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Magma
magma: G := TransitiveGroup(26, 5);
Group action invariants
Degree $n$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $5$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{13}:C_6$ | ||
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: | (1,10,4)(2,9,3)(5,19,16)(6,20,15)(7,12,22)(8,11,21)(17,24,26)(18,23,25), (1,2)(3,7,19,4,8,20)(5,13,11,6,14,12)(9,25,21,10,26,22)(15,17,23,16,18,24) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $39$: $C_{13}:C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: $C_{13}:C_3$
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
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 $ | $1$ | $1$ | $()$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $13$ | $3$ | $( 3, 8,19)( 4, 7,20)( 5,14,11)( 6,13,12)( 9,26,21)(10,25,22)(15,18,23) (16,17,24)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $13$ | $3$ | $( 3,19, 8)( 4,20, 7)( 5,11,14)( 6,12,13)( 9,21,26)(10,22,25)(15,23,18) (16,24,17)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)$ |
$ 6, 6, 6, 6, 2 $ | $13$ | $6$ | $( 1, 2)( 3, 7,19, 4, 8,20)( 5,13,11, 6,14,12)( 9,25,21,10,26,22) (15,17,23,16,18,24)$ |
$ 6, 6, 6, 6, 2 $ | $13$ | $6$ | $( 1, 2)( 3,20, 8, 4,19, 7)( 5,12,14, 6,11,13)( 9,22,26,10,21,25) (15,24,18,16,23,17)$ |
$ 26 $ | $3$ | $26$ | $( 1, 3, 6, 8,10,11,13,16,18,19,22,24,25, 2, 4, 5, 7, 9,12,14,15,17,20,21,23,26 )$ |
$ 13, 13 $ | $3$ | $13$ | $( 1, 4, 6, 7,10,12,13,15,18,20,22,23,25)( 2, 3, 5, 8, 9,11,14,16,17,19,21,24, 26)$ |
$ 26 $ | $3$ | $26$ | $( 1, 5,10,14,18,21,25, 3, 7,11,15,19,23, 2, 6, 9,13,17,22,26, 4, 8,12,16,20,24 )$ |
$ 13, 13 $ | $3$ | $13$ | $( 1, 6,10,13,18,22,25, 4, 7,12,15,20,23)( 2, 5, 9,14,17,21,26, 3, 8,11,16,19, 24)$ |
$ 26 $ | $3$ | $26$ | $( 1, 9,18,26, 7,16,23, 5,13,21, 4,11,20, 2,10,17,25, 8,15,24, 6,14,22, 3,12,19 )$ |
$ 13, 13 $ | $3$ | $13$ | $( 1,10,18,25, 7,15,23, 6,13,22, 4,12,20)( 2, 9,17,26, 8,16,24, 5,14,21, 3,11, 19)$ |
$ 13, 13 $ | $3$ | $13$ | $( 1,15, 4,18, 6,20, 7,22,10,23,12,25,13)( 2,16, 3,17, 5,19, 8,21, 9,24,11,26, 14)$ |
$ 26 $ | $3$ | $26$ | $( 1,16, 4,17, 6,19, 7,21,10,24,12,26,13, 2,15, 3,18, 5,20, 8,22, 9,23,11,25,14 )$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $78=2 \cdot 3 \cdot 13$ | 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: | 78.2 | magma: IdentifyGroup(G);
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Character table: |
2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 . . . . . . . . 13 1 . . 1 . . 1 1 1 1 1 1 1 1 1a 3a 3b 2a 6a 6b 26a 13a 26b 13b 26c 13c 13d 26d 2P 1a 3b 3a 1a 3b 3a 13b 13b 13c 13c 13d 13d 13a 13a 3P 1a 1a 1a 2a 2a 2a 26a 13a 26b 13b 26c 13c 13d 26d 5P 1a 3b 3a 2a 6b 6a 26b 13b 26c 13c 26d 13d 13a 26a 7P 1a 3a 3b 2a 6a 6b 26d 13d 26a 13a 26b 13b 13c 26c 11P 1a 3b 3a 2a 6b 6a 26d 13d 26a 13a 26b 13b 13c 26c 13P 1a 3a 3b 2a 6a 6b 2a 1a 2a 1a 2a 1a 1a 2a 17P 1a 3b 3a 2a 6b 6a 26c 13c 26d 13d 26a 13a 13b 26b 19P 1a 3a 3b 2a 6a 6b 26b 13b 26c 13c 26d 13d 13a 26a 23P 1a 3b 3a 2a 6b 6a 26c 13c 26d 13d 26a 13a 13b 26b X.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 X.3 1 A /A -1 -A -/A -1 1 -1 1 -1 1 1 -1 X.4 1 /A A -1 -/A -A -1 1 -1 1 -1 1 1 -1 X.5 1 A /A 1 A /A 1 1 1 1 1 1 1 1 X.6 1 /A A 1 /A A 1 1 1 1 1 1 1 1 X.7 3 . . 3 . . B B /C /C /B /B C C X.8 3 . . 3 . . C C B B /C /C /B /B X.9 3 . . 3 . . /C /C /B /B C C B B X.10 3 . . 3 . . /B /B C C B B /C /C X.11 3 . . -3 . . -B B -/C /C -/B /B C -C X.12 3 . . -3 . . -C C -B B -/C /C /B -/B X.13 3 . . -3 . . -/C /C -/B /B -C C B -B X.14 3 . . -3 . . -/B /B -C C -B B /C -/C A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(13)^7+E(13)^8+E(13)^11 C = E(13)^4+E(13)^10+E(13)^12 |
magma: CharacterTable(G);