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Magma
magma: G := TransitiveGroup(34, 7);
Group action invariants
Degree $n$: | $34$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $7$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{34}:C_8$ | ||
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: | (1,34,17,25,21,24,5,31)(2,33,18,26,22,23,6,32)(3,16,10,29,20,8,13,27)(4,15,9,30,19,7,14,28), (1,7,29,19,5,33,11,22)(2,8,30,20,6,34,12,21)(3,4)(9,25,28,24,32,16,14,17)(10,26,27,23,31,15,13,18) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $C_8$ x 2, $C_4\times C_2$ $16$: $C_8\times C_2$ $136$: $C_{17}:C_{8}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: $C_{17}:C_{8}$
Low degree siblings
34T7Siblings 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, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 8, 8, 8, 8, 1, 1 $ | $17$ | $8$ | $( 3, 5,10,17,34,31,27,20)( 4, 6, 9,18,33,32,28,19)( 7,14,26,15,30,23,12,22) ( 8,13,25,16,29,24,11,21)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ | $17$ | $4$ | $( 3,10,34,27)( 4, 9,33,28)( 5,17,31,20)( 6,18,32,19)( 7,26,30,12)( 8,25,29,11) (13,16,24,21)(14,15,23,22)$ |
$ 8, 8, 8, 8, 1, 1 $ | $17$ | $8$ | $( 3,17,27, 5,34,20,10,31)( 4,18,28, 6,33,19, 9,32)( 7,15,12,14,30,22,26,23) ( 8,16,11,13,29,21,25,24)$ |
$ 8, 8, 8, 8, 1, 1 $ | $17$ | $8$ | $( 3,20,27,31,34,17,10, 5)( 4,19,28,32,33,18, 9, 6)( 7,22,12,23,30,15,26,14) ( 8,21,11,24,29,16,25,13)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ | $17$ | $4$ | $( 3,27,34,10)( 4,28,33, 9)( 5,20,31,17)( 6,19,32,18)( 7,12,30,26)( 8,11,29,25) (13,21,24,16)(14,22,23,15)$ |
$ 8, 8, 8, 8, 1, 1 $ | $17$ | $8$ | $( 3,31,10,20,34, 5,27,17)( 4,32, 9,19,33, 6,28,18)( 7,23,26,22,30,14,12,15) ( 8,24,25,21,29,13,11,16)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $17$ | $2$ | $( 3,34)( 4,33)( 5,31)( 6,32)( 7,30)( 8,29)( 9,28)(10,27)(11,25)(12,26)(13,24) (14,23)(15,22)(16,21)(17,20)(18,19)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)$ |
$ 8, 8, 8, 8, 2 $ | $17$ | $8$ | $( 1, 2)( 3, 6,10,18,34,32,27,19)( 4, 5, 9,17,33,31,28,20)( 7,13,26,16,30,24, 12,21)( 8,14,25,15,29,23,11,22)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $17$ | $4$ | $( 1, 2)( 3, 9,34,28)( 4,10,33,27)( 5,18,31,19)( 6,17,32,20)( 7,25,30,11) ( 8,26,29,12)(13,15,24,22)(14,16,23,21)$ |
$ 8, 8, 8, 8, 2 $ | $17$ | $8$ | $( 1, 2)( 3,18,27, 6,34,19,10,32)( 4,17,28, 5,33,20, 9,31)( 7,16,12,13,30,21, 26,24)( 8,15,11,14,29,22,25,23)$ |
$ 8, 8, 8, 8, 2 $ | $17$ | $8$ | $( 1, 2)( 3,19,27,32,34,18,10, 6)( 4,20,28,31,33,17, 9, 5)( 7,21,12,24,30,16, 26,13)( 8,22,11,23,29,15,25,14)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $17$ | $4$ | $( 1, 2)( 3,28,34, 9)( 4,27,33,10)( 5,19,31,18)( 6,20,32,17)( 7,11,30,25) ( 8,12,29,26)(13,22,24,15)(14,21,23,16)$ |
$ 8, 8, 8, 8, 2 $ | $17$ | $8$ | $( 1, 2)( 3,32,10,19,34, 6,27,18)( 4,31, 9,20,33, 5,28,17)( 7,24,26,21,30,13, 12,16)( 8,23,25,22,29,14,11,15)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $17$ | $2$ | $( 1, 2)( 3,33)( 4,34)( 5,32)( 6,31)( 7,29)( 8,30)( 9,27)(10,28)(11,26)(12,25) (13,23)(14,24)(15,21)(16,22)(17,19)(18,20)$ |
$ 17, 17 $ | $8$ | $17$ | $( 1, 3, 5, 8,10,11,13,16,17,20,21,24,25,27,29,31,34)( 2, 4, 6, 7, 9,12,14,15, 18,19,22,23,26,28,30,32,33)$ |
$ 34 $ | $8$ | $34$ | $( 1, 4, 5, 7,10,12,13,15,17,19,21,23,25,28,29,32,34, 2, 3, 6, 8, 9,11,14,16, 18,20,22,24,26,27,30,31,33)$ |
$ 34 $ | $8$ | $34$ | $( 1, 7,13,19,25,32, 3, 9,16,22,27,33, 5,12,17,23,29, 2, 8,14,20,26,31, 4,10, 15,21,28,34, 6,11,18,24,30)$ |
$ 17, 17 $ | $8$ | $17$ | $( 1, 8,13,20,25,31, 3,10,16,21,27,34, 5,11,17,24,29)( 2, 7,14,19,26,32, 4, 9, 15,22,28,33, 6,12,18,23,30)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $272=2^{4} \cdot 17$ | 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: | 272.51 | magma: IdentifyGroup(G);
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Character table: |
2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 1 1 1 17 1 . . . . . . . 1 . . . . . . . 1 1 1 1a 8a 4a 8b 8c 4b 8d 2a 2b 8e 4c 8f 8g 4d 8h 2c 17a 34a 34b 2P 1a 4a 2a 4b 4b 2a 4a 1a 1a 4a 2a 4b 4b 2a 4a 1a 17a 17a 17b 3P 1a 8b 4b 8a 8d 4a 8c 2a 2b 8f 4d 8e 8h 4c 8g 2c 17b 34b 34a 5P 1a 8d 4a 8c 8b 4b 8a 2a 2b 8h 4c 8g 8f 4d 8e 2c 17b 34b 34a 7P 1a 8c 4b 8d 8a 4a 8b 2a 2b 8g 4d 8h 8e 4c 8f 2c 17b 34b 34a 11P 1a 8b 4b 8a 8d 4a 8c 2a 2b 8f 4d 8e 8h 4c 8g 2c 17b 34b 34a 13P 1a 8d 4a 8c 8b 4b 8a 2a 2b 8h 4c 8g 8f 4d 8e 2c 17a 34a 34b 17P 1a 8a 4a 8b 8c 4b 8d 2a 2b 8e 4c 8f 8g 4d 8h 2c 1a 2b 2b 19P 1a 8b 4b 8a 8d 4a 8c 2a 2b 8f 4d 8e 8h 4c 8g 2c 17a 34a 34b 23P 1a 8c 4b 8d 8a 4a 8b 2a 2b 8g 4d 8h 8e 4c 8f 2c 17b 34b 34a 29P 1a 8d 4a 8c 8b 4b 8a 2a 2b 8h 4c 8g 8f 4d 8e 2c 17b 34b 34a 31P 1a 8c 4b 8d 8a 4a 8b 2a 2b 8g 4d 8h 8e 4c 8f 2c 17b 34b 34a X.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 X.3 1 -1 1 -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 1 -1 -1 X.5 1 A -1 -A -A -1 A 1 -1 -A 1 A A 1 -A -1 1 -1 -1 X.6 1 -A -1 A A -1 -A 1 -1 A 1 -A -A 1 A -1 1 -1 -1 X.7 1 A -1 -A -A -1 A 1 1 A -1 -A -A -1 A 1 1 1 1 X.8 1 -A -1 A A -1 -A 1 1 -A -1 A A -1 -A 1 1 1 1 X.9 1 B -A -/B /B A -B -1 -1 -B A /B -/B -A B 1 1 -1 -1 X.10 1 -/B A B -B -A /B -1 -1 /B -A -B B A -/B 1 1 -1 -1 X.11 1 /B A -B B -A -/B -1 -1 -/B -A B -B A /B 1 1 -1 -1 X.12 1 -B -A /B -/B A B -1 -1 B A -/B /B -A -B 1 1 -1 -1 X.13 1 B -A -/B /B A -B -1 1 B -A -/B /B A -B -1 1 1 1 X.14 1 -/B A B -B -A /B -1 1 -/B A B -B -A /B -1 1 1 1 X.15 1 /B A -B B -A -/B -1 1 /B A -B B -A -/B -1 1 1 1 X.16 1 -B -A /B -/B A B -1 1 -B -A /B -/B A B -1 1 1 1 X.17 8 . . . . . . . -8 . . . . . . . C -C -*C X.18 8 . . . . . . . -8 . . . . . . . *C -*C -C X.19 8 . . . . . . . 8 . . . . . . . C C *C X.20 8 . . . . . . . 8 . . . . . . . *C *C C 2 1 17 1 17b 2P 17b 3P 17a 5P 17a 7P 17a 11P 17a 13P 17b 17P 1a 19P 17b 23P 17a 29P 17a 31P 17a X.1 1 X.2 1 X.3 1 X.4 1 X.5 1 X.6 1 X.7 1 X.8 1 X.9 1 X.10 1 X.11 1 X.12 1 X.13 1 X.14 1 X.15 1 X.16 1 X.17 *C X.18 C X.19 *C X.20 C A = -E(4) = -Sqrt(-1) = -i B = -E(8) C = E(17)^3+E(17)^5+E(17)^6+E(17)^7+E(17)^10+E(17)^11+E(17)^12+E(17)^14 = (-1-Sqrt(17))/2 = -1-b17 |
magma: CharacterTable(G);