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Magma
magma: G := TransitiveGroup(26, 28);
Group action invariants
| Degree $n$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $28$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_{13}^2:S_3$ | ||
| 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: | (1,19,11,17)(2,24,10,25)(3,16,9,20)(4,21,8,15)(5,26,7,23)(6,18)(12,22,13,14), (1,23,5,21,9,19,13,17,4,15,8,26,12,24,3,22,7,20,11,18,2,16,6,14,10,25) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $8$: $D_{4}$ $12$: $D_{6}$ $24$: $(C_6\times C_2):C_2$ 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
| 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 $ | $1$ | $1$ | $()$ | |
| $ 13, 13 $ | $24$ | $13$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$ | |
| $ 13, 13 $ | $24$ | $13$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,20,26,19,25,18,24,17,23,16,22,15, 21)$ | |
| $ 13, 13 $ | $24$ | $13$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$ | |
| $ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $13$ | $(14,25,23,21,19,17,15,26,24,22,20,18,16)$ | |
| $ 13, 13 $ | $12$ | $13$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$ | |
| $ 13, 13 $ | $12$ | $13$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$ | |
| $ 13, 13 $ | $12$ | $13$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$ | |
| $ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $13$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)$ | |
| $ 13, 13 $ | $12$ | $13$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$ | |
| $ 13, 13 $ | $12$ | $13$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$ | |
| $ 13, 13 $ | $12$ | $13$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $169$ | $2$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $338$ | $3$ | $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$ | |
| $ 6, 6, 6, 6, 1, 1 $ | $338$ | $6$ | $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,24,23,26,17,18)(16,21,19,25,20,22)$ | |
| $ 4, 4, 4, 4, 4, 4, 2 $ | $1014$ | $4$ | $( 1,19,11,17)( 2,24,10,25)( 3,16, 9,20)( 4,21, 8,15)( 5,26, 7,23)( 6,18) (12,22,13,14)$ | |
| $ 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 $ | $156$ | $26$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,17)(15,16)(18,26)(19,25)(20,24) (21,23)$ | |
| $ 13, 2, 2, 2, 2, 2, 2, 1 $ | $156$ | $26$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,20)(15,19)(16,18)(21,26)(22,25) (23,24)$ | |
| $ 6, 6, 3, 3, 3, 3, 1, 1 $ | $338$ | $6$ | $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,24,23,26,17,18)(16,21,19,25,20,22)$ | |
| $ 6, 6, 3, 3, 3, 3, 1, 1 $ | $338$ | $6$ | $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,17,23)(16,20,19)(18,26,24)(21,22,25)$ | |
| $ 26 $ | $156$ | $26$ | $( 1,19,13,14,12,22,11,17,10,25, 9,20, 8,15, 7,23, 6,18, 5,26, 4,21, 3,16, 2,24 )$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $78$ | $2$ | $( 1,22)( 2,14)( 3,19)( 4,24)( 5,16)( 6,21)( 7,26)( 8,18)( 9,23)(10,15)(11,20) (12,25)(13,17)$ | |
| $ 26 $ | $156$ | $26$ | $( 1,18, 4,20, 7,22,10,24,13,26, 3,15, 6,17, 9,19,12,21, 2,23, 5,25, 8,14,11,16 )$ | |
| $ 26 $ | $156$ | $26$ | $( 1,17, 8,26, 2,22, 9,18, 3,14,10,23, 4,19,11,15, 5,24,12,20, 6,16,13,25, 7,21 )$ | |
| $ 26 $ | $156$ | $26$ | $( 1,15, 3,25, 5,22, 7,19, 9,16,11,26,13,23, 2,20, 4,17, 6,14, 8,24,10,21,12,18 )$ | |
| $ 26 $ | $156$ | $26$ | $( 1,24, 6,23,11,22, 3,21, 8,20,13,19, 5,18,10,17, 2,16, 7,15,12,14, 4,26, 9,25 )$ | |
| $ 26 $ | $156$ | $26$ | $( 1,21, 5,15, 9,22,13,16, 4,23, 8,17,12,24, 3,18, 7,25,11,19, 2,26, 6,20,10,14 )$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $4056=2^{3} \cdot 3 \cdot 13^{2}$ | 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: | 4056.ba | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 13 P | |
| Type |
magma: CharacterTable(G);