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Magma
magma: G := TransitiveGroup(26, 36);
Group action invariants
| Degree $n$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $36$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{13}^2:D_{12}$ | ||
| 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: | (1,25,8,23,2,21,9,19,3,17,10,15,4,26,11,24,5,22,12,20,6,18,13,16,7,14), (1,20,11,18,8,16,5,14,2,25,12,23,9,21,6,19,3,17,13,15,10,26,7,24,4,22) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $8$: $D_{4}$ $12$: $D_{6}$ $24$: $D_{12}$ 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 |
| $1^{26}$ | $1$ | $1$ | $()$ | |
| $13^{2}$ | $12$ | $13$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$ | |
| $13^{2}$ | $12$ | $13$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$ | |
| $13^{2}$ | $12$ | $13$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$ | |
| $13^{2}$ | $12$ | $13$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$ | |
| $13^{2}$ | $12$ | $13$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$ | |
| $13^{2}$ | $12$ | $13$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$ | |
| $13,1^{13}$ | $24$ | $13$ | $(14,24,21,18,15,25,22,19,16,26,23,20,17)$ | |
| $13^{2}$ | $12$ | $13$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,20,26,19,25,18,24,17,23,16,22,15, 21)$ | |
| $13^{2}$ | $12$ | $13$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$ | |
| $13^{2}$ | $12$ | $13$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$ | |
| $13^{2}$ | $12$ | $13$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$ | |
| $13^{2}$ | $12$ | $13$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$ | |
| $13^{2}$ | $12$ | $13$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$ | |
| $3^{8},1^{2}$ | $338$ | $3$ | $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$ | |
| $26$ | $156$ | $26$ | $( 1,25, 8,23, 2,21, 9,19, 3,17,10,15, 4,26,11,24, 5,22,12,20, 6,18,13,16, 7,14 )$ | |
| $26$ | $156$ | $26$ | $( 1,23, 9,17, 4,24,12,18, 7,25, 2,19,10,26, 5,20,13,14, 8,21, 3,15,11,22, 6,16 )$ | |
| $26$ | $156$ | $26$ | $( 1,21,10,24, 6,14, 2,17,11,20, 7,23, 3,26,12,16, 8,19, 4,22,13,25, 9,15, 5,18 )$ | |
| $26$ | $156$ | $26$ | $( 1,17,12,25,10,20, 8,15, 6,23, 4,18, 2,26,13,21,11,16, 9,24, 7,19, 5,14, 3,22 )$ | |
| $26$ | $156$ | $26$ | $( 1,19,11,18, 8,17, 5,16, 2,15,12,14, 9,26, 6,25, 3,24,13,23,10,22, 7,21, 4,20 )$ | |
| $2^{13}$ | $78$ | $2$ | $( 1,26)( 2,22)( 3,18)( 4,14)( 5,23)( 6,19)( 7,15)( 8,24)( 9,20)(10,16)(11,25) (12,21)(13,17)$ | |
| $26$ | $156$ | $26$ | $( 1,15,13,19,12,23,11,14,10,18, 9,22, 8,26, 7,17, 6,21, 5,25, 4,16, 3,20, 2,24 )$ | |
| $6^{4},1^{2}$ | $338$ | $6$ | $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,24,23,26,17,18)(16,21,19,25,20,22)$ | |
| $2^{12},1^{2}$ | $169$ | $2$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$ | |
| $12^{2},1^{2}$ | $338$ | $12$ | $( 2, 3, 5, 9, 4, 7,13,12,10, 6,11, 8)(15,21,24,19,23,25,26,20,17,22,18,16)$ | |
| $12^{2},1^{2}$ | $338$ | $12$ | $( 2, 7,11, 9,10, 3,13, 8, 4, 6, 5,12)(15,25,18,19,17,21,26,16,23,22,24,20)$ | |
| $4^{6},1^{2}$ | $338$ | $4$ | $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,22,26,19)(16,17,25,24)(18,20,23,21)$ | |
| $26$ | $156$ | $26$ | $( 1,25,11,23, 8,21, 5,19, 2,17,12,15, 9,26, 6,24, 3,22,13,20,10,18, 7,16, 4,14 )$ | |
| $26$ | $156$ | $26$ | $( 1,23, 2,15, 3,20, 4,25, 5,17, 6,22, 7,14, 8,19, 9,24,10,16,11,21,12,26,13,18 )$ | |
| $26$ | $156$ | $26$ | $( 1,21, 6,20,11,19, 3,18, 8,17,13,16, 5,15,10,14, 2,26, 7,25,12,24, 4,23, 9,22 )$ | |
| $2^{13}$ | $78$ | $2$ | $( 1,17)( 2,22)( 3,14)( 4,19)( 5,24)( 6,16)( 7,21)( 8,26)( 9,18)(10,23)(11,15) (12,20)(13,25)$ | |
| $26$ | $156$ | $26$ | $( 1,19,10,25, 6,18, 2,24,11,17, 7,23, 3,16,12,22, 8,15, 4,21,13,14, 9,20, 5,26 )$ | |
| $26$ | $156$ | $26$ | $( 1,14, 7,18,13,22, 6,26,12,17, 5,21,11,25, 4,16,10,20, 3,24, 9,15, 2,19, 8,23 )$ | |
| $26$ | $156$ | $26$ | $( 1,16, 3,26, 5,23, 7,20, 9,17,11,14,13,24, 2,21, 4,18, 6,15, 8,25,10,22,12,19 )$ |
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.bh | magma: IdentifyGroup(G);
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| Character table: | 34 x 34 character table |
magma: CharacterTable(G);