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Magma
magma: G := TransitiveGroup(33, 35);
Group action invariants
| Degree $n$: | $33$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3\times M_{11}$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,23,30,25,4,2,24,28,26,5,3,22,29,27,6)(7,16,21,33,11,8,17,19,31,12,9,18,20,32,10)(13,15,14), (4,28,20,14)(5,29,21,15)(6,30,19,13)(7,17,23,27)(8,18,24,25)(9,16,22,26) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ $7920$: $M_{11}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 11: $M_{11}$
Low degree siblings
36T12768Siblings are shown 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 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)(31,33,32)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $165$ | $2$ | $( 1,24)( 2,22)( 3,23)( 4,33)( 5,31)( 6,32)(10,15)(11,13)(12,14)(16,26)(17,27) (18,25)$ |
| $ 6, 6, 6, 6, 3, 3, 3 $ | $165$ | $6$ | $( 1,22, 3,24, 2,23)( 4,31, 6,33, 5,32)( 7, 8, 9)(10,13,12,15,11,14) (16,27,18,26,17,25)(19,20,21)(28,29,30)$ |
| $ 6, 6, 6, 6, 3, 3, 3 $ | $165$ | $6$ | $( 1,23, 2,24, 3,22)( 4,32, 5,33, 6,31)( 7, 9, 8)(10,14,11,15,12,13) (16,25,17,26,18,27)(19,21,20)(28,30,29)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $440$ | $3$ | $( 1, 6,27)( 2, 4,25)( 3, 5,26)( 7,29,19)( 8,30,20)( 9,28,21)(10,11,12) (13,14,15)(16,23,31)(17,24,32)(18,22,33)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $440$ | $3$ | $( 1, 4,26)( 2, 5,27)( 3, 6,25)( 7,30,21)( 8,28,19)( 9,29,20)(10,12,11) (13,15,14)(16,24,33)(17,22,31)(18,23,32)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $440$ | $3$ | $( 1, 5,25)( 2, 6,26)( 3, 4,27)( 7,28,20)( 8,29,21)( 9,30,19)(16,22,32) (17,23,33)(18,24,31)$ |
| $ 6, 6, 6, 6, 3, 3, 3 $ | $1320$ | $6$ | $( 1,17, 6,24,27,32)( 2,18, 4,22,25,33)( 3,16, 5,23,26,31)( 7,19,29)( 8,20,30) ( 9,21,28)(10,14,11,15,12,13)$ |
| $ 6, 6, 6, 3, 3, 3, 2, 2, 2 $ | $1320$ | $6$ | $( 1,18, 5,24,25,31)( 2,16, 6,22,26,32)( 3,17, 4,23,27,33)( 7,20,28)( 8,21,29) ( 9,19,30)(10,15)(11,13)(12,14)$ |
| $ 6, 6, 6, 6, 3, 3, 3 $ | $1320$ | $6$ | $( 1,16, 4,24,26,33)( 2,17, 5,22,27,31)( 3,18, 6,23,25,32)( 7,21,30)( 8,19,28) ( 9,20,29)(10,13,12,15,11,14)$ |
| $ 15, 15, 3 $ | $1584$ | $15$ | $( 1,17,19,24,33, 2,18,20,22,31, 3,16,21,23,32)( 4,13,25,12,30, 5,14,26,10,28, 6,15,27,11,29)( 7, 9, 8)$ |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $1584$ | $5$ | $( 1,18,21,24,31)( 2,16,19,22,32)( 3,17,20,23,33)( 4,14,27,12,28) ( 5,15,25,10,29)( 6,13,26,11,30)$ |
