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Magma
magma: G := TransitiveGroup(31, 1);
Group action invariants
| Degree $n$: | $31$ | 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: | $C_{31}$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | yes | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $31$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| 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) | magma: Generators(G);
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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 $ | $1$ | $1$ | $()$ |
| $ 31 $ | $1$ | $31$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25, 26,27,28,29,30,31)$ |
| $ 31 $ | $1$ | $31$ | $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25,27,29,31, 2, 4, 6, 8,10,12,14,16,18, 20,22,24,26,28,30)$ |
| $ 31 $ | $1$ | $31$ | $( 1, 4, 7,10,13,16,19,22,25,28,31, 3, 6, 9,12,15,18,21,24,27,30, 2, 5, 8,11, 14,17,20,23,26,29)$ |
| $ 31 $ | $1$ | $31$ | $( 1, 5, 9,13,17,21,25,29, 2, 6,10,14,18,22,26,30, 3, 7,11,15,19,23,27,31, 4, 8,12,16,20,24,28)$ |
| $ 31 $ | $1$ | $31$ | $( 1, 6,11,16,21,26,31, 5,10,15,20,25,30, 4, 9,14,19,24,29, 3, 8,13,18,23,28, 2, 7,12,17,22,27)$ |
| $ 31 $ | $1$ | $31$ | $( 1, 7,13,19,25,31, 6,12,18,24,30, 5,11,17,23,29, 4,10,16,22,28, 3, 9,15,21, 27, 2, 8,14,20,26)$ |
| $ 31 $ | $1$ | $31$ | $( 1, 8,15,22,29, 5,12,19,26, 2, 9,16,23,30, 6,13,20,27, 3,10,17,24,31, 7,14, 21,28, 4,11,18,25)$ |
| $ 31 $ | $1$ | $31$ | $( 1, 9,17,25, 2,10,18,26, 3,11,19,27, 4,12,20,28, 5,13,21,29, 6,14,22,30, 7, 15,23,31, 8,16,24)$ |
| $ 31 $ | $1$ | $31$ | $( 1,10,19,28, 6,15,24, 2,11,20,29, 7,16,25, 3,12,21,30, 8,17,26, 4,13,22,31, 9,18,27, 5,14,23)$ |
| $ 31 $ | $1$ | $31$ | $( 1,11,21,31,10,20,30, 9,19,29, 8,18,28, 7,17,27, 6,16,26, 5,15,25, 4,14,24, 3,13,23, 2,12,22)$ |
| $ 31 $ | $1$ | $31$ | $( 1,12,23, 3,14,25, 5,16,27, 7,18,29, 9,20,31,11,22, 2,13,24, 4,15,26, 6,17, 28, 8,19,30,10,21)$ |
| $ 31 $ | $1$ | $31$ | $( 1,13,25, 6,18,30,11,23, 4,16,28, 9,21, 2,14,26, 7,19,31,12,24, 5,17,29,10, 22, 3,15,27, 8,20)$ |
| $ 31 $ | $1$ | $31$ | $( 1,14,27, 9,22, 4,17,30,12,25, 7,20, 2,15,28,10,23, 5,18,31,13,26, 8,21, 3, 16,29,11,24, 6,19)$ |
| $ 31 $ | $1$ | $31$ | $( 1,15,29,12,26, 9,23, 6,20, 3,17,31,14,28,11,25, 8,22, 5,19, 2,16,30,13,27, 10,24, 7,21, 4,18)$ |
| $ 31 $ | $1$ | $31$ | $( 1,16,31,15,30,14,29,13,28,12,27,11,26,10,25, 9,24, 8,23, 7,22, 6,21, 5,20, 4,19, 3,18, 2,17)$ |
| $ 31 $ | $1$ | $31$ | $( 1,17, 2,18, 3,19, 4,20, 5,21, 6,22, 7,23, 8,24, 9,25,10,26,11,27,12,28,13, 29,14,30,15,31,16)$ |
| $ 31 $ | $1$ | $31$ | $( 1,18, 4,21, 7,24,10,27,13,30,16, 2,19, 5,22, 8,25,11,28,14,31,17, 3,20, 6, 23, 9,26,12,29,15)$ |
| $ 31 $ | $1$ | $31$ | $( 1,19, 6,24,11,29,16, 3,21, 8,26,13,31,18, 5,23,10,28,15, 2,20, 7,25,12,30, 17, 4,22, 9,27,14)$ |
| $ 31 $ | $1$ | $31$ | $( 1,20, 8,27,15, 3,22,10,29,17, 5,24,12,31,19, 7,26,14, 2,21, 9,28,16, 4,23, 11,30,18, 6,25,13)$ |
| $ 31 $ | $1$ | $31$ | $( 1,21,10,30,19, 8,28,17, 6,26,15, 4,24,13, 2,22,11,31,20, 9,29,18, 7,27,16, 5,25,14, 3,23,12)$ |
| $ 31 $ | $1$ | $31$ | $( 1,22,12, 2,23,13, 3,24,14, 4,25,15, 5,26,16, 6,27,17, 7,28,18, 8,29,19, 9, 30,20,10,31,21,11)$ |
| $ 31 $ | $1$ | $31$ | $( 1,23,14, 5,27,18, 9,31,22,13, 4,26,17, 8,30,21,12, 3,25,16, 7,29,20,11, 2, 24,15, 6,28,19,10)$ |
| $ 31 $ | $1$ | $31$ | $( 1,24,16, 8,31,23,15, 7,30,22,14, 6,29,21,13, 5,28,20,12, 4,27,19,11, 3,26, 18,10, 2,25,17, 9)$ |
| $ 31 $ | $1$ | $31$ | $( 1,25,18,11, 4,28,21,14, 7,31,24,17,10, 3,27,20,13, 6,30,23,16, 9, 2,26,19, 12, 5,29,22,15, 8)$ |
| $ 31 $ | $1$ | $31$ | $( 1,26,20,14, 8, 2,27,21,15, 9, 3,28,22,16,10, 4,29,23,17,11, 5,30,24,18,12, 6,31,25,19,13, 7)$ |
| $ 31 $ | $1$ | $31$ | $( 1,27,22,17,12, 7, 2,28,23,18,13, 8, 3,29,24,19,14, 9, 4,30,25,20,15,10, 5, 31,26,21,16,11, 6)$ |
| $ 31 $ | $1$ | $31$ | $( 1,28,24,20,16,12, 8, 4,31,27,23,19,15,11, 7, 3,30,26,22,18,14,10, 6, 2,29, 25,21,17,13, 9, 5)$ |
| $ 31 $ | $1$ | $31$ | $( 1,29,26,23,20,17,14,11, 8, 5, 2,30,27,24,21,18,15,12, 9, 6, 3,31,28,25,22, 19,16,13,10, 7, 4)$ |
| $ 31 $ | $1$ | $31$ | $( 1,30,28,26,24,22,20,18,16,14,12,10, 8, 6, 4, 2,31,29,27,25,23,21,19,17,15, 13,11, 9, 7, 5, 3)$ |
| $ 31 $ | $1$ | $31$ | $( 1,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $31$ (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: | 31.1 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);