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Magma
magma: G := TransitiveGroup(33, 49);
Group action invariants
| Degree $n$: | $33$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $49$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3^5:C_{33}:C_{10}$ | ||
| 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: | (1,6,33,10,20,3,5,32,12,19,2,4,31,11,21)(7,26,23,30,16)(8,27,24,28,17)(9,25,22,29,18), (1,4,15,8,22)(2,6,13,7,23,3,5,14,9,24)(10,32,28,20,25,11,31,29,19,26)(12,33,30,21,27)(16,17) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $5$: $C_5$ $6$: $S_3$ $10$: $C_{10}$ $30$: $S_3 \times C_5$ $55$: $C_{11}:C_5$ $110$: 22T5 $330$: 33T7 $26730$: 33T36 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Degree 11: $C_{11}:C_5$
Low degree siblings
33T49Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| Label | 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, 1, 1, 1, 1, 1, 1 $ | $110$ | $3$ | $( 4, 6, 5)( 7, 8, 9)(10,12,11)(13,14,15)(19,20,21)(22,24,23)(25,26,27) (28,29,30)(31,32,33)$ | |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $110$ | $3$ | $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(13,15,14)(16,17,18)(25,27,26)$ | |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $22$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(19,20,21)(25,26,27)(28,29,30)$ | |
| $ 11, 11, 11 $ | $1215$ | $11$ | $( 1,27,17, 8,31,22,14, 4,29,19,12)( 2,25,18, 9,32,23,15, 5,30,20,10) ( 3,26,16, 7,33,24,13, 6,28,21,11)$ | |
| $ 11, 11, 11 $ | $1215$ | $11$ | $( 1,17,31,14,29,12,27, 8,22, 4,19)( 2,18,32,15,30,10,25, 9,23, 5,20) ( 3,16,33,13,28,11,26, 7,24, 6,21)$ | |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $891$ | $5$ | $( 1,29,11,22,27)( 2,30,12,23,25)( 3,28,10,24,26)( 7,14,33,20,18) ( 8,15,31,21,16)( 9,13,32,19,17)$ | |
| $ 15, 5, 5, 5, 3 $ | $1782$ | $15$ | $( 1,30,11,24,27)( 2,28,12,22,25)( 3,29,10,23,26)( 4, 6, 5)( 7,15,32,20,18, 8, 13,33,21,16, 9,14,31,19,17)$ | |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $891$ | $5$ | $( 1,11,27,29,22)( 2,12,25,30,23)( 3,10,26,28,24)( 7,33,18,14,20) ( 8,31,16,15,21)( 9,32,17,13,19)$ | |
| $ 15, 5, 5, 5, 3 $ | $1782$ | $15$ | $( 1,10,27,30,22)( 2,11,25,28,23)( 3,12,26,29,24)( 4, 6, 5)( 7,31,16,13,20, 8, 32,17,14,21, 9,33,18,15,19)$ | |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $891$ | $5$ | $( 1,27,22,11,29)( 2,25,23,12,30)( 3,26,24,10,28)( 7,18,20,33,14) ( 8,16,21,31,15)( 9,17,19,32,13)$ | |
| $ 15, 5, 5, 5, 3 $ | $1782$ | $15$ | $( 1,25,22,10,29)( 2,26,23,11,30)( 3,27,24,12,28)( 4, 6, 5)( 7,18,21,32,14, 8, 16,19,33,15, 9,17,20,31,13)$ | |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $891$ | $5$ | $( 1,22,29,27,11)( 2,23,30,25,12)( 3,24,28,26,10)( 7,20,14,18,33) ( 8,21,15,16,31)( 9,19,13,17,32)$ | |
| $ 15, 5, 5, 5, 3 $ | $1782$ | $15$ | $( 1,24,29,25,11)( 2,22,30,26,12)( 3,23,28,27,10)( 4, 6, 5)( 7,21,13,17,33, 8, 19,14,18,31, 9,20,15,16,32)$ | |
| $ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $110$ | $3$ | $(16,17,18)(22,24,23)(25,26,27)(28,30,29)(31,33,32)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $110$ | $3$ | $( 4, 6, 5)( 7, 8, 9)(10,12,11)(13,14,15)(16,17,18)(19,20,21)(22,23,24) (25,27,26)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $110$ | $3$ | $( 4, 5, 6)( 7, 9, 8)(10,11,12)(13,15,14)(16,17,18)(19,21,20)(28,29,30) (31,32,33)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $110$ | $3$ | $( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(13,15,14)(16,18,17)(22,24,23)(28,30,29) (31,33,32)$ | |
