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Magma
magma: G := TransitiveGroup(26, 1);
Group action invariants
| Degree $n$: | $26$ | 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: | $C_{26}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $26$ | 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) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $13$: $C_{13}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: $C_{13}$
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 $ | $1$ | $1$ | $()$ |
| $ 26 $ | $1$ | $26$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26 )$ |
| $ 13, 13 $ | $1$ | $13$ | $( 1, 3, 5, 7, 9,11,13,15,17,19,21,23,25)( 2, 4, 6, 8,10,12,14,16,18,20,22,24, 26)$ |
| $ 26 $ | $1$ | $26$ | $( 1, 4, 7,10,13,16,19,22,25, 2, 5, 8,11,14,17,20,23,26, 3, 6, 9,12,15,18,21,24 )$ |
| $ 13, 13 $ | $1$ | $13$ | $( 1, 5, 9,13,17,21,25, 3, 7,11,15,19,23)( 2, 6,10,14,18,22,26, 4, 8,12,16,20, 24)$ |
| $ 26 $ | $1$ | $26$ | $( 1, 6,11,16,21,26, 5,10,15,20,25, 4, 9,14,19,24, 3, 8,13,18,23, 2, 7,12,17,22 )$ |
| $ 13, 13 $ | $1$ | $13$ | $( 1, 7,13,19,25, 5,11,17,23, 3, 9,15,21)( 2, 8,14,20,26, 6,12,18,24, 4,10,16, 22)$ |
| $ 26 $ | $1$ | $26$ | $( 1, 8,15,22, 3,10,17,24, 5,12,19,26, 7,14,21, 2, 9,16,23, 4,11,18,25, 6,13,20 )$ |
| $ 13, 13 $ | $1$ | $13$ | $( 1, 9,17,25, 7,15,23, 5,13,21, 3,11,19)( 2,10,18,26, 8,16,24, 6,14,22, 4,12, 20)$ |
| $ 26 $ | $1$ | $26$ | $( 1,10,19, 2,11,20, 3,12,21, 4,13,22, 5,14,23, 6,15,24, 7,16,25, 8,17,26, 9,18 )$ |
| $ 13, 13 $ | $1$ | $13$ | $( 1,11,21, 5,15,25, 9,19, 3,13,23, 7,17)( 2,12,22, 6,16,26,10,20, 4,14,24, 8, 18)$ |
| $ 26 $ | $1$ | $26$ | $( 1,12,23, 8,19, 4,15,26,11,22, 7,18, 3,14,25,10,21, 6,17, 2,13,24, 9,20, 5,16 )$ |
| $ 13, 13 $ | $1$ | $13$ | $( 1,13,25,11,23, 9,21, 7,19, 5,17, 3,15)( 2,14,26,12,24,10,22, 8,20, 6,18, 4, 16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,14)( 2,15)( 3,16)( 4,17)( 5,18)( 6,19)( 7,20)( 8,21)( 9,22)(10,23)(11,24) (12,25)(13,26)$ |
| $ 13, 13 $ | $1$ | $13$ | $( 1,15, 3,17, 5,19, 7,21, 9,23,11,25,13)( 2,16, 4,18, 6,20, 8,22,10,24,12,26, 14)$ |
| $ 26 $ | $1$ | $26$ | $( 1,16, 5,20, 9,24,13, 2,17, 6,21,10,25,14, 3,18, 7,22,11,26,15, 4,19, 8,23,12 )$ |
| $ 13, 13 $ | $1$ | $13$ | $( 1,17, 7,23,13, 3,19, 9,25,15, 5,21,11)( 2,18, 8,24,14, 4,20,10,26,16, 6,22, 12)$ |
| $ 26 $ | $1$ | $26$ | $( 1,18, 9,26,17, 8,25,16, 7,24,15, 6,23,14, 5,22,13, 4,21,12, 3,20,11, 2,19,10 )$ |
| $ 13, 13 $ | $1$ | $13$ | $( 1,19,11, 3,21,13, 5,23,15, 7,25,17, 9)( 2,20,12, 4,22,14, 6,24,16, 8,26,18, 10)$ |
| $ 26 $ | $1$ | $26$ | $( 1,20,13, 6,25,18,11, 4,23,16, 9, 2,21,14, 7,26,19,12, 5,24,17,10, 3,22,15, 8 )$ |
| $ 13, 13 $ | $1$ | $13$ | $( 1,21,15, 9, 3,23,17,11, 5,25,19,13, 7)( 2,22,16,10, 4,24,18,12, 6,26,20,14, 8)$ |
| $ 26 $ | $1$ | $26$ | $( 1,22,17,12, 7, 2,23,18,13, 8, 3,24,19,14, 9, 4,25,20,15,10, 5,26,21,16,11, 6 )$ |
| $ 13, 13 $ | $1$ | $13$ | $( 1,23,19,15,11, 7, 3,25,21,17,13, 9, 5)( 2,24,20,16,12, 8, 4,26,22,18,14,10, 6)$ |
| $ 26 $ | $1$ | $26$ | $( 1,24,21,18,15,12, 9, 6, 3,26,23,20,17,14,11, 8, 5, 2,25,22,19,16,13,10, 7, 4 )$ |
| $ 13, 13 $ | $1$ | $13$ | $( 1,25,23,21,19,17,15,13,11, 9, 7, 5, 3)( 2,26,24,22,20,18,16,14,12,10, 8, 6, 4)$ |
| $ 26 $ | $1$ | $26$ | $( 1,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2 )$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $26=2 \cdot 13$ | 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: | 26.2 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);