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Magma
magma: G := TransitiveGroup(33, 30);
Group action invariants
| Degree $n$: | $33$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $30$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3^5:C_{11}:C_5$ | ||
| 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: | (1,28,22,16,11,5,31,25,20,13,8)(2,29,23,17,12,6,32,26,21,14,9)(3,30,24,18,10,4,33,27,19,15,7), (1,30,4,9,21)(2,28,5,7,19)(3,29,6,8,20)(10,33,18,22,14)(11,31,16,23,15)(12,32,17,24,13) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $5$: $C_5$ $55$: $C_{11}:C_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Degree 11: $C_{11}:C_5$
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 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $55$ | $3$ | $( 4, 5, 6)( 7, 9, 8)(10,12,11)(13,14,15)(16,18,17)(19,20,21)(22,24,23) (25,27,26)(31,33,32)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $55$ | $3$ | $( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,15,14)(16,17,18)(19,21,20)(22,23,24) (25,26,27)(31,32,33)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $11$ | $3$ | $( 1, 2, 3)( 4, 5, 6)(16,17,18)(22,23,24)(25,26,27)(31,32,33)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $3$ | $( 1, 2, 3)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,14,15)(19,20,21)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $3$ | $( 1, 3, 2)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,15,14)(19,21,20)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $11$ | $3$ | $( 1, 3, 2)( 4, 6, 5)(16,18,17)(22,24,23)(25,27,26)(31,33,32)$ |
| $ 11, 11, 11 $ | $1215$ | $11$ | $( 1,28,22,16,11, 5,31,25,20,13, 8)( 2,29,23,17,12, 6,32,26,21,14, 9) ( 3,30,24,18,10, 4,33,27,19,15, 7)$ |
| $ 11, 11, 11 $ | $1215$ | $11$ | $( 1,22,11,31,20, 8,28,16, 5,25,13)( 2,23,12,32,21, 9,29,17, 6,26,14) ( 3,24,10,33,19, 7,30,18, 4,27,15)$ |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $891$ | $5$ | $( 1,26,22,11,28)( 2,27,23,12,29)( 3,25,24,10,30)( 7,17,20,32,15) ( 8,18,21,33,13)( 9,16,19,31,14)$ |
| $ 15, 5, 5, 5, 3 $ | $891$ | $15$ | $( 1,25,23,11,28)( 2,26,24,12,29)( 3,27,22,10,30)( 4, 5, 6)( 7,16,20,31,15, 9, 18,19,33,14, 8,17,21,32,13)$ |
| $ 15, 5, 5, 5, 3 $ | $891$ | $15$ | $( 1,27,24,11,28)( 2,25,22,12,29)( 3,26,23,10,30)( 4, 6, 5)( 7,18,20,33,15, 8, 16,21,31,13, 9,17,19,32,14)$ |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $891$ | $5$ | $( 1,22,28,26,11)( 2,23,29,27,12)( 3,24,30,25,10)( 7,20,15,17,32) ( 8,21,13,18,33)( 9,19,14,16,31)$ |
| $ 15, 5, 5, 5, 3 $ | $891$ | $15$ | $( 1,24,30,27,11)( 2,22,28,25,12)( 3,23,29,26,10)( 4, 5, 6)( 7,21,14,18,32, 9, 20,13,17,31, 8,19,15,16,33)$ |
| $ 15, 5, 5, 5, 3 $ | $891$ | $15$ | $( 1,23,29,25,11)( 2,24,30,26,12)( 3,22,28,27,10)( 4, 6, 5)( 7,19,13,16,32, 8, 20,14,17,33, 9,21,15,18,31)$ |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $891$ | $5$ | $( 1,28,11,22,26)( 2,29,12,23,27)( 3,30,10,24,25)( 7,15,32,20,17) ( 8,13,33,21,18)( 9,14,31,19,16)$ |
| $ 15, 5, 5, 5, 3 $ | $891$ | $15$ | $( 1,28,10,23,26)( 2,29,11,24,27)( 3,30,12,22,25)( 4, 5, 6)( 7,13,32,21,17, 9, 15,31,20,16, 8,14,33,19,18)$ |
| $ 15, 5, 5, 5, 3 $ | $891$ | $15$ | $( 1,28,12,24,26)( 2,29,10,22,27)( 3,30,11,23,25)( 4, 6, 5)( 7,14,32,19,17, 8, 15,33,20,18, 9,13,31,21,16)$ |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $891$ | $5$ | $( 1,11,26,28,22)( 2,12,27,29,23)( 3,10,25,30,24)( 7,32,17,15,20) ( 8,33,18,13,21)( 9,31,16,14,19)$ |
| $ 15, 5, 5, 5, 3 $ | $891$ | $15$ | $( 1,10,27,29,22)( 2,11,25,30,23)( 3,12,26,28,24)( 4, 5, 6)( 7,31,18,14,20, 9, 33,17,13,19, 8,32,16,15,21)$ |
| $ 15, 5, 5, 5, 3 $ | $891$ | $15$ | $( 1,12,25,30,22)( 2,10,26,28,23)( 3,11,27,29,24)( 4, 6, 5)( 7,33,16,13,20, 8, 31,17,14,21, 9,32,18,15,19)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $13365=3^{5} \cdot 5 \cdot 11$ | 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: | 13365.a | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);