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Magma
magma: G := TransitiveGroup(33, 11);
Group action invariants
| Degree $n$: | $33$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $11$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $S_3\times F_{11}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,18,19,13,25,3,16,21,14,27)(2,17,20,15,26)(4,12,33,23,9,6,10,32,24,8)(5,11,31,22,7)(28,30), (1,17)(2,16)(3,18)(4,15)(5,14)(6,13)(7,10)(8,12)(9,11)(19,31)(20,33)(21,32)(22,28)(23,30)(24,29)(25,26) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $5$: $C_5$ $6$: $S_3$ $10$: $C_{10}$ x 3 $12$: $D_{6}$ $20$: 20T3 $30$: $S_3 \times C_5$ $60$: 30T12 $110$: $F_{11}$ $220$: 22T6 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 11: $F_{11}$
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 $ | $1$ | $1$ | $()$ |
| $ 11, 11, 11 $ | $10$ | $11$ | $( 1,24,10,33,19, 9,28,16, 4,25,14)( 2,22,11,31,20, 7,29,17, 5,26,15) ( 3,23,12,32,21, 8,30,18, 6,27,13)$ |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $11$ | $5$ | $( 4,14,16,28,10)( 5,15,17,29,11)( 6,13,18,30,12)( 7,26,31,22,20) ( 8,27,32,23,21)( 9,25,33,24,19)$ |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $11$ | $5$ | $( 4,16,10,14,28)( 5,17,11,15,29)( 6,18,12,13,30)( 7,31,20,26,22) ( 8,32,21,27,23)( 9,33,19,25,24)$ |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $11$ | $5$ | $( 4,10,28,16,14)( 5,11,29,17,15)( 6,12,30,18,13)( 7,20,22,31,26) ( 8,21,23,32,27)( 9,19,24,33,25)$ |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $11$ | $5$ | $( 4,28,14,10,16)( 5,29,15,11,17)( 6,30,13,12,18)( 7,22,26,20,31) ( 8,23,27,21,32)( 9,24,25,19,33)$ |
| $ 33 $ | $20$ | $33$ | $( 1,22,12,33,20, 8,28,17, 6,25,15, 3,24,11,32,19, 7,30,16, 5,27,14, 2,23,10, 31,21, 9,29,18, 4,26,13)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)$ |
| $ 15, 15, 3 $ | $22$ | $15$ | $( 1,20,27,16,15, 3,19,26,18,14, 2,21,25,17,13)( 4,31, 8,10,22, 6,33, 7,12,24, 5,32, 9,11,23)(28,29,30)$ |
| $ 15, 15, 3 $ | $22$ | $15$ | $( 1, 7, 6,24,15, 3, 9, 5,23,14, 2, 8, 4,22,13)(10,20,32,25,29,12,19,31,27,28, 11,21,33,26,30)(16,17,18)$ |
| $ 15, 15, 3 $ | $22$ | $15$ | $( 1,31,23,28,15, 3,33,22,30,14, 2,32,24,29,13)( 4, 7,18,10,26, 6, 9,17,12,25, 5, 8,16,11,27)(19,20,21)$ |
| $ 15, 15, 3 $ | $22$ | $15$ | $( 1,26,12, 9,15, 3,25,11, 8,14, 2,27,10, 7,13)( 4,20,23,16,29, 6,19,22,18,28, 5,21,24,17,30)(31,32,33)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 2, 3)( 5, 6)( 7, 8)(11,12)(13,15)(17,18)(20,21)(22,23)(26,27)(29,30)(31,32)$ |
| $ 22, 11 $ | $30$ | $22$ | $( 1,24,10,33,19, 9,28,16, 4,25,14)( 2,23,11,32,20, 8,29,18, 5,27,15, 3,22,12, 31,21, 7,30,17, 6,26,13)$ |
| $ 10, 10, 5, 5, 2, 1 $ | $33$ | $10$ | $( 2, 3)( 4,14,16,28,10)( 5,13,17,30,11, 6,15,18,29,12)( 7,27,31,23,20, 8,26, 32,22,21)( 9,25,33,24,19)$ |