| $ 15, 15, 3 $ | $1584$ | $15$ | $( 1,16,20,24,32, 3,18,19,23,31, 2,17,21,22,33)( 4,15,26,12,29, 6,14,25,11,28, 5,13,27,10,30)( 7, 8, 9)$ |
| $ 12, 12, 3, 3, 3 $ | $990$ | $12$ | $( 1, 3, 2)( 4,22,31,14, 6,24,33,13, 5,23,32,15)( 7,30,25,17, 9,29,27,16, 8,28, 26,18)(10,12,11)(19,21,20)$ |
| $ 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $990$ | $4$ | $( 4,23,33,14)( 5,24,31,15)( 6,22,32,13)( 7,28,27,17)( 8,29,25,18)( 9,30,26,16)$ |
| $ 12, 12, 3, 3, 3 $ | $990$ | $12$ | $( 1, 2, 3)( 4,24,32,14, 5,22,33,15, 6,23,31,13)( 7,29,26,17, 8,30,27,18, 9,28, 25,16)(10,11,12)(19,20,21)$ |
| $ 24, 6, 3 $ | $990$ | $24$ | $( 1,12, 2,10, 3,11)( 4, 9,24,28,32,25,14,16, 5, 7,22,29,33,26,15,17, 6, 8,23, 30,31,27,13,18)(19,21,20)$ |
| $ 8, 8, 8, 2, 2, 2, 1, 1, 1 $ | $990$ | $8$ | $( 1,10)( 2,11)( 3,12)( 4, 7,23,28,33,27,14,17)( 5, 8,24,29,31,25,15,18) ( 6, 9,22,30,32,26,13,16)$ |
| $ 24, 6, 3 $ | $990$ | $24$ | $( 1,11, 3,10, 2,12)( 4, 8,22,28,31,26,14,18, 6, 7,24,30,33,25,13,17, 5, 9,23, 29,32,27,15,16)(19,20,21)$ |
| $ 8, 8, 8, 2, 2, 2, 1, 1, 1 $ | $990$ | $8$ | $( 1,10)( 2,11)( 3,12)( 4,17,14,27,33,28,23, 7)( 5,18,15,25,31,29,24, 8) ( 6,16,13,26,32,30,22, 9)$ |
| $ 24, 6, 3 $ | $990$ | $24$ | $( 1,11, 3,10, 2,12)( 4,18,13,27,31,30,23, 8, 6,17,15,26,33,29,22, 7, 5,16,14, 25,32,28,24, 9)(19,20,21)$ |
| $ 24, 6, 3 $ | $990$ | $24$ | $( 1,12, 2,10, 3,11)( 4,16,15,27,32,29,23, 9, 5,17,13,25,33,30,24, 7, 6,18,14, 26,31,28,22, 8)(19,21,20)$ |
| $ 33 $ | $720$ | $33$ | $( 1,19, 4, 8,32,27,24,11,14,18,30, 3,21, 6, 7,31,26,23,10,13,17,29, 2,20, 5, 9,33,25,22,12,15,16,28)$ |
| $ 33 $ | $720$ | $33$ | $( 1,20, 6, 8,33,26,24,12,13,18,28, 2,21, 4, 9,31,27,22,10,14,16,29, 3,19, 5, 7,32,25,23,11,15,17,30)$ |
| $ 11, 11, 11 $ | $720$ | $11$ | $( 1,21, 5, 8,31,25,24,10,15,18,29)( 2,19, 6, 9,32,26,22,11,13,16,30) ( 3,20, 4, 7,33,27,23,12,14,17,28)$ |
| $ 33 $ | $720$ | $33$ | $( 1,30,17,15,11,23,25,32, 7, 5,19, 3,29,16,14,10,22,27,31, 9, 4,21, 2,28,18, 13,12,24,26,33, 8, 6,20)$ |
| $ 33 $ | $720$ | $33$ | $( 1,28,16,15,12,22,25,33, 9, 5,20, 2,29,17,13,10,23,26,31, 7, 6,21, 3,30,18, 14,11,24,27,32, 8, 4,19)$ |
| $ 11, 11, 11 $ | $720$ | $11$ | $( 1,29,18,15,10,24,25,31, 8, 5,21)( 2,30,16,13,11,22,26,32, 9, 6,19) ( 3,28,17,14,12,23,27,33, 7, 4,20)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $23760=2^{4} \cdot 3^{3} \cdot 5 \cdot 11$ | 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: | no | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 23760.b | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);