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Magma
magma: G := TransitiveGroup(26, 4);
Group action invariants
Degree $n$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $4$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{13}:C_4$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,3,5,7,9,12,14,16,18,20,22,24,25)(2,4,6,8,10,11,13,15,17,19,21,23,26), (1,10,24,15)(2,9,23,16)(3,19,22,6)(4,20,21,5)(7,13,18,11)(8,14,17,12)(25,26) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: $C_{13}:C_4$
Low degree siblings
13T4Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Cycle Type | Size | Order | Representative |
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $13$ | $2$ | $( 3,25)( 4,26)( 5,24)( 6,23)( 7,22)( 8,21)( 9,20)(10,19)(11,17)(12,18)(13,15) (14,16)$ |
$ 4, 4, 4, 4, 4, 4, 2 $ | $13$ | $4$ | $( 1, 2)( 3,11,25,17)( 4,12,26,18)( 5,21,24, 8)( 6,22,23, 7)( 9,15,20,13) (10,16,19,14)$ |
$ 4, 4, 4, 4, 4, 4, 2 $ | $13$ | $4$ | $( 1, 2)( 3,17,25,11)( 4,18,26,12)( 5, 8,24,21)( 6, 7,23,22)( 9,13,20,15) (10,14,19,16)$ |
$ 13, 13 $ | $4$ | $13$ | $( 1, 3, 5, 7, 9,12,14,16,18,20,22,24,25)( 2, 4, 6, 8,10,11,13,15,17,19,21,23, 26)$ |
$ 13, 13 $ | $4$ | $13$ | $( 1, 5, 9,14,18,22,25, 3, 7,12,16,20,24)( 2, 6,10,13,17,21,26, 4, 8,11,15,19, 23)$ |
$ 13, 13 $ | $4$ | $13$ | $( 1, 9,18,25, 7,16,24, 5,14,22, 3,12,20)( 2,10,17,26, 8,15,23, 6,13,21, 4,11, 19)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $52=2^{2} \cdot 13$ | 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: | 52.3 | magma: IdentifyGroup(G);
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Character table: |
2 2 2 2 2 . . . 13 1 . . . 1 1 1 1a 2a 4a 4b 13a 13b 13c 2P 1a 1a 2a 2a 13b 13c 13a 3P 1a 2a 4b 4a 13b 13c 13a 5P 1a 2a 4a 4b 13a 13b 13c 7P 1a 2a 4b 4a 13c 13a 13b 11P 1a 2a 4b 4a 13b 13c 13a 13P 1a 2a 4a 4b 1a 1a 1a X.1 1 1 1 1 1 1 1 X.2 1 1 -1 -1 1 1 1 X.3 1 -1 A -A 1 1 1 X.4 1 -1 -A A 1 1 1 X.5 4 . . . B C D X.6 4 . . . C D B X.7 4 . . . D B C A = -E(4) = -Sqrt(-1) = -i B = E(13)^2+E(13)^3+E(13)^10+E(13)^11 C = E(13)^4+E(13)^6+E(13)^7+E(13)^9 D = E(13)+E(13)^5+E(13)^8+E(13)^12 |
magma: CharacterTable(G);