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Magma
magma: G := TransitiveGroup(33, 2);
Group action invariants
| Degree $n$: | $33$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $2$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $S_3\times C_{11}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $11$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,20,4,22,9,26,11,28,15,33,18,3,19,6,24,8,25,10,30,14,32,17,2,21,5,23,7,27,12,29,13,31,16), (1,31,30,25,22,20,17,14,11,9,5,2,33,28,27,23,19,18,13,12,8,6)(3,32,29,26,24,21,16,15,10,7,4) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ $11$: $C_{11}$ $22$: 22T1 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 11: $C_{11}$
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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 2, 3)( 4, 6)( 7, 9)(10,12)(14,15)(16,18)(20,21)(23,24)(25,26)(28,29)(31,32)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)$ |
| $ 33 $ | $2$ | $33$ | $( 1, 4, 9,11,15,18,19,24,25,30,32, 2, 5, 7,12,13,16,20,22,26,28,33, 3, 6, 8, 10,14,17,21,23,27,29,31)$ |
| $ 22, 11 $ | $3$ | $22$ | $( 1, 4, 8,10,13,16,19,24,27,29,33, 3, 5, 7,11,15,17,21,22,26,30,32) ( 2, 6, 9,12,14,18,20,23,25,28,31)$ |
| $ 11, 11, 11 $ | $1$ | $11$ | $( 1, 5, 8,11,13,17,19,22,27,30,33)( 2, 6, 9,12,14,18,20,23,25,28,31) ( 3, 4, 7,10,15,16,21,24,26,29,32)$ |
| $ 33 $ | $2$ | $33$ | $( 1, 7,14,19,26,31, 5,10,18,22,29, 2, 8,15,20,27,32, 6,11,16,23,30, 3, 9,13, 21,25,33, 4,12,17,24,28)$ |
| $ 22, 11 $ | $3$ | $22$ | $( 1, 7,13,21,27,32, 5,10,17,24,30, 3, 8,15,19,26,33, 4,11,16,22,29) ( 2, 9,14,20,25,31, 6,12,18,23,28)$ |
| $ 11, 11, 11 $ | $1$ | $11$ | $( 1, 8,13,19,27,33, 5,11,17,22,30)( 2, 9,14,20,25,31, 6,12,18,23,28) ( 3, 7,15,21,26,32, 4,10,16,24,29)$ |
| $ 33 $ | $2$ | $33$ | $( 1,10,20,30, 4,14,22,32, 9,17,26, 2,11,21,28, 5,15,23,33, 7,18,27, 3,12,19, 29, 6,13,24,31, 8,16,25)$ |
| $ 22, 11 $ | $3$ | $22$ | $( 1,10,19,29, 5,15,22,32, 8,16,27, 3,11,21,30, 4,13,24,33, 7,17,26) ( 2,12,20,28, 6,14,23,31, 9,18,25)$ |
| $ 11, 11, 11 $ | $1$ | $11$ | $( 1,11,19,30, 5,13,22,33, 8,17,27)( 2,12,20,28, 6,14,23,31, 9,18,25) ( 3,10,21,29, 4,15,24,32, 7,16,26)$ |
| $ 11, 11, 11 $ | $1$ | $11$ | $( 1,13,27, 5,17,30, 8,19,33,11,22)( 2,14,25, 6,18,28, 9,20,31,12,23) ( 3,15,26, 4,16,29, 7,21,32,10,24)$ |
| $ 22, 11 $ | $3$ | $22$ | $( 1,13,27, 5,17,30, 8,19,33,11,22)( 2,15,25, 4,18,29, 9,21,31,10,23, 3,14,26, 6,16,28, 7,20,32,12,24)$ |
| $ 33 $ | $2$ | $33$ | $( 1,14,26, 5,18,29, 8,20,32,11,23, 3,13,25, 4,17,28, 7,19,31,10,22, 2,15,27, 6,16,30, 9,21,33,12,24)$ |
| $ 33 $ | $2$ | $33$ | $( 1,16,31,13,29,12,27, 7,23, 5,21, 2,17,32,14,30,10,25, 8,24, 6,19, 3,18,33, 15,28,11,26, 9,22, 4,20)$ |
