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Magma
magma: G := TransitiveGroup(26, 50);
Group invariants
| Abstract group: | $C_{13}^2:C_3^2:Q_8$ | magma: IdentifyGroup(G);
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| Order: | $12168=2^{3} \cdot 3^{2} \cdot 13^{2}$ | 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 | magma: NilpotencyClass(G);
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Group action invariants
| Degree $n$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $50$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | $(1,23,7,25,5,20,10,26,4,24,6,16)(2,19,11,22,8,21,9,17,13,14,3,15)(12,18)$, $(1,19,4,25,8,20,9,22,6,16,2,21)(3,23,11,26,13,17,7,18,12,15,10,24)(5,14)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $S_3$, $C_6$ x 3 $8$: $Q_8$ $12$: $D_{6}$, $C_6\times C_2$ $18$: $S_3\times C_3$ $24$: 24T4, 24T5 $36$: $C_6\times S_3$ $72$: 24T64 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 | Index | Representative |
| 1A | $1^{26}$ | $1$ | $1$ | $0$ | $()$ |
| 2A | $2^{12},1^{2}$ | $169$ | $2$ | $12$ | $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,13)(10,12)(14,24)(15,23)(16,22)(17,21)(18,20)(25,26)$ |
| 3A1 | $3^{4},1^{14}$ | $26$ | $3$ | $8$ | $(14,22,20)(15,25,16)(17,18,21)(19,24,26)$ |
| 3A-1 | $3^{4},1^{14}$ | $26$ | $3$ | $8$ | $(14,20,22)(15,16,25)(17,21,18)(19,26,24)$ |
| 3B1 | $3^{8},1^{2}$ | $169$ | $3$ | $16$ | $( 1,12, 7)( 2, 8,10)( 3, 4,13)( 5, 9, 6)(14,26,17)(15,22,20)(16,18,23)(21,24,25)$ |
| 3B-1 | $3^{8},1^{2}$ | $169$ | $3$ | $16$ | $( 1, 7,12)( 2,10, 8)( 3,13, 4)( 5, 6, 9)(14,17,26)(15,20,22)(16,23,18)(21,25,24)$ |
| 3C | $3^{8},1^{2}$ | $338$ | $3$ | $16$ | $( 1,12, 7)( 2, 8,10)( 3, 4,13)( 5, 9, 6)(14,16,22)(15,19,18)(17,25,23)(20,21,24)$ |
| 4A | $4^{6},1^{2}$ | $338$ | $4$ | $18$ | $( 1, 9, 8,13)( 2, 4, 7, 5)( 3,12, 6,10)(14,18,25,21)(15,23,24,16)(17,20,22,19)$ |
| 4B | $4^{6},2$ | $1014$ | $4$ | $19$ | $( 1,26, 8,25)( 2,24, 7,14)( 3,22, 6,16)( 4,20, 5,18)( 9,23,13,15)(10,21,12,17)(11,19)$ |
| 4C | $4^{6},2$ | $1014$ | $4$ | $19$ | $( 1,21,12,16)( 2,17,11,20)( 3,26,10,24)( 4,22, 9,15)( 5,18, 8,19)( 6,14, 7,23)(13,25)$ |
| 6A1 | $6^{4},1^{2}$ | $169$ | $6$ | $20$ | $( 1, 2,12, 8, 7,10)( 3, 9, 4, 6,13, 5)(14,21,26,24,17,25)(15,18,22,23,20,16)$ |
| 6A-1 | $6^{4},1^{2}$ | $169$ | $6$ | $20$ | $( 1,10, 7, 8,12, 2)( 3, 5,13, 6, 4, 9)(14,25,17,24,26,21)(15,16,20,23,22,18)$ |
| 6B | $6^{4},1^{2}$ | $338$ | $6$ | $20$ | $( 1, 2,12, 8, 7,10)( 3, 9, 4, 6,13, 5)(14,17,16,25,22,23)(15,21,19,24,18,20)$ |
| 6C1 | $6^{2},2^{6},1^{2}$ | $338$ | $6$ | $16$ | $( 1, 3)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)(14,21,23,18,24,22)(15,25,26,17,20,19)$ |
| 6C-1 | $6^{2},2^{6},1^{2}$ | $338$ | $6$ | $16$ | $( 1, 3)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)(14,22,24,18,23,21)(15,19,20,17,26,25)$ |
| 12A1 | $12^{2},1^{2}$ | $338$ | $12$ | $22$ | $( 1, 3, 2, 9,12, 4, 8, 6, 7,13,10, 5)(14,24,17,18,16,20,25,15,22,21,23,19)$ |
| 12A5 | $12^{2},1^{2}$ | $338$ | $12$ | $22$ | $( 1, 4,10, 9, 7, 3, 8, 5,12,13, 2, 6)(14,20,23,18,22,24,25,19,16,21,17,15)$ |
| 12B1 | $12^{2},1^{2}$ | $338$ | $12$ | $22$ | $( 1, 6, 3,10,11,13, 4,12, 2, 8, 7, 5)(14,15,26,17,22,25,19,18,20,16,24,21)$ |
| 12B-1 | $12^{2},1^{2}$ | $338$ | $12$ | $22$ | $( 1, 5, 7, 8, 2,12, 4,13,11,10, 3, 6)(14,21,24,16,20,18,19,25,22,17,26,15)$ |
| 12C1 | $12,4^{3},1^{2}$ | $338$ | $12$ | $20$ | $( 1,10, 3, 7)( 4,12,13, 5)( 6, 9,11, 8)(14,20,21,19,23,15,18,25,24,26,22,17)$ |
| 12C-1 | $12,4^{3},1^{2}$ | $338$ | $12$ | $20$ | $( 1, 7, 3,10)( 4, 5,13,12)( 6, 8,11, 9)(14,17,22,26,24,25,18,15,23,19,21,20)$ |
| 12C5 | $12,4^{3},1^{2}$ | $338$ | $12$ | $20$ | $( 1,10, 3, 7)( 4,12,13, 5)( 6, 9,11, 8)(14,15,22,19,24,20,18,17,23,26,21,25)$ |
| 12C-5 | $12,4^{3},1^{2}$ | $338$ | $12$ | $20$ | $( 1, 7, 3,10)( 4, 5,13,12)( 6, 8,11, 9)(14,25,21,26,23,17,18,20,24,19,22,15)$ |
| 12D1 | $12^{2},2$ | $1014$ | $12$ | $23$ | $( 1,21, 2,26,12,24, 8,17, 7,25,10,14)( 3,18, 9,22, 4,23, 6,20,13,16, 5,15)(11,19)$ |
| 12D-1 | $12^{2},2$ | $1014$ | $12$ | $23$ | $( 1,14,10,25, 7,17, 8,24,12,26, 2,21)( 3,15, 5,16,13,20, 6,23, 4,22, 9,18)(11,19)$ |
| 12E1 | $12^{2},2$ | $1014$ | $12$ | $23$ | $( 1,24, 4,21, 3,22,12,26, 9,16,10,15)( 2,23, 8,17, 6,19,11,14, 5,20, 7,18)(13,25)$ |
| 12E-1 | $12^{2},2$ | $1014$ | $12$ | $23$ | $( 1,15,10,16, 9,26,12,22, 3,21, 4,24)( 2,18, 7,20, 5,14,11,19, 6,17, 8,23)(13,25)$ |
| 13A | $13,1^{13}$ | $24$ | $13$ | $12$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)$ |
| 13B | $13^{2}$ | $72$ | $13$ | $24$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,22,17,25,20,15,23,18,26,21,16,24,19)$ |
| 13C | $13^{2}$ | $72$ | $13$ | $24$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,17,20,23,26,16,19,22,25,15,18,21,24)$ |
| 39A1 | $13,3^{4},1$ | $312$ | $39$ | $20$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,22,20)(15,25,16)(17,18,21)(19,24,26)$ |
| 39A-1 | $13,3^{4},1$ | $312$ | $39$ | $20$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,20,22)(15,16,25)(17,21,18)(19,26,24)$ |
Malle's constant $a(G)$: $1/8$
magma: ConjugacyClasses(G);
Character table
32 x 32 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed