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Magma
magma: G := TransitiveGroup(26, 25);
Group invariants
| Abstract group: | $D_{13}^2.C_2^2$ | magma: IdentifyGroup(G);
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| Order: | $2704=2^{4} \cdot 13^{2}$ | 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 $t$: | $25$ | 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: | $(1,3,5,7,9,11,13,2,4,6,8,10,12)(14,18,22,26,17,21,25,16,20,24,15,19,23)$, $(1,12)(2,11)(3,10)(4,9)(5,8)(6,7)$, $(1,5,12,8)(2,10,11,3)(4,7,9,6)(14,21,25,18)(15,16,24,23)(17,19,22,20)$, $(1,16,5,19)(2,20,4,15)(3,24)(6,23,13,25)(7,14,12,21)(8,18,11,17)(9,22,10,26)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $8$: $C_2^3$ $16$: $Q_8:C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: None
Low degree siblings
26T25 x 2Siblings 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 | $2^{6},1^{14}$ | $26$ | $2$ | $6$ | $( 1, 4)( 2, 3)( 5,13)( 6,12)( 7,11)( 8,10)$ |
| 2B | $2^{13}$ | $26$ | $2$ | $13$ | $( 1,23)( 2,14)( 3,18)( 4,22)( 5,26)( 6,17)( 7,21)( 8,25)( 9,16)(10,20)(11,24)(12,15)(13,19)$ |
| 2C | $2^{13}$ | $26$ | $2$ | $13$ | $( 1,23)( 2,17)( 3,24)( 4,18)( 5,25)( 6,19)( 7,26)( 8,20)( 9,14)(10,21)(11,15)(12,22)(13,16)$ |
| 2D | $2^{12},1^{2}$ | $169$ | $2$ | $12$ | $( 1, 5)( 2, 4)( 6,13)( 7,12)( 8,11)( 9,10)(14,21)(15,20)(16,19)(17,18)(22,26)(23,25)$ |
| 4A1 | $4^{6},1^{2}$ | $169$ | $4$ | $18$ | $( 1,13, 5, 6)( 2, 8, 4,11)( 7, 9,12,10)(14,22,21,26)(15,17,20,18)(16,25,19,23)$ |
| 4A-1 | $4^{6},1^{2}$ | $169$ | $4$ | $18$ | $( 1, 6, 5,13)( 2,11, 4, 8)( 7,10,12, 9)(14,26,21,22)(15,18,20,17)(16,23,19,25)$ |
| 4B | $4^{6},2$ | $338$ | $4$ | $19$ | $( 1,19, 5,16)( 2,15, 4,20)( 3,24)( 6,25,13,23)( 7,21,12,14)( 8,17,11,18)( 9,26,10,22)$ |
| 4C | $4^{6},2$ | $338$ | $4$ | $19$ | $( 1,18, 3,17)( 2,24)( 4,23,13,25)( 5,16,12,19)( 6,22,11,26)( 7,15,10,20)( 8,21, 9,14)$ |
| 4D | $4^{6},1^{2}$ | $338$ | $4$ | $18$ | $( 1, 8,12, 5)( 2, 3,11,10)( 4, 6, 9, 7)(14,17,19,16)(15,22,18,24)(20,21,26,25)$ |
| 13A1 | $13,1^{13}$ | $8$ | $13$ | $12$ | $(14,23,19,15,24,20,16,25,21,17,26,22,18)$ |
| 13A2 | $13,1^{13}$ | $8$ | $13$ | $12$ | $(14,25,23,21,19,17,15,26,24,22,20,18,16)$ |
| 13A4 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,26,25,24,23,22,21,20,19,18,17,16,15)$ |
| 13B1 | $13,1^{13}$ | $8$ | $13$ | $12$ | $(14,26,25,24,23,22,21,20,19,18,17,16,15)$ |
| 13B2 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,24,21,18,15,25,22,19,16,26,23,20,17)$ |
| 13B4 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,25,23,21,19,17,15,26,24,22,20,18,16)$ |
| 13C1 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,22,17,25,20,15,23,18,26,21,16,24,19)$ |
| 13C2 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,23,19,15,24,20,16,25,21,17,26,22,18)$ |
| 13C4 | $13^{2}$ | $8$ | $13$ | $24$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,21,15,22,16,23,17,24,18,25,19,26,20)$ |
| 13D1 | $13^{2}$ | $16$ | $13$ | $24$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,26,25,24,23,22,21,20,19,18,17,16,15)$ |
| 13D2 | $13^{2}$ | $16$ | $13$ | $24$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,24,21,18,15,25,22,19,16,26,23,20,17)$ |
| 13D4 | $13^{2}$ | $16$ | $13$ | $24$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,25,23,21,19,17,15,26,24,22,20,18,16)$ |
| 13E1 | $13^{2}$ | $16$ | $13$ | $24$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,22,17,25,20,15,23,18,26,21,16,24,19)$ |
| 13E2 | $13^{2}$ | $16$ | $13$ | $24$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,24,21,18,15,25,22,19,16,26,23,20,17)$ |
| 13E4 | $13^{2}$ | $16$ | $13$ | $24$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,25,23,21,19,17,15,26,24,22,20,18,16)$ |
| 26A1 | $26$ | $104$ | $26$ | $25$ | $( 1,25, 5,14, 9,16,13,18, 4,20, 8,22,12,24, 3,26, 7,15,11,17, 2,19, 6,21,10,23)$ |
| 26A3 | $26$ | $104$ | $26$ | $25$ | $( 1,17, 2,24, 3,18, 4,25, 5,19, 6,26, 7,20, 8,14, 9,21,10,15,11,22,12,16,13,23)$ |
| 26A7 | $26$ | $104$ | $26$ | $25$ | $( 1,24, 3,25, 5,26, 7,14, 9,15,11,16,13,17, 2,18, 4,19, 6,20, 8,21,10,22,12,23)$ |
| 26B1 | $13,2^{6},1$ | $104$ | $26$ | $18$ | $( 1, 4)( 2, 3)( 5,13)( 6,12)( 7,11)( 8,10)(14,17,20,23,26,16,19,22,25,15,18,21,24)$ |
| 26B3 | $13,2^{6},1$ | $104$ | $26$ | $18$ | $( 1, 4)( 2, 3)( 5,13)( 6,12)( 7,11)( 8,10)(14,20,26,19,25,18,24,17,23,16,22,15,21)$ |
| 26B7 | $13,2^{6},1$ | $104$ | $26$ | $18$ | $( 1, 4)( 2, 3)( 5,13)( 6,12)( 7,11)( 8,10)(14,19,24,16,21,26,18,23,15,20,25,17,22)$ |
| 26C1 | $26$ | $104$ | $26$ | $25$ | $( 1,19,13,15,12,24,11,20,10,16, 9,25, 8,21, 7,17, 6,26, 5,22, 4,18, 3,14, 2,23)$ |
| 26C3 | $26$ | $104$ | $26$ | $25$ | $( 1,24,11,25, 8,26, 5,14, 2,15,12,16, 9,17, 6,18, 3,19,13,20,10,21, 7,22, 4,23)$ |
| 26C7 | $26$ | $104$ | $26$ | $25$ | $( 1,20,10,17, 6,14, 2,24,11,21, 7,18, 3,15,12,25, 8,22, 4,19,13,16, 9,26, 5,23)$ |
Malle's constant $a(G)$: $1/6$
magma: ConjugacyClasses(G);
Character table
34 x 34 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed