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Magma
magma: G := TransitiveGroup(37, 8);
Group action invariants
Degree $n$: | $37$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{37}:C_{18}$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | yes | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,4,16,27,34,25,26,30,9,36,33,21,10,3,12,11,7,28)(2,8,32,17,31,13,15,23,18,35,29,5,20,6,24,22,14,19), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $9$: $C_9$ $18$: $C_{18}$ Resolvents shown for degrees $\leq 47$
Subfields
Prime degree - 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
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, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 37 $ | $18$ | $37$ | $( 1, 5, 9,13,17,21,25,29,33,37, 4, 8,12,16,20,24,28,32,36, 3, 7,11,15,19,23, 27,31,35, 2, 6,10,14,18,22,26,30,34)$ |
$ 37 $ | $18$ | $37$ | $( 1, 9,17,25,33, 4,12,20,28,36, 7,15,23,31, 2,10,18,26,34, 5,13,21,29,37, 8, 16,24,32, 3,11,19,27,35, 6,14,22,30)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $37$ | $3$ | $( 2,27,11)( 3,16,21)( 4, 5,31)( 6,20,14)( 7, 9,24)( 8,35,34)(10,13,17) (12,28,37)(15,32,30)(18,36,23)(19,25,33)(22,29,26)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ | $37$ | $3$ | $( 2,11,27)( 3,21,16)( 4,31, 5)( 6,14,20)( 7,24, 9)( 8,34,35)(10,17,13) (12,37,28)(15,30,32)(18,23,36)(19,33,25)(22,26,29)$ |
$ 9, 9, 9, 9, 1 $ | $37$ | $9$ | $( 2,17,35,27,10,34,11,13, 8)( 3,33,32,16,19,30,21,25,15)( 4,12,29, 5,28,26,31, 37,22)( 6, 7,23,20, 9,18,14,24,36)$ |
$ 9, 9, 9, 9, 1 $ | $37$ | $9$ | $( 2,10, 8,27,13,35,11,17,34)( 3,19,15,16,25,32,21,33,30)( 4,28,22, 5,37,29,31, 12,26)( 6, 9,36,20,24,23,14, 7,18)$ |
$ 9, 9, 9, 9, 1 $ | $37$ | $9$ | $( 2,13,34,27,17, 8,11,10,35)( 3,25,30,16,33,15,21,19,32)( 4,37,26, 5,12,22,31, 28,29)( 6,24,18,20, 7,36,14, 9,23)$ |
$ 9, 9, 9, 9, 1 $ | $37$ | $9$ | $( 2,35,10,11, 8,17,27,34,13)( 3,32,19,21,15,33,16,30,25)( 4,29,28,31,22,12, 5, 26,37)( 6,23, 9,14,36, 7,20,18,24)$ |
$ 9, 9, 9, 9, 1 $ | $37$ | $9$ | $( 2,34,17,11,35,13,27, 8,10)( 3,30,33,21,32,25,16,15,19)( 4,26,12,31,29,37, 5, 22,28)( 6,18, 7,14,23,24,20,36, 9)$ |
$ 9, 9, 9, 9, 1 $ | $37$ | $9$ | $( 2, 8,13,11,34,10,27,35,17)( 3,15,25,21,30,19,16,32,33)( 4,22,37,31,26,28, 5, 29,12)( 6,36,24,14,18, 9,20,23, 7)$ |
$ 18, 18, 1 $ | $37$ | $18$ | $( 2, 5,17,28,35,26,27,31,10,37,34,22,11, 4,13,12, 8,29)( 3, 9,33,18,32,14,16, 24,19,36,30, 6,21, 7,25,23,15,20)$ |
$ 18, 18, 1 $ | $37$ | $18$ | $( 2,31,13,28,34,29,27, 4,17,37, 8,26,11, 5,10,12,35,22)( 3,24,25,18,30,20,16, 7,33,36,15,14,21, 9,19,23,32, 6)$ |
$ 18, 18, 1 $ | $37$ | $18$ | $( 2, 4,10,28, 8,22,27, 5,13,37,35,29,11,31,17,12,34,26)( 3, 7,19,18,15, 6,16, 9,25,36,32,20,21,24,33,23,30,14)$ |
$ 6, 6, 6, 6, 6, 6, 1 $ | $37$ | $6$ | $( 2,28,27,37,11,12)( 3,18,16,36,21,23)( 4, 8, 5,35,31,34)( 6,25,20,33,14,19) ( 7,15, 9,32,24,30)(10,22,13,29,17,26)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $37$ | $2$ | $( 2,37)( 3,36)( 4,35)( 5,34)( 6,33)( 7,32)( 8,31)( 9,30)(10,29)(11,28)(12,27) (13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)$ |
$ 6, 6, 6, 6, 6, 6, 1 $ | $37$ | $6$ | $( 2,12,11,37,27,28)( 3,23,21,36,16,18)( 4,34,31,35, 5, 8)( 6,19,14,33,20,25) ( 7,30,24,32, 9,15)(10,26,17,29,13,22)$ |
