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Magma
magma: G := TransitiveGroup(26, 18);
Group action invariants
| Degree $n$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_{13}^2.C_2$ | ||
| 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: | (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,15)(16,26)(17,25)(18,24)(19,23)(20,22), (1,22,7,18)(2,17,6,23)(3,25,5,15)(4,20)(8,26,13,14)(9,21,12,19)(10,16,11,24), (1,12,2,4)(3,9,13,7)(5,6,11,10)(14,19,18,26)(15,24,17,21)(20,23,25,22) | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $C_4\times C_2$ $52$: $C_{13}:C_4$ x 2 $104$: 26T7 x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: None
Low degree siblings
26T18 x 5Siblings 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 | $2^{13}$ | $13$ | $2$ | $13$ | $( 1,14)( 2,15)( 3,16)( 4,17)( 5,18)( 6,19)( 7,20)( 8,21)( 9,22)(10,23)(11,24)(12,25)(13,26)$ |
| 2B | $2^{13}$ | $13$ | $2$ | $13$ | $( 1,18)( 2,17)( 3,16)( 4,15)( 5,14)( 6,26)( 7,25)( 8,24)( 9,23)(10,22)(11,21)(12,20)(13,19)$ |
| 2C | $2^{12},1^{2}$ | $169$ | $2$ | $12$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)$ |
| 4A1 | $4^{6},1^{2}$ | $169$ | $4$ | $18$ | $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,22,26,19)(16,17,25,24)(18,20,23,21)$ |
| 4A-1 | $4^{6},1^{2}$ | $169$ | $4$ | $18$ | $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,19,26,22)(16,24,25,17)(18,21,23,20)$ |
| 4B1 | $4^{6},2$ | $169$ | $4$ | $19$ | $( 1,20,11,18)( 2,25,10,26)( 3,17, 9,21)( 4,22, 8,16)( 5,14, 7,24)( 6,19)(12,23,13,15)$ |
| 4B-1 | $4^{6},2$ | $169$ | $4$ | $19$ | $( 1,21,12,18)( 2,16,11,23)( 3,24,10,15)( 4,19, 9,20)( 5,14, 8,25)( 6,22, 7,17)(13,26)$ |
| 13A1 | $13^{2}$ | $4$ | $13$ | $24$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,24,21,18,15,25,22,19,16,26,23,20,17)$ |
| 13A2 | $13^{2}$ | $4$ | $13$ | $24$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,21,15,22,16,23,17,24,18,25,19,26,20)$ |
| 13A4 | $13^{2}$ | $4$ | $13$ | $24$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,19,24,16,21,26,18,23,15,20,25,17,22)$ |
| 13B1 | $13^{2}$ | $4$ | $13$ | $24$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,26,25,24,23,22,21,20,19,18,17,16,15)$ |
| 13B2 | $13^{2}$ | $4$ | $13$ | $24$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,25,23,21,19,17,15,26,24,22,20,18,16)$ |
| 13B4 | $13^{2}$ | $4$ | $13$ | $24$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,23,19,15,24,20,16,25,21,17,26,22,18)$ |
| 13C1 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,15,16,17,18,19,20,21,22,23,24,25,26)$ |
| 13C2 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,22,17,25,20,15,23,18,26,21,16,24,19)$ |
| 13C4 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,20,26,19,25,18,24,17,23,16,22,15,21)$ |
| 13D1 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,18,22,26,17,21,25,16,20,24,15,19,23)$ |
| 13D2 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,18,22,26,17,21,25,16,20,24,15,19,23)$ |
| 13D4 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,17,20,23,26,16,19,22,25,15,18,21,24)$ |
| 13E1 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,20,26,19,25,18,24,17,23,16,22,15,21)$ |
| 13E2 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,19,24,16,21,26,18,23,15,20,25,17,22)$ |
| 13E4 | $13,1^{13}$ | $8$ | $13$ | $12$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)$ |
| 13F1 | $13,1^{13}$ | $8$ | $13$ | $12$ | $(14,21,15,22,16,23,17,24,18,25,19,26,20)$ |
| 13F2 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,16,18,20,22,24,26,15,17,19,21,23,25)$ |
| 13F4 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,18,22,26,17,21,25,16,20,24,15,19,23)$ |
| 13G1 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,26,25,24,23,22,21,20,19,18,17,16,15)$ |
| 13G2 | $13,1^{13}$ | $8$ | $13$ | $12$ | $(14,15,16,17,18,19,20,21,22,23,24,25,26)$ |
| 13G4 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)(14,17,20,23,26,16,19,22,25,15,18,21,24)$ |
| 13H1 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,16,18,20,22,24,26,15,17,19,21,23,25)$ |
| 13H2 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,23,19,15,24,20,16,25,21,17,26,22,18)$ |
| 13H4 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,19,24,16,21,26,18,23,15,20,25,17,22)$ |
| 26A1 | $26$ | $52$ | $26$ | $25$ | $( 1,17, 3,15, 5,26, 7,24, 9,22,11,20,13,18, 2,16, 4,14, 6,25, 8,23,10,21,12,19)$ |
| 26A3 | $26$ | $52$ | $26$ | $25$ | $( 1,15, 3,17, 5,19, 7,21, 9,23,11,25,13,14, 2,16, 4,18, 6,20, 8,22,10,24,12,26)$ |
| 26A7 | $26$ | $52$ | $26$ | $25$ | $( 1,14, 9,19, 4,24,12,16, 7,21, 2,26,10,18, 5,23,13,15, 8,20, 3,25,11,17, 6,22)$ |
| 26B1 | $26$ | $52$ | $26$ | $25$ | $( 1,23, 6,15,11,20, 3,25, 8,17,13,22, 5,14,10,19, 2,24, 7,16,12,21, 4,26, 9,18)$ |
| 26B3 | $26$ | $52$ | $26$ | $25$ | $( 1,16, 5,25, 9,21,13,17, 4,26, 8,22,12,18, 3,14, 7,23,11,19, 2,15, 6,24,10,20)$ |
| 26B7 | $26$ | $52$ | $26$ | $25$ | $( 1,25,10,21, 6,17, 2,26,11,22, 7,18, 3,14,12,23, 8,19, 4,15,13,24, 9,20, 5,16)$ |
Malle's constant $a(G)$: $1/12$
magma: ConjugacyClasses(G);
Group invariants
| 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 | ||
| Label: | 1352.41 | magma: IdentifyGroup(G);
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| Character table: | 38 x 38 character table |
magma: CharacterTable(G);