13,24,14,22,15,23)(16,19,17,20,18,21)$ |
| $ 11, 11, 11 $ | $2$ | $11$ | $( 1, 4, 8,12,13,18,19,22,27,30,33)( 2, 5, 9,10,14,16,20,23,25,28,31) ( 3, 6, 7,11,15,17,21,24,26,29,32)$ |
| $ 33 $ | $2$ | $33$ | $( 1, 5, 7,12,14,17,19,23,26,30,31, 3, 4, 9,11,13,16,21,22,25,29,33, 2, 6, 8, 10,15,18,20,24,27,28,32)$ |
| $ 33 $ | $2$ | $33$ | $( 1, 6, 9,12,15,16,19,24,25,30,32, 2, 4, 7,10,13,17,20,22,26,28,33, 3, 5, 8, 11,14,18,21,23,27,29,31)$ |
| $ 33 $ | $2$ | $33$ | $( 1, 7,14,19,26,31, 4,11,16,22,29, 2, 8,15,20,27,32, 5,12,17,23,30, 3, 9,13, 21,25,33, 6,10,18,24,28)$ |
| $ 11, 11, 11 $ | $2$ | $11$ | $( 1, 8,13,19,27,33, 4,12,18,22,30)( 2, 9,14,20,25,31, 5,10,16,23,28) ( 3, 7,15,21,26,32, 6,11,17,24,29)$ |
| $ 33 $ | $2$ | $33$ | $( 1, 9,15,19,25,32, 4,10,17,22,28, 3, 8,14,21,27,31, 6,12,16,24,30, 2, 7,13, 20,26,33, 5,11,18,23,29)$ |
| $ 33 $ | $2$ | $33$ | $( 1,10,21,30, 5,15,22,31, 7,18,25, 3,12,20,29, 4,14,24,33, 9,17,27, 2,11,19, 28, 6,13,23,32, 8,16,26)$ |
| $ 33 $ | $2$ | $33$ | $( 1,11,20,30, 6,14,22,32, 9,18,26, 2,12,21,28, 4,15,23,33, 7,16,27, 3,10,19, 29, 5,13,24,31, 8,17,25)$ |
| $ 11, 11, 11 $ | $2$ | $11$ | $( 1,12,19,30, 4,13,22,33, 8,18,27)( 2,10,20,28, 5,14,23,31, 9,16,25) ( 3,11,21,29, 6,15,24,32, 7,17,26)$ |
| $ 11, 11, 11 $ | $2$ | $11$ | $( 1,13,27, 4,18,30, 8,19,33,12,22)( 2,14,25, 5,16,28, 9,20,31,10,23) ( 3,15,26, 6,17,29, 7,21,32,11,24)$ |
| $ 33 $ | $2$ | $33$ | $( 1,14,26, 4,16,29, 8,20,32,12,23, 3,13,25, 6,18,28, 7,19,31,11,22, 2,15,27, 5,17,30, 9,21,33,10,24)$ |
| $ 33 $ | $2$ | $33$ | $( 1,15,25, 4,17,28, 8,21,31,12,24, 2,13,26, 5,18,29, 9,19,32,10,22, 3,14,27, 6,16,30, 7,20,33,11,23)$ |
| $ 33 $ | $2$ | $33$ | $( 1,16,32,13,28,11,27, 9,24, 4,20, 3,18,31,15,30,10,26, 8,23, 6,19, 2,17,33, 14,29,12,25, 7,22, 5,21)$ |
| $ 33 $ | $2$ | $33$ | $( 1,17,31,13,29,10,27, 7,23, 4,21, 2,18,32,14,30,11,25, 8,24, 5,19, 3,16,33, 15,28,12,26, 9,22, 6,20)$ |
| $ 11, 11, 11 $ | $2$ | $11$ | $( 1,18,33,13,30,12,27, 8,22, 4,19)( 2,16,31,14,28,10,25, 9,23, 5,20) ( 3,17,32,15,29,11,26, 7,24, 6,21)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $66=2 \cdot 3 \cdot 11$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 66.2 | magma: IdentifyGroup(G);
|
| Character table: not available. |
magma: CharacterTable(G);