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Magma
magma: G := TransitiveGroup(26, 16);
Group invariants
| Abstract group: | $D_{13}\wr C_2$ | magma: IdentifyGroup(G);
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| Order: | $1352=2^{3} \cdot 13^{2}$ | 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 | 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 $t$: | $16$ | 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: | $(1,19)(2,21,13,17)(3,23,12,15)(4,25,11,26)(5,14,10,24)(6,16,9,22)(7,18,8,20)$, $(1,23,4,16,7,22,10,15,13,21,3,14,6,20,9,26,12,19,2,25,5,18,8,24,11,17)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $D_{4}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: None
Low degree siblings
26T16, 26T19Siblings 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 | Index | Representative |
| 1A | $1^{26}$ | $1$ | $1$ | $0$ | $()$ |
| 2A | $2^{13}$ | $26$ | $2$ | $13$ | $( 1,19)( 2,17)( 3,15)( 4,26)( 5,24)( 6,22)( 7,20)( 8,18)( 9,16)(10,14)(11,25)(12,23)(13,21)$ |
| 2B | $2^{6},1^{14}$ | $26$ | $2$ | $6$ | $(14,24)(15,23)(16,22)(17,21)(18,20)(25,26)$ |
| 2C | $2^{12},1^{2}$ | $169$ | $2$ | $12$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(14,24)(15,23)(16,22)(17,21)(18,20)(25,26)$ |
| 4A | $4^{6},2$ | $338$ | $4$ | $19$ | $( 1,19)( 2,21,13,17)( 3,23,12,15)( 4,25,11,26)( 5,14,10,24)( 6,16, 9,22)( 7,18, 8,20)$ |
| 13A1 | $13,1^{13}$ | $4$ | $13$ | $12$ | $(14,23,19,15,24,20,16,25,21,17,26,22,18)$ |
| 13A2 | $13,1^{13}$ | $4$ | $13$ | $12$ | $(14,17,20,23,26,16,19,22,25,15,18,21,24)$ |
| 13A3 | $13,1^{13}$ | $4$ | $13$ | $12$ | $(14,22,17,25,20,15,23,18,26,21,16,24,19)$ |
| 13A4 | $13^{2}$ | $4$ | $13$ | $24$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,15,16,17,18,19,20,21,22,23,24,25,26)$ |
| 13A5 | $13^{2}$ | $4$ | $13$ | $24$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,23,19,15,24,20,16,25,21,17,26,22,18)$ |
| 13A6 | $13,1^{13}$ | $4$ | $13$ | $12$ | $(14,15,16,17,18,19,20,21,22,23,24,25,26)$ |
| 13B1 | $13^{2}$ | $4$ | $13$ | $24$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,20,26,19,25,18,24,17,23,16,22,15,21)$ |
| 13B2 | $13^{2}$ | $4$ | $13$ | $24$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,17,20,23,26,16,19,22,25,15,18,21,24)$ |
| 13B3 | $13^{2}$ | $4$ | $13$ | $24$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,16,18,20,22,24,26,15,17,19,21,23,25)$ |
| 13B4 | $13,1^{13}$ | $4$ | $13$ | $12$ | $(14,20,26,19,25,18,24,17,23,16,22,15,21)$ |
| 13B5 | $13^{2}$ | $4$ | $13$ | $24$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,19,24,16,21,26,18,23,15,20,25,17,22)$ |
| 13B6 | $13,1^{13}$ | $4$ | $13$ | $12$ | $(14,25,23,21,19,17,15,26,24,22,20,18,16)$ |
| 13C1 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,16,18,20,22,24,26,15,17,19,21,23,25)$ |
| 13C2 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,22,17,25,20,15,23,18,26,21,16,24,19)$ |
| 13C4 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,15,16,17,18,19,20,21,22,23,24,25,26)$ |
| 13D1 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,25,23,21,19,17,15,26,24,22,20,18,16)$ |
| 13D2 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,15,16,17,18,19,20,21,22,23,24,25,26)$ |
| 13D3 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,23,19,15,24,20,16,25,21,17,26,22,18)$ |
| 13D4 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,23,19,15,24,20,16,25,21,17,26,22,18)$ |
| 13D5 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,19,24,16,21,26,18,23,15,20,25,17,22)$ |
| 13D6 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,19,24,16,21,26,18,23,15,20,25,17,22)$ |
| 13E1 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,25,23,21,19,17,15,26,24,22,20,18,16)$ |
| 13E2 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,19,24,16,21,26,18,23,15,20,25,17,22)$ |
| 13E3 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,20,26,19,25,18,24,17,23,16,22,15,21)$ |
| 13E4 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,25,23,21,19,17,15,26,24,22,20,18,16)$ |
| 13E5 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,19,24,16,21,26,18,23,15,20,25,17,22)$ |
| 13E6 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,15,16,17,18,19,20,21,22,23,24,25,26)$ |
| 26A1 | $26$ | $52$ | $26$ | $25$ | $( 1,21, 6,24,11,14, 3,17, 8,20,13,23, 5,26,10,16, 2,19, 7,22,12,25, 4,15, 9,18)$ |
| 26A3 | $26$ | $52$ | $26$ | $25$ | $( 1,18, 5,23, 9,15,13,20, 4,25, 8,17,12,22, 3,14, 7,19,11,24, 2,16, 6,21,10,26)$ |
| 26A5 | $26$ | $52$ | $26$ | $25$ | $( 1,15, 4,22, 7,16,10,23,13,17, 3,24, 6,18, 9,25,12,19, 2,26, 5,20, 8,14,11,21)$ |
| 26A7 | $26$ | $52$ | $26$ | $25$ | $( 1,25, 3,21, 5,17, 7,26, 9,22,11,18,13,14, 2,23, 4,19, 6,15, 8,24,10,20,12,16)$ |
| 26A9 | $26$ | $52$ | $26$ | $25$ | $( 1,22, 2,20, 3,18, 4,16, 5,14, 6,25, 7,23, 8,21, 9,19,10,17,11,15,12,26,13,24)$ |
| 26A11 | $26$ | $52$ | $26$ | $25$ | $( 1,24, 7,25,13,26, 6,14,12,15, 5,16,11,17, 4,18,10,19, 3,20, 9,21, 2,22, 8,23)$ |
| 26B1 | $13,2^{6},1$ | $52$ | $26$ | $18$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,15)(16,26)(17,25)(18,24)(19,23)(20,22)$ |
| 26B3 | $13,2^{6},1$ | $52$ | $26$ | $18$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,18)(15,17)(19,26)(20,25)(21,24)(22,23)$ |
| 26B5 | $13,2^{6},1$ | $52$ | $26$ | $18$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,21)(15,20)(16,19)(17,18)(22,26)(23,25)$ |
| 26B7 | $13,2^{6},1$ | $52$ | $26$ | $18$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,26)(15,25)(16,24)(17,23)(18,22)(19,21)$ |
| 26B9 | $13,2^{6},1$ | $52$ | $26$ | $18$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,23)(15,22)(16,21)(17,20)(18,19)(24,26)$ |
| 26B11 | $13,2^{6},1$ | $52$ | $26$ | $18$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,16)(17,26)(18,25)(19,24)(20,23)(21,22)$ |
Malle's constant $a(G)$: $1/6$
magma: ConjugacyClasses(G);
Character table
44 x 44 character tablemagma: CharacterTable(G);