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Magma
magma: G := TransitiveGroup(26, 46);
Group action invariants
| Degree $n$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $46$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_{13}^2.D_6$ | ||
| 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: | (1,17,11,16,8,15,5,14,2,26,12,25,9,24,6,23,3,22,13,21,10,20,7,19,4,18), (1,12,11,4,7,2,6,8,9,3,13,5)(14,24,18,19,21,25,20,23,16,15,26,22), (1,2,10,9)(3,5,8,6)(4,13,7,11)(14,22,21,26)(15,17,20,18)(16,25,19,23) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $6$: $S_3$ $8$: $C_2^3$ $12$: $D_{6}$ x 3 $16$: $Q_8:C_2$ $24$: $S_3 \times C_2^2$ $48$: 24T19 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: 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 $ | $1$ | $1$ | $()$ |
| $ 13, 13 $ | $24$ | $13$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$ |
| $ 13, 13 $ | $24$ | $13$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$ |
| $ 13, 13 $ | $24$ | $13$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$ |
| $ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $24$ | $13$ | $(14,19,24,16,21,26,18,23,15,20,25,17,22)$ |
| $ 13, 13 $ | $24$ | $13$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$ |
| $ 13, 13 $ | $24$ | $13$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$ |
| $ 13, 13 $ | $24$ | $13$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $338$ | $3$ | $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$ |
| $ 26 $ | $312$ | $26$ | $( 1,17,11,16, 8,15, 5,14, 2,26,12,25, 9,24, 6,23, 3,22,13,21,10,20, 7,19, 4,18 )$ |
| $ 26 $ | $312$ | $26$ | $( 1,16, 5,26, 9,23,13,20, 4,17, 8,14,12,24, 3,21, 7,18,11,15, 2,25, 6,22,10,19 )$ |
| $ 26 $ | $312$ | $26$ | $( 1,26,13,17,12,21,11,25,10,16, 9,20, 8,24, 7,15, 6,19, 5,23, 4,14, 3,18, 2,22 )$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $78$ | $2$ | $( 1,24)( 2,20)( 3,16)( 4,25)( 5,21)( 6,17)( 7,26)( 8,22)( 9,18)(10,14)(11,23) (12,19)(13,15)$ |
| $ 6, 6, 6, 6, 1, 1 $ | $338$ | $6$ | $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,24,23,26,17,18)(16,21,19,25,20,22)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $169$ | $2$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$ |
| $ 12, 12, 1, 1 $ | $338$ | $12$ | $( 2, 8,11, 6,10,12,13, 7, 4, 9, 5, 3)(15,16,18,22,17,20,26,25,23,19,24,21)$ |
| $ 4, 4, 4, 4, 4, 4, 1, 1 $ | $338$ | $4$ | $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,19,26,22)(16,24,25,17)(18,21,23,20)$ |
| $ 12, 12, 1, 1 $ | $338$ | $12$ | $( 2,12, 5, 6, 4, 8,13, 3,10, 9,11, 7)(15,20,24,22,23,16,26,21,17,19,18,25)$ |
| $ 26 $ | $312$ | $26$ | $( 1,17, 7,18,13,19, 6,20,12,21, 5,22,11,23, 4,24,10,25, 3,26, 9,14, 2,15, 8,16 )$ |
| $ 26 $ | $312$ | $26$ | $( 1,16,11,22, 8,15, 5,21, 2,14,12,20, 9,26, 6,19, 3,25,13,18,10,24, 7,17, 4,23 )$ |
| $ 26 $ | $312$ | $26$ | $( 1,15, 2,26, 3,24, 4,22, 5,20, 6,18, 7,16, 8,14, 9,25,10,23,11,21,12,19,13,17 )$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $78$ | $2$ | $( 1,25)( 2,23)( 3,21)( 4,19)( 5,17)( 6,15)( 7,26)( 8,24)( 9,22)(10,20)(11,18) (12,16)(13,14)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $26$ | $2$ | $(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)$ |
| $ 13, 2, 2, 2, 2, 2, 2, 1 $ | $312$ | $26$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,26)(15,25)(16,24)(17,23)(18,22) (19,21)$ |
| $ 6, 6, 3, 3, 3, 3, 1, 1 $ | $338$ | $6$ | $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,18,17,26,23,24)(16,22,20,25,19,21)$ |
| $ 6, 6, 3, 3, 3, 3, 1, 1 $ | $338$ | $6$ | $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,24,23,26,17,18)(16,21,19,25,20,22)$ |
| $ 4, 4, 4, 4, 4, 4, 2 $ | $1014$ | $4$ | $( 1,17, 6,23)( 2,26, 5,14)( 3,22, 4,18)( 7,19,13,21)( 8,15,12,25)( 9,24,11,16) (10,20)$ |
| $ 12, 12, 1, 1 $ | $338$ | $12$ | $( 2, 8,11, 6,10,12,13, 7, 4, 9, 5, 3)(15,25,18,19,17,21,26,16,23,22,24,20)$ |
| $ 4, 4, 4, 4, 4, 4, 1, 1 $ | $169$ | $4$ | $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,22,26,19)(16,17,25,24)(18,20,23,21)$ |
| $ 4, 4, 4, 4, 4, 4, 2 $ | $1014$ | $4$ | $( 1,17,10,25)( 2,15, 9,14)( 3,26, 8,16)( 4,24, 7,18)( 5,22, 6,20)(11,23,13,19) (12,21)$ |
| $ 12, 12, 1, 1 $ | $338$ | $12$ | $( 2, 3, 5, 9, 4, 7,13,12,10, 6,11, 8)(15,20,24,22,23,16,26,21,17,19,18,25)$ |
| $ 4, 4, 4, 4, 4, 4, 1, 1 $ | $169$ | $4$ | $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,19,26,22)(16,24,25,17)(18,21,23,20)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $8112=2^{4} \cdot 3 \cdot 13^{2}$ | 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: | 8112.bg | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);