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Magma
magma: G := TransitiveGroup(26, 3);
Group action invariants
Degree $n$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $3$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $D_{26}$ | ||
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: | (1,22,15,10,3,23,18,12,5,26,20,13,7,2,21,16,9,4,24,17,11,6,25,19,14,8), (1,21)(2,22)(3,20)(4,19)(5,18)(6,17)(7,15)(8,16)(9,14)(10,13)(23,26)(24,25) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $26$: $D_{13}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: $D_{13}$
Low degree siblings
26T3Siblings 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 $ | $13$ | $2$ | $( 3,25)( 4,26)( 5,24)( 6,23)( 7,21)( 8,22)( 9,20)(10,19)(11,18)(12,17)(13,16) (14,15)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $13$ | $2$ | $( 1, 2)( 3,26)( 4,25)( 5,23)( 6,24)( 7,22)( 8,21)( 9,19)(10,20)(11,17)(12,18) (13,15)(14,16)$ |
$ 13, 13 $ | $2$ | $13$ | $( 1, 3, 5, 7, 9,11,14,15,18,20,21,24,25)( 2, 4, 6, 8,10,12,13,16,17,19,22,23, 26)$ |
$ 26 $ | $2$ | $26$ | $( 1, 4, 5, 8, 9,12,14,16,18,19,21,23,25, 2, 3, 6, 7,10,11,13,15,17,20,22,24,26 )$ |
$ 13, 13 $ | $2$ | $13$ | $( 1, 5, 9,14,18,21,25, 3, 7,11,15,20,24)( 2, 6,10,13,17,22,26, 4, 8,12,16,19, 23)$ |
$ 26 $ | $2$ | $26$ | $( 1, 6, 9,13,18,22,25, 4, 7,12,15,19,24, 2, 5,10,14,17,21,26, 3, 8,11,16,20,23 )$ |
$ 13, 13 $ | $2$ | $13$ | $( 1, 7,14,20,25, 5,11,18,24, 3, 9,15,21)( 2, 8,13,19,26, 6,12,17,23, 4,10,16, 22)$ |
$ 26 $ | $2$ | $26$ | $( 1, 8,14,19,25, 6,11,17,24, 4, 9,16,21, 2, 7,13,20,26, 5,12,18,23, 3,10,15,22 )$ |
$ 13, 13 $ | $2$ | $13$ | $( 1, 9,18,25, 7,15,24, 5,14,21, 3,11,20)( 2,10,17,26, 8,16,23, 6,13,22, 4,12, 19)$ |
$ 26 $ | $2$ | $26$ | $( 1,10,18,26, 7,16,24, 6,14,22, 3,12,20, 2, 9,17,25, 8,15,23, 5,13,21, 4,11,19 )$ |
$ 13, 13 $ | $2$ | $13$ | $( 1,11,21, 5,15,25, 9,20, 3,14,24, 7,18)( 2,12,22, 6,16,26,10,19, 4,13,23, 8, 17)$ |
$ 26 $ | $2$ | $26$ | $( 1,12,21, 6,15,26, 9,19, 3,13,24, 8,18, 2,11,22, 5,16,25,10,20, 4,14,23, 7,17 )$ |
$ 26 $ | $2$ | $26$ | $( 1,13,25,12,24,10,21, 8,20, 6,18, 4,15, 2,14,26,11,23, 9,22, 7,19, 5,17, 3,16 )$ |
$ 13, 13 $ | $2$ | $13$ | $( 1,14,25,11,24, 9,21, 7,20, 5,18, 3,15)( 2,13,26,12,23,10,22, 8,19, 6,17, 4, 16)$ |
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: | 52.4 | magma: IdentifyGroup(G);
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Character table: |
2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 13 1 . 1 . 1 1 1 1 1 1 1 1 1 1 1 1 1a 2a 2b 2c 13a 26a 13b 26b 13c 26c 13d 26d 13e 26e 26f 13f 2P 1a 1a 1a 1a 13b 13b 13d 13d 13f 13f 13e 13e 13c 13c 13a 13a 3P 1a 2a 2b 2c 13c 26c 13f 26f 13d 26d 13a 26a 13b 26b 26e 13e 5P 1a 2a 2b 2c 13e 26e 13c 26c 13b 26b 13f 26f 13a 26a 26d 13d 7P 1a 2a 2b 2c 13f 26f 13a 26a 13e 26e 13b 26b 13d 26d 26c 13c 11P 1a 2a 2b 2c 13b 26b 13d 26d 13f 26f 13e 26e 13c 26c 26a 13a 13P 1a 2a 2b 2c 1a 2b 1a 2b 1a 2b 1a 2b 1a 2b 2b 1a 17P 1a 2a 2b 2c 13d 26d 13e 26e 13a 26a 13c 26c 13f 26f 26b 13b 19P 1a 2a 2b 2c 13f 26f 13a 26a 13e 26e 13b 26b 13d 26d 26c 13c 23P 1a 2a 2b 2c 13c 26c 13f 26f 13d 26d 13a 26a 13b 26b 26e 13e X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 X.3 1 -1 1 -1 1 1 1 1 1 1 1 1 1 1 1 1 X.4 1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 X.5 2 . 2 . A A C C B B D D F F E E X.6 2 . 2 . B B E E D D A A C C F F X.7 2 . 2 . C C D D E E F F B B A A X.8 2 . 2 . D D F F A A B B E E C C X.9 2 . 2 . E E A A F F C C D D B B X.10 2 . 2 . F F B B C C E E A A D D X.11 2 . -2 . A -A C -C B -B D -D F -F -E E X.12 2 . -2 . B -B E -E D -D A -A C -C -F F X.13 2 . -2 . C -C D -D E -E F -F B -B -A A X.14 2 . -2 . D -D F -F A -A B -B E -E -C C X.15 2 . -2 . E -E A -A F -F C -C D -D -B B X.16 2 . -2 . F -F B -B C -C E -E A -A -D D A = E(13)^3+E(13)^10 B = E(13)^4+E(13)^9 C = E(13)^6+E(13)^7 D = E(13)+E(13)^12 E = E(13)^5+E(13)^8 F = E(13)^2+E(13)^11 |
magma: CharacterTable(G);