| $ 10, 10, 5, 5, 2, 1 $ | $33$ | $10$ | $( 2, 3)( 4,16,10,14,28)( 5,18,11,13,29, 6,17,12,15,30)( 7,32,20,27,22, 8,31, 21,26,23)( 9,33,19,25,24)$ |
| $ 10, 10, 5, 5, 2, 1 $ | $33$ | $10$ | $( 2, 3)( 4,10,28,16,14)( 5,12,29,18,15, 6,11,30,17,13)( 7,21,22,32,26, 8,20, 23,31,27)( 9,19,24,33,25)$ |
| $ 10, 10, 5, 5, 2, 1 $ | $33$ | $10$ | $( 2, 3)( 4,28,14,10,16)( 5,30,15,12,17, 6,29,13,11,18)( 7,23,26,21,31, 8,22, 27,20,32)( 9,24,25,19,33)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $33$ | $2$ | $( 2, 3)( 4,33)( 5,32)( 6,31)( 7,30)( 8,29)( 9,28)(10,25)(11,27)(12,26)(13,22) (14,24)(15,23)(16,19)(17,21)(18,20)$ |
| $ 10, 10, 10, 2, 1 $ | $33$ | $10$ | $( 2, 3)( 4,24,16, 9,10,33,14,19,28,25)( 5,23,17, 8,11,32,15,21,29,27) ( 6,22,18, 7,12,31,13,20,30,26)$ |
| $ 10, 10, 10, 2, 1 $ | $33$ | $10$ | $( 2, 3)( 4,19,10,24,28,33,16,25,14, 9)( 5,21,11,23,29,32,17,27,15, 8) ( 6,20,12,22,30,31,18,26,13, 7)$ |
| $ 10, 10, 10, 2, 1 $ | $33$ | $10$ | $( 2, 3)( 4,25,28,19,14,33,10, 9,16,24)( 5,27,29,21,15,32,11, 8,17,23) ( 6,26,30,20,13,31,12, 7,18,22)$ |
| $ 10, 10, 10, 2, 1 $ | $33$ | $10$ | $( 2, 3)( 4, 9,14,25,16,33,28,24,10,19)( 5, 8,15,27,17,32,29,23,11,21) ( 6, 7,13,26,18,31,30,22,12,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $11$ | $2$ | $( 4,33)( 5,31)( 6,32)( 7,29)( 8,30)( 9,28)(10,25)(11,26)(12,27)(13,23)(14,24) (15,22)(16,19)(17,20)(18,21)$ |
| $ 10, 10, 10, 1, 1, 1 $ | $11$ | $10$ | $( 4,24,16, 9,10,33,14,19,28,25)( 5,22,17, 7,11,31,15,20,29,26)( 6,23,18, 8,12, 32,13,21,30,27)$ |
| $ 10, 10, 10, 1, 1, 1 $ | $11$ | $10$ | $( 4,19,10,24,28,33,16,25,14, 9)( 5,20,11,22,29,31,17,26,15, 7)( 6,21,12,23,30, 32,18,27,13, 8)$ |
| $ 10, 10, 10, 1, 1, 1 $ | $11$ | $10$ | $( 4,25,28,19,14,33,10, 9,16,24)( 5,26,29,20,15,31,11, 7,17,22)( 6,27,30,21,13, 32,12, 8,18,23)$ |
| $ 10, 10, 10, 1, 1, 1 $ | $11$ | $10$ | $( 4, 9,14,25,16,33,28,24,10,19)( 5, 7,15,26,17,31,29,22,11,20)( 6, 8,13,27,18, 32,30,23,12,21)$ |
| $ 6, 6, 6, 6, 6, 3 $ | $22$ | $6$ | $( 1,22, 3,24, 2,23)( 4,20, 6,19, 5,21)( 7,18, 9,17, 8,16)(10,15,12,14,11,13) (25,31,27,33,26,32)(28,29,30)$ |
| $ 30, 3 $ | $22$ | $30$ | $( 1,20,13, 4, 7,30,10,17,27,24, 2,21,14, 5, 8,28,11,18,25,22, 3,19,15, 6, 9, 29,12,16,26,23)(31,32,33)$ |
| $ 30, 3 $ | $22$ | $30$ | $( 1, 7,12,28, 5,27,19,17,32,24, 2, 8,10,29, 6,25,20,18,33,22, 3, 9,11,30, 4, 26,21,16,31,23)(13,14,15)$ |
| $ 30, 3 $ | $22$ | $30$ | $( 1,31, 8,14,29,18,19,11, 6,24, 2,32, 9,15,30,16,20,12, 4,22, 3,33, 7,13,28, 17,21,10, 5,23)(25,26,27)$ |
| $ 30, 3 $ | $22$ | $30$ | $( 1,26, 8, 4,31,21,28,15,18,24, 2,27, 9, 5,32,19,29,13,16,22, 3,25, 7, 6,33, 20,30,14,17,23)(10,11,12)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $660=2^{2} \cdot 3 \cdot 5 \cdot 11$ | 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: | yes | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 660.15 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);