| $ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $22$ | $3$ | $( 1, 2, 3)( 4, 5, 6)(10,11,12)(19,20,21)(31,32,33)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $22$ | $3$ | $( 1, 3, 2)( 4, 5, 6)( 7, 9, 8)(10,12,11)(13,15,14)(16,17,18)(19,20,21) (22,23,24)(25,27,26)(28,29,30)(31,33,32)$ | |
| $ 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 $ | $2430$ | $33$ | $( 1,27,18, 7,33,23,14, 4,29,21,11, 3,26,17, 9,32,22,13, 6,28,20,10, 2,25,16, 8,31,24,15, 5,30,19,12)$ | |
| $ 33 $ | $2430$ | $33$ | $( 1,17,32,14,29,11,26, 8,22, 6,21, 3,16,31,13,28,10,25, 7,24, 5,20, 2,18,33, 15,30,12,27, 9,23, 4,19)$ | |
| $ 15, 5, 5, 5, 1, 1, 1 $ | $1782$ | $15$ | $( 1,29,10,24,25, 3,28,12,23,27, 2,30,11,22,26)( 7,14,33,19,17)( 8,15,31,20,18) ( 9,13,32,21,16)$ | |
| $ 15, 15, 3 $ | $1782$ | $15$ | $( 1,30,10,23,25, 3,29,12,22,27, 2,28,11,24,26)( 4, 6, 5)( 7,15,32,19,17, 8,13, 33,20,18, 9,14,31,21,16)$ | |
| $ 15, 15, 3 $ | $1782$ | $15$ | $( 1,28,10,22,25, 3,30,12,24,27, 2,29,11,23,26)( 4, 5, 6)( 7,13,31,19,17, 9,15, 33,21,16, 8,14,32,20,18)$ | |
| $ 15, 5, 5, 5, 1, 1, 1 $ | $1782$ | $15$ | $( 1,11,27,30,22, 3,10,26,29,24, 2,12,25,28,23)( 7,33,17,14,20)( 8,31,18,15,21) ( 9,32,16,13,19)$ | |
| $ 15, 15, 3 $ | $1782$ | $15$ | $( 1,10,27,28,22, 3,12,26,30,24, 2,11,25,29,23)( 4, 6, 5)( 7,31,18,13,20, 8,32, 16,14,21, 9,33,17,15,19)$ | |
| $ 15, 15, 3 $ | $1782$ | $15$ | $( 1,12,27,29,22, 3,11,26,28,24, 2,10,25,30,23)( 4, 5, 6)( 7,32,16,15,20, 9,31, 18,14,19, 8,33,17,13,21)$ | |
| $ 15, 5, 5, 5, 1, 1, 1 $ | $1782$ | $15$ | $( 1,27,23,11,29, 3,26,22,10,28, 2,25,24,12,30)( 7,18,21,31,14)( 8,16,19,32,15) ( 9,17,20,33,13)$ | |
| $ 15, 15, 3 $ | $1782$ | $15$ | $( 1,25,23,10,29, 3,27,22,12,28, 2,26,24,11,30)( 4, 6, 5)( 7,18,19,33,14, 8,16, 20,31,15, 9,17,21,32,13)$ | |
| $ 15, 15, 3 $ | $1782$ | $15$ | $( 1,26,23,12,29, 3,25,22,11,28, 2,27,24,10,30)( 4, 5, 6)( 7,18,20,32,14, 9,17, 19,31,13, 8,16,21,33,15)$ | |
| $ 15, 5, 5, 5, 1, 1, 1 $ | $1782$ | $15$ | $( 1,22,28,25,10, 3,24,30,27,12, 2,23,29,26,11)( 7,20,14,18,31)( 8,21,15,16,32) ( 9,19,13,17,33)$ | |
| $ 15, 15, 3 $ | $1782$ | $15$ | $( 1,24,28,26,10, 3,23,30,25,12, 2,22,29,27,11)( 4, 6, 5)( 7,21,13,17,31, 8,19, 14,18,32, 9,20,15,16,33)$ | |
| $ 15, 15, 3 $ | $1782$ | $15$ | $( 1,23,28,27,10, 3,22,30,26,12, 2,24,29,25,11)( 4, 5, 6)( 7,19,15,16,31, 9,21, 14,18,33, 8,20,13,17,32)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $729$ | $2$ | $( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(19,21)(23,24)(25,27)(28,29)(31,32)$ | |
| $ 22, 11 $ | $3645$ | $22$ | $( 1,27,18, 8,32,22,14, 5,28,19,11)( 2,26,16, 7,33,24,15, 4,29,21,12, 3,25,17, 9,31,23,13, 6,30,20,10)$ | |
| $ 22, 11 $ | $3645$ | $22$ | $( 1,17,32,14,30,10,25, 8,23, 6,20, 2,16,33,13,28,12,26, 7,24, 5,21) ( 3,18,31,15,29,11,27, 9,22, 4,19)$ | |
| $ 10, 10, 5, 5, 2, 1 $ | $8019$ | $10$ | $( 1,29,10,24,25)( 2,28,11,23,26, 3,30,12,22,27)( 5, 6)( 7,14,31,19,16, 8,13, 32,21,17)( 9,15,33,20,18)$ | |
| $ 10, 10, 5, 5, 2, 1 $ | $8019$ | $10$ | $( 1,11,25,29,24, 2,10,26,28,22)( 3,12,27,30,23)( 5, 6)( 7,33,18,13,19, 8,32, 16,15,20)( 9,31,17,14,21)$ | |
| $ 10, 10, 5, 5, 2, 1 $ | $8019$ | $10$ | $( 1,27,23,10,28)( 2,26,24,12,29, 3,25,22,11,30)( 5, 6)( 7,18,19,31,13, 9,16, 21,32,15)( 8,17,20,33,14)$ | |
| $ 10, 10, 5, 5, 2, 1 $ | $8019$ | $10$ | $( 1,22,29,26,10, 3,23,28,27,12)( 2,24,30,25,11)( 5, 6)( 7,20,14,16,31, 9,21, 13,17,33)( 8,19,15,18,32)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $80190=2 \cdot 3^{6} \cdot 5 \cdot 11$ | 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: | 80190.a | magma: IdentifyGroup(G);
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| Character table: | 42 x 42 character table |
magma: CharacterTable(G);