$ 18, 18, 1 $ | $37$ | $18$ | $( 2,26,34,12,17,31,11,29,35,37,13, 5,27,22, 8,28,10, 4)( 3,14,30,23,33,24,21, 20,32,36,25, 9,16, 6,15,18,19, 7)$ |
$ 18, 18, 1 $ | $37$ | $18$ | $( 2,22,35,12,10, 5,11,26, 8,37,17, 4,27,29,34,28,13,31)( 3, 6,32,23,19, 9,21, 14,15,36,33, 7,16,20,30,18,25,24)$ |
$ 18, 18, 1 $ | $37$ | $18$ | $( 2,29, 8,12,13, 4,11,22,34,37,10,31,27,26,35,28,17, 5)( 3,20,15,23,25, 7,21, 6,30,36,19,24,16,14,32,18,33, 9)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $666=2 \cdot 3^{2} \cdot 37$ | 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: | 666.7 | magma: IdentifyGroup(G);
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Character table: |
2 1 . . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 2 . . 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 37 1 1 1 . . . . . . . . . . . . . . . . . 1a 37a 37b 3a 3b 9a 9b 9c 9d 9e 9f 18a 18b 18c 6a 2a 6b 18d 18e 18f 2P 1a 37b 37a 3b 3a 9d 9f 9e 9b 9a 9c 9a 9c 9b 3a 1a 3b 9e 9d 9f 3P 1a 37a 37b 1a 1a 3a 3a 3a 3b 3b 3b 6a 6a 6a 2a 2a 2a 6b 6b 6b 5P 1a 37b 37a 3b 3a 9e 9d 9f 9a 9c 9b 18d 18f 18e 6b 2a 6a 18b 18a 18c 7P 1a 37a 37b 3a 3b 9c 9a 9b 9e 9f 9d 18b 18c 18a 6a 2a 6b 18f 18d 18e 11P 1a 37a 37b 3b 3a 9d 9f 9e 9b 9a 9c 18e 18d 18f 6b 2a 6a 18a 18c 18b 13P 1a 37b 37a 3a 3b 9b 9c 9a 9f 9d 9e 18c 18a 18b 6a 2a 6b 18e 18f 18d 17P 1a 37b 37a 3b 3a 9f 9e 9d 9c 9b 9a 18f 18e 18d 6b 2a 6a 18c 18b 18a 19P 1a 37b 37a 3a 3b 9a 9b 9c 9d 9e 9f 18a 18b 18c 6a 2a 6b 18d 18e 18f 23P 1a 37b 37a 3b 3a 9e 9d 9f 9a 9c 9b 18d 18f 18e 6b 2a 6a 18b 18a 18c 29P 1a 37b 37a 3b 3a 9d 9f 9e 9b 9a 9c 18e 18d 18f 6b 2a 6a 18a 18c 18b 31P 1a 37b 37a 3a 3b 9b 9c 9a 9f 9d 9e 18c 18a 18b 6a 2a 6b 18e 18f 18d 37P 1a 1a 1a 3a 3b 9a 9b 9c 9d 9e 9f 18a 18b 18c 6a 2a 6b 18d 18e 18f X.1 1 1 1 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 -1 -1 -1 -1 X.3 1 1 1 1 1 B B B /B /B /B -/B -/B -/B -1 -1 -1 -B -B -B X.4 1 1 1 1 1 /B /B /B B B B -B -B -B -1 -1 -1 -/B -/B -/B X.5 1 1 1 1 1 B B B /B /B /B /B /B /B 1 1 1 B B B X.6 1 1 1 1 1 /B /B /B B B B B B B 1 1 1 /B /B /B X.7 1 1 1 B /B C D E /E /D /C -/D -/C -/E -/B -1 -B -E -C -D X.8 1 1 1 B /B D E C /C /E /D -/E -/D -/C -/B -1 -B -C -D -E X.9 1 1 1 B /B E C D /D /C /E -/C -/E -/D -/B -1 -B -D -E -C X.10 1 1 1 /B B /C /D /E E D C -D -C -E -B -1 -/B -/E -/C -/D X.11 1 1 1 /B B /E /C /D D C E -C -E -D -B -1 -/B -/D -/E -/C X.12 1 1 1 /B B /D /E /C C E D -E -D -C -B -1 -/B -/C -/D -/E X.13 1 1 1 B /B C D E /E /D /C /D /C /E /B 1 B E C D X.14 1 1 1 B /B D E C /C /E /D /E /D /C /B 1 B C D E X.15 1 1 1 B /B E C D /D /C /E /C /E /D /B 1 B D E C X.16 1 1 1 /B B /C /D /E E D C D C E B 1 /B /E /C /D X.17 1 1 1 /B B /E /C /D D C E C E D B 1 /B /D /E /C X.18 1 1 1 /B B /D /E /C C E D E D C B 1 /B /C /D /E X.19 18 A *A . . . . . . . . . . . . . . . . . X.20 18 *A A . . . . . . . . . . . . . . . . . A = E(37)+E(37)^3+E(37)^4+E(37)^7+E(37)^9+E(37)^10+E(37)^11+E(37)^12+E(37)^16+E(37)^21+E(37)^25+E(37)^26+E(37)^27+E(37)^28+E(37)^30+E(37)^33+E(37)^34+E(37)^36 = (-1+Sqrt(37))/2 = b37 B = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 C = -E(9)^2-E(9)^5 D = E(9)^5 E = E(9)^2 |
magma: CharacterTable(G);