| $ 22, 11 $ | $3$ | $22$ | $( 1,16,33,15,30,10,27, 7,22, 4,19, 3,17,32,13,29,11,26, 8,24, 5,21) ( 2,18,31,14,28,12,25, 9,23, 6,20)$ |
| $ 11, 11, 11 $ | $1$ | $11$ | $( 1,17,33,13,30,11,27, 8,22, 5,19)( 2,18,31,14,28,12,25, 9,23, 6,20) ( 3,16,32,15,29,10,26, 7,24, 4,21)$ |
| $ 11, 11, 11 $ | $1$ | $11$ | $( 1,19, 5,22, 8,27,11,30,13,33,17)( 2,20, 6,23, 9,25,12,28,14,31,18) ( 3,21, 4,24, 7,26,10,29,15,32,16)$ |
| $ 22, 11 $ | $3$ | $22$ | $( 1,19, 5,22, 8,27,11,30,13,33,17)( 2,21, 6,24, 9,26,12,29,14,32,18, 3,20, 4, 23, 7,25,10,28,15,31,16)$ |
| $ 33 $ | $2$ | $33$ | $( 1,20, 4,22, 9,26,11,28,15,33,18, 3,19, 6,24, 8,25,10,30,14,32,17, 2,21, 5, 23, 7,27,12,29,13,31,16)$ |
| $ 11, 11, 11 $ | $1$ | $11$ | $( 1,22,11,33,19, 8,30,17, 5,27,13)( 2,23,12,31,20, 9,28,18, 6,25,14) ( 3,24,10,32,21, 7,29,16, 4,26,15)$ |
| $ 22, 11 $ | $3$ | $22$ | $( 1,22,11,33,19, 8,30,17, 5,27,13)( 2,24,12,32,20, 7,28,16, 6,26,14, 3,23,10, 31,21, 9,29,18, 4,25,15)$ |
| $ 33 $ | $2$ | $33$ | $( 1,23,10,33,20, 7,30,18, 4,27,14, 3,22,12,32,19, 9,29,17, 6,26,13, 2,24,11, 31,21, 8,28,16, 5,25,15)$ |
| $ 33 $ | $2$ | $33$ | $( 1,25,16, 8,31,24,13, 6,29,19,12, 3,27,18, 7,33,23,15, 5,28,21,11, 2,26,17, 9,32,22,14, 4,30,20,10)$ |
| $ 22, 11 $ | $3$ | $22$ | $( 1,25,17, 9,33,23,13, 6,30,20,11, 2,27,18, 8,31,22,14, 5,28,19,12) ( 3,26,16, 7,32,24,15, 4,29,21,10)$ |
| $ 11, 11, 11 $ | $1$ | $11$ | $( 1,27,17, 8,33,22,13, 5,30,19,11)( 2,25,18, 9,31,23,14, 6,28,20,12) ( 3,26,16, 7,32,24,15, 4,29,21,10)$ |
| $ 33 $ | $2$ | $33$ | $( 1,28,24,17,12, 4,33,25,21,13, 9, 3,30,23,16,11, 6,32,27,20,15, 8, 2,29,22, 18,10, 5,31,26,19,14, 7)$ |
| $ 22, 11 $ | $3$ | $22$ | $( 1,28,22,18,11, 6,33,25,19,14, 8, 2,30,23,17,12, 5,31,27,20,13, 9) ( 3,29,24,16,10, 4,32,26,21,15, 7)$ |
| $ 11, 11, 11 $ | $1$ | $11$ | $( 1,30,22,17,11, 5,33,27,19,13, 8)( 2,28,23,18,12, 6,31,25,20,14, 9) ( 3,29,24,16,10, 4,32,26,21,15, 7)$ |
| $ 33 $ | $2$ | $33$ | $( 1,31,29,27,23,21,17,14,10, 8, 6, 3,33,28,26,22,20,16,13,12, 7, 5, 2,32,30, 25,24,19,18,15,11, 9, 4)$ |
| $ 22, 11 $ | $3$ | $22$ | $( 1,31,30,25,22,20,17,14,11, 9, 5, 2,33,28,27,23,19,18,13,12, 8, 6) ( 3,32,29,26,24,21,16,15,10, 7, 4)$ |
| $ 11, 11, 11 $ | $1$ | $11$ | $( 1,33,30,27,22,19,17,13,11, 8, 5)( 2,31,28,25,23,20,18,14,12, 9, 6) ( 3,32,29,26,24,21,16,15,10, 7, 4)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $66=2 \cdot 3 \cdot 11$ | magma: Order(G);
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| Cyclic: | no | magma: IsCyclic(G);
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| Abelian: | no | magma: IsAbelian(G);
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| Solvable: | yes | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 66.1 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);