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Magma
magma: G := TransitiveGroup(33, 20);
Group action invariants
| Degree $n$: | $33$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_5^4:D_4$ | ||
| 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: | (1,26,16)(2,31,20)(3,25,13)(4,30,17)(5,24,21)(6,29,14)(7,23,18)(8,28,22)(9,33,15)(10,27,19)(11,32,12), (1,2,9,3,5,8,7,11,6,4)(12,23,20,27,16,25,18,26,17,31)(13,29,14,24,19,32,22,28,15,30)(21,33) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $5$: $C_5$ $6$: $S_3$ $10$: $C_{10}$ $30$: $S_3 \times C_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 11: None
Low degree siblings
33T21Siblings 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 $ | $1$ | $1$ | $()$ | |
| $ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $15$ | $11$ | $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,22,21,20,19,18,17,16,15,14,13)$ | |
| $ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $15$ | $11$ | $( 1, 7, 2, 8, 3, 9, 4,10, 5,11, 6)(12,21,19,17,15,13,22,20,18,16,14)$ | |
| $ 11, 11, 11 $ | $30$ | $11$ | $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,21,19,17,15,13,22,20,18,16,14) (23,27,31,24,28,32,25,29,33,26,30)$ | |
| $ 11, 11, 11 $ | $30$ | $11$ | $( 1, 7, 2, 8, 3, 9, 4,10, 5,11, 6)(12,20,17,14,22,19,16,13,21,18,15) (23,27,31,24,28,32,25,29,33,26,30)$ | |
| $ 11, 11, 11 $ | $30$ | $11$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,18,13,19,14,20,15,21,16,22,17) (23,27,31,24,28,32,25,29,33,26,30)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $242$ | $3$ | $( 1,26,16)( 2,31,20)( 3,25,13)( 4,30,17)( 5,24,21)( 6,29,14)( 7,23,18) ( 8,28,22)( 9,33,15)(10,27,19)(11,32,12)$ | |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $121$ | $5$ | $( 2, 6, 4, 5,10)( 3,11, 7, 9, 8)(13,17,15,16,21)(14,22,18,20,19) (23,33,28,25,32)(24,27,31,29,30)$ | |
| $ 15, 15, 3 $ | $242$ | $15$ | $( 1,26,16, 5,27,20,10,31,14, 8,25,12,11,23,15)( 2,29,17, 9,28,13, 4,24,19, 6, 30,21, 3,32,18)( 7,33,22)$ | |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $121$ | $5$ | $( 2, 4,10, 6, 5)( 3, 7, 8,11, 9)(13,15,21,17,16)(14,18,19,22,20) (23,28,32,33,25)(24,31,30,27,29)$ | |
| $ 15, 15, 3 $ | $242$ | $15$ | $( 1,26,16, 3,23,22, 2,30,19, 8,32,15, 5,31,17)( 4,27,14, 7,28,12,11,33,13, 9, 25,18,10,29,21)( 6,24,20)$ | |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $121$ | $5$ | $( 2,10, 5, 4, 6)( 3, 8, 9, 7,11)(13,21,16,15,17)(14,19,20,18,22) (23,32,25,28,33)(24,30,29,31,27)$ | |
| $ 15, 15, 3 $ | $242$ | $15$ | $( 1,26,16, 9,23,12,11,25,22, 6,31,19, 2,27,21)( 3,28,15, 4,29,20, 7,32,13, 5, 30,14,10,24,17)( 8,33,18)$ | |
| $ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $121$ | $5$ | $( 2, 5, 6,10, 4)( 3, 9,11, 8, 7)(13,16,17,21,15)(14,20,22,19,18) (23,25,33,32,28)(24,29,27,30,31)$ | |
| $ 15, 15, 3 $ | $242$ | $15$ | $( 1,26,16, 4,31,21, 9,32,22,10,30,20, 8,23,13)( 2,24,14)( 3,33,12,11,28,18, 6, 27,17, 5,29,19, 7,25,15)$ | |
| $ 10, 10, 10, 2, 1 $ | $363$ | $10$ | $( 2, 8, 6, 3, 4,11, 5, 7,10, 9)(12,23,13,29,18,26,21,33,14,24)(15,30,17,31,16, 25,22,28,19,32)(20,27)$ | |
| $ 10, 10, 10, 2, 1 $ | $363$ | $10$ | $( 2, 3, 5, 9, 6,11,10, 8, 4, 7)(12,33,22,25,18,26,13,30,15,24)(14,27,19,23,17, 29,20,31,21,28)(16,32)$ | |
| $ 22, 2, 2, 2, 2, 2, 1 $ | $165$ | $22$ | $( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(12,28,22,32,21,25,20,29,19,33,18,26,17,30, 16,23,15,27,14,31,13,24)$ | |
| $ 22, 2, 2, 2, 2, 2, 1 $ | $165$ | $22$ | $( 1, 4)( 2, 3)( 5,11)( 6,10)( 7, 9)(12,28,21,25,19,33,17,30,15,27,13,24,22,32, 20,29,18,26,16,23,14,31)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $33$ | $2$ | $( 1, 9)( 2, 8)( 3, 7)( 4, 6)(10,11)(12,28)(13,24)(14,31)(15,27)(16,23)(17,30) (18,26)(19,33)(20,29)(21,25)(22,32)$ | |
| $ 10, 10, 10, 2, 1 $ | $363$ | $10$ | $( 2, 9,10, 7, 5,11, 4, 3, 6, 8)(12,32,17,27,18,26,16,28,20,24)(13,31,15,29,22, 33,19,25,14,30)(21,23)$ | |
| $ 10, 10, 10, 2, 1 $ | $363$ | $10$ | $( 2, 7, 4, 8,10,11, 6, 9, 5, 3)(12,25,19,28,18,26,15,31,17,24)(13,27,22,23,16, 33,20,30,21,32)(14,29)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $3630=2 \cdot 3 \cdot 5 \cdot 11^{2}$ | 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: | 3630.p | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 5 P | |
| 11 P | |
| Type |
magma: CharacterTable(G);