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Magma
magma: G := TransitiveGroup(26, 32);
Group action invariants
| Degree $n$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $32$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_{13}^2:C_6$ | ||
| 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: | (1,23,12,26,4,25,11,21,13,18,8,19)(2,15,3,20,7,14,10,16,9,24,5,17)(6,22), (1,23,5,18,4,16)(2,25,8,24,13,21)(3,14,11,17,9,26)(6,20,7,22,10,15)(12,19) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $C_6$ x 3 $8$: $D_{4}$ $12$: $C_6\times C_2$ $24$: $D_4 \times C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: None
Low degree siblings
26T32, 26T37Siblings are shown 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 $ | $12$ | $13$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$ | |
| $ 13, 13 $ | $12$ | $13$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$ | |
| $ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $13$ | $(14,23,19,15,24,20,16,25,21,17,26,22,18)$ | |
| $ 13, 13 $ | $24$ | $13$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,18,22,26,17,21,25,16,20,24,15,19, 23)$ | |
| $ 13, 13 $ | $24$ | $13$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$ | |
| $ 13, 13 $ | $24$ | $13$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$ | |
| $ 13, 13 $ | $24$ | $13$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$ | |
| $ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $13$ | $(14,19,24,16,21,26,18,23,15,20,25,17,22)$ | |
| $ 13, 13 $ | $24$ | $13$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $169$ | $3$ | $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $169$ | $3$ | $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$ | |
| $ 6, 6, 6, 6, 1, 1 $ | $169$ | $6$ | $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,18,17,26,23,24)(16,22,20,25,19,21)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $169$ | $2$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$ | |
| $ 6, 6, 6, 6, 1, 1 $ | $169$ | $6$ | $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,24,23,26,17,18)(16,21,19,25,20,22)$ | |
| $ 12, 12, 2 $ | $338$ | $12$ | $( 1,23,12,26, 4,25,11,21,13,18, 8,19)( 2,15, 3,20, 7,14,10,16, 9,24, 5,17) ( 6,22)$ | |
| $ 12, 12, 2 $ | $338$ | $12$ | $( 1,15, 7,14, 2,17, 4,21,11,22, 3,19)( 5,23, 8,16,12,24,13,26,10,20, 6,25) ( 9,18)$ | |
| $ 4, 4, 4, 4, 4, 4, 2 $ | $338$ | $4$ | $( 1,17,10,19)( 2,23, 9,26)( 3,16, 8,20)( 4,22, 7,14)( 5,15, 6,21)(11,25,13,24) (12,18)$ | |
| $ 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 $ | $156$ | $26$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,22)(15,21)(16,20)(17,19)(23,26) (24,25)$ | |
| $ 13, 2, 2, 2, 2, 2, 2, 1 $ | $156$ | $26$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,17)(15,16)(18,26)(19,25)(20,24) (21,23)$ | |
| $ 6, 6, 3, 3, 3, 3, 1, 1 $ | $338$ | $6$ | $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,24,23,26,17,18)(16,21,19,25,20,22)$ | |
| $ 6, 6, 3, 3, 3, 3, 1, 1 $ | $338$ | $6$ | $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,18,17,26,23,24)(16,22,20,25,19,21)$ | |
| $ 6, 6, 6, 6, 2 $ | $338$ | $6$ | $( 1,23, 8,19, 6,22)( 2,15, 4,25, 9,24)( 3,20,13,18,12,26)( 5,17) ( 7,14,10,16,11,21)$ | |
| $ 6, 6, 6, 6, 2 $ | $338$ | $6$ | $( 1,15,10,20,11,22)( 2,17,13,26, 7,14)( 3,19)( 4,21, 6,25,12,24) ( 5,23, 9,18, 8,16)$ | |
| $ 26 $ | $156$ | $26$ | $( 1,17,11,25, 8,20, 5,15, 2,23,12,18, 9,26, 6,21, 3,16,13,24,10,19, 7,14, 4,22 )$ | |
| $ 26 $ | $156$ | $26$ | $( 1,20,12,21,10,22, 8,23, 6,24, 4,25, 2,26,13,14,11,15, 9,16, 7,17, 5,18, 3,19 )$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $26$ | $2$ | $( 1,26)( 2,19)( 3,25)( 4,18)( 5,24)( 6,17)( 7,23)( 8,16)( 9,22)(10,15)(11,21) (12,14)(13,20)$ |
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.bd | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 13 P | |
| Type |
magma: CharacterTable(G);