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Magma
magma: G := TransitiveGroup(26, 44);
Group action invariants
| Degree $n$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $44$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{13}^2:C_3:\OD_{16}$ | ||
| 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: | (1,9,11,5,10,8)(2,6,7,4,13,12)(14,15,24)(16,20,17)(18,25,23)(19,21,26), (1,17,3,24,13,20,11,26)(2,14,8,22,12,23,6,15)(4,21,5,18,10,16,9,19)(7,25) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $6$: $S_3$ $8$: $C_4\times C_2$ $12$: $D_{6}$, $C_3 : C_4$ x 2 $16$: $C_8:C_2$ $24$: 24T6 $48$: 24T20 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 $ | $48$ | $13$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$ | |
| $ 13, 13 $ | $48$ | $13$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$ | |
| $ 13, 13 $ | $48$ | $13$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$ | |
| $ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $24$ | $13$ | $(14,20,26,19,25,18,24,17,23,16,22,15,21)$ | |
| $ 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, 3, 3, 3, 3, 1, 1 $ | $338$ | $6$ | $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,23,17)(16,19,20)(18,24,26)(21,25,22)$ | |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $26$ | $2$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$ | |
| $ 13, 2, 2, 2, 2, 2, 2, 1 $ | $312$ | $26$ | $( 1,13)( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)(14,21,15,22,16,23,17,24,18,25,19, 26,20)$ | |
| $ 6, 6, 3, 3, 3, 3, 1, 1 $ | $338$ | $6$ | $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,17,23)(16,20,19)(18,26,24)(21,22,25)$ | |
| $ 6, 6, 6, 6, 1, 1 $ | $338$ | $6$ | $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,18,17,26,23,24)(16,22,20,25,19,21)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $169$ | $2$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$ | |
| $ 4, 4, 4, 4, 4, 4, 1, 1 $ | $169$ | $4$ | $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,19,26,22)(16,24,25,17)(18,21,23,20)$ | |
| $ 12, 12, 1, 1 $ | $338$ | $12$ | $( 2, 7,11, 9,10, 3,13, 8, 4, 6, 5,12)(15,16,18,22,17,20,26,25,23,19,24,21)$ | |
| $ 12, 12, 1, 1 $ | $338$ | $12$ | $( 2,12, 5, 6, 4, 8,13, 3,10, 9,11, 7)(15,20,24,22,23,16,26,21,17,19,18,25)$ | |
| $ 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 $ | $338$ | $12$ | $( 2, 8,11, 6,10,12,13, 7, 4, 9, 5, 3)(15,16,18,22,17,20,26,25,23,19,24,21)$ | |
| $ 12, 12, 1, 1 $ | $338$ | $12$ | $( 2,12, 5, 6, 4, 8,13, 3,10, 9,11, 7)(15,21,24,19,23,25,26,20,17,22,18,16)$ | |
| $ 4, 4, 4, 4, 4, 4, 1, 1 $ | $169$ | $4$ | $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,22,26,19)(16,17,25,24)(18,20,23,21)$ | |
| $ 8, 8, 8, 2 $ | $1014$ | $8$ | $( 1,17, 3,24,13,20,11,26)( 2,14, 8,22,12,23, 6,15)( 4,21, 5,18,10,16, 9,19) ( 7,25)$ | |
| $ 8, 8, 8, 2 $ | $1014$ | $8$ | $( 1,15,11,18,13,16, 3,26)( 2,14, 6,23,12,17, 8,21)( 4,25, 9,20,10,19, 5,24) ( 7,22)$ | |
| $ 8, 8, 8, 2 $ | $1014$ | $8$ | $( 1,16, 2,14,10,24, 9,26)( 3,25, 5,21, 8,15, 6,19)( 4,23,13,18, 7,17,11,22) (12,20)$ | |
| $ 8, 8, 8, 2 $ | $1014$ | $8$ | $( 1,19, 8,23, 4,17,10,26)( 2,14,13,24, 3,22, 5,25)( 6,20, 7,15,12,16,11,21) ( 9,18)$ |
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.be | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 13 P | |
| Type |
magma: CharacterTable(G);