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Magma
magma: G := TransitiveGroup(37, 1);
Group action invariants
| Degree $n$: | $37$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $1$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{37}$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | yes | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $37$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37) | magma: Generators(G);
|
Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Prime degree - none
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, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 37 $ | $1$ | $37$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37)$ |
| $ 37 $ | $1$ | $37$ | $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31,33,35,37, 2, 4, 6, 8,10,12, 14,16,18,20,22,24,26,28,30,32,34,36)$ |
| $ 37 $ | $1$ | $37$ | $( 1, 4, 7,10,13,16,19,22,25,28,31,34,37, 3, 6, 9,12,15,18,21,24,27,30,33,36, 2, 5, 8,11,14,17,20,23,26,29,32,35)$ |
| $ 37 $ | $1$ | $37$ | $( 1, 5, 9,13,17,21,25,29,33,37, 4, 8,12,16,20,24,28,32,36, 3, 7,11,15,19,23, 27,31,35, 2, 6,10,14,18,22,26,30,34)$ |
| $ 37 $ | $1$ | $37$ | $( 1, 6,11,16,21,26,31,36, 4, 9,14,19,24,29,34, 2, 7,12,17,22,27,32,37, 5,10, 15,20,25,30,35, 3, 8,13,18,23,28,33)$ |
| $ 37 $ | $1$ | $37$ | $( 1, 7,13,19,25,31,37, 6,12,18,24,30,36, 5,11,17,23,29,35, 4,10,16,22,28,34, 3, 9,15,21,27,33, 2, 8,14,20,26,32)$ |
| $ 37 $ | $1$ | $37$ | $( 1, 8,15,22,29,36, 6,13,20,27,34, 4,11,18,25,32, 2, 9,16,23,30,37, 7,14,21, 28,35, 5,12,19,26,33, 3,10,17,24,31)$ |
| $ 37 $ | $1$ | $37$ | $( 1, 9,17,25,33, 4,12,20,28,36, 7,15,23,31, 2,10,18,26,34, 5,13,21,29,37, 8, 16,24,32, 3,11,19,27,35, 6,14,22,30)$ |
| $ 37 $ | $1$ | $37$ | $( 1,10,19,28,37, 9,18,27,36, 8,17,26,35, 7,16,25,34, 6,15,24,33, 5,14,23,32, 4,13,22,31, 3,12,21,30, 2,11,20,29)$ |
| $ 37 $ | $1$ | $37$ | $( 1,11,21,31, 4,14,24,34, 7,17,27,37,10,20,30, 3,13,23,33, 6,16,26,36, 9,19, 29, 2,12,22,32, 5,15,25,35, 8,18,28)$ |
| $ 37 $ | $1$ | $37$ | $( 1,12,23,34, 8,19,30, 4,15,26,37,11,22,33, 7,18,29, 3,14,25,36,10,21,32, 6, 17,28, 2,13,24,35, 9,20,31, 5,16,27)$ |
| $ 37 $ | $1$ | $37$ | $( 1,13,25,37,12,24,36,11,23,35,10,22,34, 9,21,33, 8,20,32, 7,19,31, 6,18,30, 5,17,29, 4,16,28, 3,15,27, 2,14,26)$ |
| $ 37 $ | $1$ | $37$ | $( 1,14,27, 3,16,29, 5,18,31, 7,20,33, 9,22,35,11,24,37,13,26, 2,15,28, 4,17, 30, 6,19,32, 8,21,34,10,23,36,12,25)$ |
| $ 37 $ | $1$ | $37$ | $( 1,15,29, 6,20,34,11,25, 2,16,30, 7,21,35,12,26, 3,17,31, 8,22,36,13,27, 4, 18,32, 9,23,37,14,28, 5,19,33,10,24)$ |
| $ 37 $ | $1$ | $37$ | $( 1,16,31, 9,24, 2,17,32,10,25, 3,18,33,11,26, 4,19,34,12,27, 5,20,35,13,28, 6,21,36,14,29, 7,22,37,15,30, 8,23)$ |
| $ 37 $ | $1$ | $37$ | $( 1,17,33,12,28, 7,23, 2,18,34,13,29, 8,24, 3,19,35,14,30, 9,25, 4,20,36,15, 31,10,26, 5,21,37,16,32,11,27, 6,22)$ |
| $ 37 $ | $1$ | $37$ | $( 1,18,35,15,32,12,29, 9,26, 6,23, 3,20,37,17,34,14,31,11,28, 8,25, 5,22, 2, 19,36,16,33,13,30,10,27, 7,24, 4,21)$ |
| $ 37 $ | $1$ | $37$ | $( 1,19,37,18,36,17,35,16,34,15,33,14,32,13,31,12,30,11,29,10,28, 9,27, 8,26, 7,25, 6,24, 5,23, 4,22, 3,21, 2,20)$ |
| $ 37 $ | $1$ | $37$ | $( 1,20, 2,21, 3,22, 4,23, 5,24, 6,25, 7,26, 8,27, 9,28,10,29,11,30,12,31,13, 32,14,33,15,34,16,35,17,36,18,37,19)$ |
| $ 37 $ | $1$ | $37$ | $( 1,21, 4,24, 7,27,10,30,13,33,16,36,19, 2,22, 5,25, 8,28,11,31,14,34,17,37, 20, 3,23, 6,26, 9,29,12,32,15,35,18)$ |
| $ 37 $ | $1$ | $37$ | $( 1,22, 6,27,11,32,16,37,21, 5,26,10,31,15,36,20, 4,25, 9,30,14,35,19, 3,24, 8,29,13,34,18, 2,23, 7,28,12,33,17)$ |
| $ 37 $ | $1$ | $37$ | $( 1,23, 8,30,15,37,22, 7,29,14,36,21, 6,28,13,35,20, 5,27,12,34,19, 4,26,11, 33,18, 3,25,10,32,17, 2,24, 9,31,16)$ |
| $ 37 $ | $1$ | $37$ | $( 1,24,10,33,19, 5,28,14,37,23, 9,32,18, 4,27,13,36,22, 8,31,17, 3,26,12,35, 21, 7,30,16, 2,25,11,34,20, 6,29,15)$ |
| $ 37 $ | $1$ | $37$ | $( 1,25,12,36,23,10,34,21, 8,32,19, 6,30,17, 4,28,15, 2,26,13,37,24,11,35,22, 9,33,20, 7,31,18, 5,29,16, 3,27,14)$ |
| $ 37 $ | $1$ | $37$ | $( 1,26,14, 2,27,15, 3,28,16, 4,29,17, 5,30,18, 6,31,19, 7,32,20, 8,33,21, 9, 34,22,10,35,23,11,36,24,12,37,25,13)$ |
| $ 37 $ | $1$ | $37$ | $( 1,27,16, 5,31,20, 9,35,24,13, 2,28,17, 6,32,21,10,36,25,14, 3,29,18, 7,33, 22,11,37,26,15, 4,30,19, 8,34,23,12)$ |
| $ 37 $ | $1$ | $37$ | $( 1,28,18, 8,35,25,15, 5,32,22,12, 2,29,19, 9,36,26,16, 6,33,23,13, 3,30,20, 10,37,27,17, 7,34,24,14, 4,31,21,11)$ |
| $ 37 $ | $1$ | $37$ | $( 1,29,20,11, 2,30,21,12, 3,31,22,13, 4,32,23,14, 5,33,24,15, 6,34,25,16, 7, 35,26,17, 8,36,27,18, 9,37,28,19,10)$ |
| $ 37 $ | $1$ | $37$ | $( 1,30,22,14, 6,35,27,19,11, 3,32,24,16, 8,37,29,21,13, 5,34,26,18,10, 2,31, 23,15, 7,36,28,20,12, 4,33,25,17, 9)$ |
| $ 37 $ | $1$ | $37$ | $( 1,31,24,17,10, 3,33,26,19,12, 5,35,28,21,14, 7,37,30,23,16, 9, 2,32,25,18, 11, 4,34,27,20,13, 6,36,29,22,15, 8)$ |
| $ 37 $ | $1$ | $37$ | $( 1,32,26,20,14, 8, 2,33,27,21,15, 9, 3,34,28,22,16,10, 4,35,29,23,17,11, 5, 36,30,24,18,12, 6,37,31,25,19,13, 7)$ |
| $ 37 $ | $1$ | $37$ | $( 1,33,28,23,18,13, 8, 3,35,30,25,20,15,10, 5,37,32,27,22,17,12, 7, 2,34,29, 24,19,14, 9, 4,36,31,26,21,16,11, 6)$ |
| $ 37 $ | $1$ | $37$ | $( 1,34,30,26,22,18,14,10, 6, 2,35,31,27,23,19,15,11, 7, 3,36,32,28,24,20,16, 12, 8, 4,37,33,29,25,21,17,13, 9, 5)$ |
| $ 37 $ | $1$ | $37$ | $( 1,35,32,29,26,23,20,17,14,11, 8, 5, 2,36,33,30,27,24,21,18,15,12, 9, 6, 3, 37,34,31,28,25,22,19,16,13,10, 7, 4)$ |
| $ 37 $ | $1$ | $37$ | $( 1,36,34,32,30,28,26,24,22,20,18,16,14,12,10, 8, 6, 4, 2,37,35,33,31,29,27, 25,23,21,19,17,15,13,11, 9, 7, 5, 3)$ |
| $ 37 $ | $1$ | $37$ | $( 1,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14, 13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $37$ (is prime) | magma: Order(G);
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| Cyclic: | yes | magma: IsCyclic(G);
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| Abelian: | yes | magma: IsAbelian(G);
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| Solvable: | yes | magma: IsSolvable(G);
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| Nilpotency class: | $1$ | ||
| Label: | 37.1 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);