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Magma
magma: G := TransitiveGroup(26, 35);
Group action invariants
| Degree $n$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{13}^2:C_3:Q_8$ | ||
| 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: | (1,15,2,21)(3,14,13,22)(4,20,12,16)(5,26,11,23)(6,19,10,17)(7,25,9,24)(8,18), (1,16,2,15)(3,14,13,17)(4,26,12,18)(5,25,11,19)(6,24,10,20)(7,23,9,21)(8,22) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $8$: $Q_8$ $12$: $D_{6}$ $24$: 24T5 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 |
| $1^{26}$ | $1$ | $1$ | $()$ | |
| $13^{2}$ | $24$ | $13$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$ | |
| $13^{2}$ | $24$ | $13$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$ | |
| $13^{2}$ | $24$ | $13$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$ | |
| $13,1^{13}$ | $24$ | $13$ | $(14,26,25,24,23,22,21,20,19,18,17,16,15)$ | |
| $13^{2}$ | $24$ | $13$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$ | |
| $13^{2}$ | $24$ | $13$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$ | |
| $13^{2}$ | $24$ | $13$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$ | |
| $2^{12},1^{2}$ | $169$ | $2$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$ | |
| $3^{8},1^{2}$ | $338$ | $3$ | $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$ | |
| $6^{4},1^{2}$ | $338$ | $6$ | $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,18,17,26,23,24)(16,22,20,25,19,21)$ | |
| $4^{6},2$ | $1014$ | $4$ | $( 1,15, 2,21)( 3,14,13,22)( 4,20,12,16)( 5,26,11,23)( 6,19,10,17)( 7,25, 9,24) ( 8,18)$ | |
| $12^{2},1^{2}$ | $338$ | $12$ | $( 2,12, 5, 6, 4, 8,13, 3,10, 9,11, 7)(15,20,24,22,23,16,26,21,17,19,18,25)$ | |
| $12^{2},1^{2}$ | $338$ | $12$ | $( 2, 3, 5, 9, 4, 7,13,12,10, 6,11, 8)(15,21,24,19,23,25,26,20,17,22,18,16)$ | |
| $4^{6},1^{2}$ | $338$ | $4$ | $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,19,26,22)(16,24,25,17)(18,21,23,20)$ | |
| $4^{6},2$ | $1014$ | $4$ | $( 1,15,12,26)( 2,16,11,25)( 3,17,10,24)( 4,18, 9,23)( 5,19, 8,22)( 6,20, 7,21) (13,14)$ |
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.bg | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 13 P | |
| Type |
magma: CharacterTable(G);