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Magma
magma: G := TransitiveGroup(26, 45);
Group invariants
| Abstract group: | $D_{13}^2.C_{12}$ | magma: IdentifyGroup(G);
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| Order: | $8112=2^{4} \cdot 3 \cdot 13^{2}$ | 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 | 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 $t$: | $45$ | 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: | $(1,19,13,22,11,15,7,14,12,25,9,21,3,26,4,23,6,17,10,18,5,20,8,24)(2,16)$, $(1,21,3,25,7,20,2,23,5,16,11,15,10,26,8,22,4,14,9,24,6,18,13,19)(12,17)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_4$ x 2, $C_2^2$ $6$: $C_6$ x 3 $8$: $C_4\times C_2$ $12$: $C_{12}$ x 2, $C_6\times C_2$ $16$: $C_8:C_2$ $24$: 24T2 $48$: 24T16 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 | $2^{6},1^{14}$ | $26$ | $2$ | $6$ | $(14,18)(15,17)(19,26)(20,25)(21,24)(22,23)$ |
| 2B | $2^{12},1^{2}$ | $169$ | $2$ | $12$ | $( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)$ |
| 3A1 | $3^{8},1^{2}$ | $169$ | $3$ | $16$ | $( 1, 9, 3)( 2, 5, 6)( 4,10,12)( 7,11, 8)(15,23,17)(16,19,20)(18,24,26)(21,25,22)$ |
| 3A-1 | $3^{8},1^{2}$ | $169$ | $3$ | $16$ | $( 1, 3, 9)( 2, 6, 5)( 4,12,10)( 7, 8,11)(15,17,23)(16,20,19)(18,26,24)(21,22,25)$ |
| 4A1 | $4^{6},1^{2}$ | $169$ | $4$ | $18$ | $( 1, 4, 6, 3)( 2, 9, 5,11)( 7, 8,13,12)(14,18,25,21)(15,23,24,16)(17,20,22,19)$ |
| 4A-1 | $4^{6},1^{2}$ | $169$ | $4$ | $18$ | $( 1, 7, 3,10)( 4, 5,13,12)( 6, 8,11, 9)(14,18,24,20)(15,26,23,25)(16,21,22,17)$ |
| 4B | $4^{6},1^{2}$ | $338$ | $4$ | $18$ | $( 1, 8,12, 5)( 2, 3,11,10)( 4, 6, 9, 7)(15,19,26,22)(16,24,25,17)(18,21,23,20)$ |
| 6A1 | $6^{4},1^{2}$ | $169$ | $6$ | $20$ | $( 1,10, 9,12, 3, 4)( 2, 7, 5,11, 6, 8)(15,24,23,26,17,18)(16,21,19,25,20,22)$ |
| 6A-1 | $6^{4},1^{2}$ | $169$ | $6$ | $20$ | $( 1, 4, 3,12, 9,10)( 2, 8, 6,11, 5, 7)(15,18,17,26,23,24)(16,22,20,25,19,21)$ |
| 6B1 | $6^{2},3^{4},1^{2}$ | $338$ | $6$ | $18$ | $( 1, 2, 5)( 3, 8,10)( 4,11, 6)( 9,13,12)(14,18,19,16,25,24)(17,22,20,26,21,23)$ |
| 6B-1 | $6^{2},3^{4},1^{2}$ | $338$ | $6$ | $18$ | $( 1, 8, 6)( 2, 4, 9)( 3,13,12)( 7,10,11)(14,18,21,20,16,26)(15,22,24,19,25,23)$ |
| 8A1 | $8^{3},2$ | $338$ | $8$ | $22$ | $( 1,19, 9,15,10,21, 2,25)( 3,18, 6,23, 8,22, 5,17)( 4,24,11,14, 7,16,13,26)(12,20)$ |
| 8A-1 | $8^{3},2$ | $338$ | $8$ | $22$ | $( 1,20,12,25, 9,26,11,21)( 2,24, 7,18, 8,22, 3,15)( 4,19,10,17, 6,14,13,16)( 5,23)$ |
| 8B1 | $8^{3},2$ | $338$ | $8$ | $22$ | $( 1,23)( 2,17, 9,14,13,16, 6,19)( 3,24, 4,18,12,22,11,15)( 5,25, 7,26,10,21, 8,20)$ |
| 8B-1 | $8^{3},2$ | $338$ | $8$ | $22$ | $( 1,17)( 2,21, 6,24,13,26, 9,23)( 3,25,11,18,12,22, 4,16)( 5,20, 8,19,10,14, 7,15)$ |
| 12A1 | $12^{2},1^{2}$ | $169$ | $12$ | $22$ | $( 2, 3, 5, 9, 4, 7,13,12,10, 6,11, 8)(14,18,26,16,22,21,19,15,20,17,24,25)$ |
| 12A-1 | $12^{2},1^{2}$ | $169$ | $12$ | $22$ | $( 1, 6, 2,13,12, 5, 8, 3, 7, 9,10, 4)(14,18,20,21,15,25,17,26,24,23,16,19)$ |
| 12A5 | $12^{2},1^{2}$ | $169$ | $12$ | $22$ | $( 1, 5, 3, 4,10, 7, 2,11,13,12, 6, 9)(14,18,16,17,23,20,15,24,26,25,19,22)$ |
| 12A-5 | $12^{2},1^{2}$ | $169$ | $12$ | $22$ | $( 1,10, 5, 2, 8, 9, 7,11, 3, 6,13,12)(14,18,23,26,20,19,21,17,25,22,15,16)$ |
| 12B1 | $12^{2},1^{2}$ | $338$ | $12$ | $22$ | $( 1, 7,10, 5, 9,11,12, 6, 3, 8, 4, 2)(15,20,24,22,23,16,26,21,17,19,18,25)$ |
| 12B-1 | $12^{2},1^{2}$ | $338$ | $12$ | $22$ | $( 1,11, 4, 5, 3, 7,12, 2, 9, 8,10, 6)(15,16,18,22,17,20,26,25,23,19,24,21)$ |
| 13A | $13,1^{13}$ | $24$ | $13$ | $12$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)$ |
| 13B | $13^{2}$ | $48$ | $13$ | $24$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,21,15,22,16,23,17,24,18,25,19,26,20)$ |
| 13C | $13^{2}$ | $48$ | $13$ | $24$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,24,21,18,15,25,22,19,16,26,23,20,17)$ |
| 13D | $13^{2}$ | $48$ | $13$ | $24$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,24,21,18,15,25,22,19,16,26,23,20,17)$ |
| 24A1 | $24,2$ | $338$ | $24$ | $24$ | $( 1,26, 8,18, 7,21, 9,15, 5,14,13,16,10,25, 3,20, 4,17, 2,23, 6,24,11,22)(12,19)$ |
| 24A-1 | $24,2$ | $338$ | $24$ | $24$ | $( 1,24,12,21,11,26, 4,22, 7,20, 2,19, 6,25, 8,15, 9,23, 3,14,13,16, 5,17)(10,18)$ |
| 24A5 | $24,2$ | $338$ | $24$ | $24$ | $( 1,18, 4,20, 9,19,13,26,11,16,12,21, 5,25, 2,23,10,24, 6,17, 8,14, 7,22)( 3,15)$ |
| 24A-5 | $24,2$ | $338$ | $24$ | $24$ | $( 1,25, 6,20,10,16, 8,18, 9,17, 2,24,12,14, 7,19, 3,23, 5,21, 4,22,11,15)(13,26)$ |
| 24B1 | $24,2$ | $338$ | $24$ | $24$ | $( 1,15, 6,23, 2,14,13,16,12,17, 5,24, 8,21, 3,26, 7,22, 9,20,10,19, 4,25)(11,18)$ |
| 24B-1 | $24,2$ | $338$ | $24$ | $24$ | $( 1,23)( 2,20, 3,17, 5,24, 9,25, 4,14, 7,18,13,26,12,16,10,22, 6,21,11,19, 8,15)$ |
| 24B5 | $24,2$ | $338$ | $24$ | $24$ | $( 1,15, 8,16, 9,18,11,22, 2,17,10,20,13,26, 6,25, 5,23, 3,19,12,24, 4,21)( 7,14)$ |
| 24B-5 | $24,2$ | $338$ | $24$ | $24$ | $( 1,14,13,16, 2,25,11,20, 6,17, 3,23, 9,24,10,22, 8,26,12,18, 4,21, 7,15)( 5,19)$ |
| 26A | $13,2^{6},1$ | $312$ | $26$ | $18$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,18)(15,17)(19,26)(20,25)(21,24)(22,23)$ |
Malle's constant $a(G)$: $1/6$
magma: ConjugacyClasses(G);
Character table
35 x 35 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed