Show commands:
Magma
magma: G := TransitiveGroup(34, 8);
Group action invariants
Degree $n$: | $34$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
Transitive number $t$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
Group: | $F_{17}$ | ||
Parity: | $-1$ | magma: IsEven(G);
| |
Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
Generators: | (1,16,20,25,17,6,22,12,14,33,30,23,31,10,27,4)(2,15,19,26,18,5,21,11,13,34,29,24,32,9,28,3)(7,8), (1,31,30,11,20,24,26,9)(2,32,29,12,19,23,25,10)(3,15,22,8,17,5,34,14)(4,16,21,7,18,6,33,13) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $8$: $C_8$ $16$: $C_{16}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: $F_{17}$
Low degree siblings
17T5Siblings 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 $ | $17$ | $8$ | $( 3, 5, 9,17,34,31,27,20)( 4, 6,10,18,33,32,28,19)( 7,13,25,16,29,23,12,21) ( 8,14,26,15,30,24,11,22)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ | $17$ | $4$ | $( 3, 9,34,27)( 4,10,33,28)( 5,17,31,20)( 6,18,32,19)( 7,25,29,12)( 8,26,30,11) (13,16,23,21)(14,15,24,22)$ |
$ 8, 8, 8, 8, 1, 1 $ | $17$ | $8$ | $( 3,17,27, 5,34,20, 9,31)( 4,18,28, 6,33,19,10,32)( 7,16,12,13,29,21,25,23) ( 8,15,11,14,30,22,26,24)$ |
$ 8, 8, 8, 8, 1, 1 $ | $17$ | $8$ | $( 3,20,27,31,34,17, 9, 5)( 4,19,28,32,33,18,10, 6)( 7,21,12,23,29,16,25,13) ( 8,22,11,24,30,15,26,14)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ | $17$ | $4$ | $( 3,27,34, 9)( 4,28,33,10)( 5,20,31,17)( 6,19,32,18)( 7,12,29,25)( 8,11,30,26) (13,21,23,16)(14,22,24,15)$ |
$ 8, 8, 8, 8, 1, 1 $ | $17$ | $8$ | $( 3,31, 9,20,34, 5,27,17)( 4,32,10,19,33, 6,28,18)( 7,23,25,21,29,13,12,16) ( 8,24,26,22,30,14,11,15)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $17$ | $2$ | $( 3,34)( 4,33)( 5,31)( 6,32)( 7,29)( 8,30)( 9,27)(10,28)(11,26)(12,25)(13,23) (14,24)(15,22)(16,21)(17,20)(18,19)$ |
$ 16, 16, 2 $ | $17$ | $16$ | $( 1, 2)( 3, 7,20,21,27,12,31,23,34,29,17,16, 9,25, 5,13)( 4, 8,19,22,28,11,32, 24,33,30,18,15,10,26, 6,14)$ |
$ 16, 16, 2 $ | $17$ | $16$ | $( 1, 2)( 3,12,17,13,27,29, 5,21,34,25,20,23, 9, 7,31,16)( 4,11,18,14,28,30, 6, 22,33,26,19,24,10, 8,32,15)$ |
$ 16, 16, 2 $ | $17$ | $16$ | $( 1, 2)( 3,13, 5,25, 9,16,17,29,34,23,31,12,27,21,20, 7)( 4,14, 6,26,10,15,18, 30,33,24,32,11,28,22,19, 8)$ |
$ 16, 16, 2 $ | $17$ | $16$ | $( 1, 2)( 3,16,31, 7, 9,23,20,25,34,21, 5,29,27,13,17,12)( 4,15,32, 8,10,24,19, 26,33,22, 6,30,28,14,18,11)$ |
$ 16, 16, 2 $ | $17$ | $16$ | $( 1, 2)( 3,21,31,29, 9,13,20,12,34,16, 5, 7,27,23,17,25)( 4,22,32,30,10,14,19, 11,33,15, 6, 8,28,24,18,26)$ |
$ 16, 16, 2 $ | $17$ | $16$ | $( 1, 2)( 3,23, 5,12, 9,21,17, 7,34,13,31,25,27,16,20,29)( 4,24, 6,11,10,22,18, 8,33,14,32,26,28,15,19,30)$ |
$ 16, 16, 2 $ | $17$ | $16$ | $( 1, 2)( 3,25,17,23,27, 7, 5,16,34,12,20,13, 9,29,31,21)( 4,26,18,24,28, 8, 6, 15,33,11,19,14,10,30,32,22)$ |
$ 16, 16, 2 $ | $17$ | $16$ | $( 1, 2)( 3,29,20,16,27,25,31,13,34, 7,17,21, 9,12, 5,23)( 4,30,19,15,28,26,32, 14,33, 8,18,22,10,11, 6,24)$ |
$ 17, 17 $ | $16$ | $17$ | $( 1, 3, 5, 8, 9,11,14,15,17,20,22,24,26,27,30,31,34)( 2, 4, 6, 7,10,12,13,16, 18,19,21,23,25,28,29,32,33)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $272=2^{4} \cdot 17$ | magma: Order(G);
| |
Cyclic: | no | magma: IsCyclic(G);
| |
Abelian: | no | magma: IsAbelian(G);
| |
Solvable: | yes | magma: IsSolvable(G);
| |
Nilpotency class: | not nilpotent | ||
Label: | 272.50 | magma: IdentifyGroup(G);
|
Character table: |
2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 . 17 1 . . . . . . . . . . . . . . . 1 1a 8a 4a 8b 8c 4b 8d 2a 16a 16b 16c 16d 16e 16f 16g 16h 17a 2P 1a 4a 2a 4b 4b 2a 4a 1a 8c 8b 8a 8d 8d 8a 8b 8c 17a 3P 1a 8b 4b 8a 8d 4a 8c 2a 16e 16c 16g 16a 16h 16b 16f 16d 17a 5P 1a 8d 4a 8c 8b 4b 8a 2a 16b 16h 16d 16f 16c 16e 16a 16g 17a 7P 1a 8c 4b 8d 8a 4a 8b 2a 16f 16e 16h 16g 16b 16a 16d 16c 17a 11P 1a 8b 4b 8a 8d 4a 8c 2a 16d 16f 16b 16h 16a 16g 16c 16e 17a 13P 1a 8d 4a 8c 8b 4b 8a 2a 16g 16a 16e 16c 16f 16d 16h 16b 17a 17P 1a 8a 4a 8b 8c 4b 8d 2a 16a 16b 16c 16d 16e 16f 16g 16h 1a X.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 X.3 1 -1 1 -1 -1 1 -1 1 A A -A -A -A -A A A 1 X.4 1 -1 1 -1 -1 1 -1 1 -A -A A A A A -A -A 1 X.5 1 A -1 -A -A -1 A 1 B -B /B -/B -/B /B -B B 1 X.6 1 A -1 -A -A -1 A 1 -B B -/B /B /B -/B B -B 1 X.7 1 -A -1 A A -1 -A 1 -/B /B -B B B -B /B -/B 1 X.8 1 -A -1 A A -1 -A 1 /B -/B B -B -B B -/B /B 1 X.9 1 B -A -/B /B A -B -1 C /D /C D -D -/C -/D -C 1 X.10 1 B -A -/B /B A -B -1 -C -/D -/C -D D /C /D C 1 X.11 1 -/B A B -B -A /B -1 D -/C /D -C C -/D /C -D 1 X.12 1 -/B A B -B -A /B -1 -D /C -/D C -C /D -/C D 1 X.13 1 /B A -B B -A -/B -1 -/C -D -C -/D /D C D /C 1 X.14 1 /B A -B B -A -/B -1 /C D C /D -/D -C -D -/C 1 X.15 1 -B -A /B -/B A B -1 -/D C -D /C -/C D -C /D 1 X.16 1 -B -A /B -/B A B -1 /D -C D -/C /C -D C -/D 1 X.17 16 . . . . . . . . . . . . . . . -1 A = -E(4) = -Sqrt(-1) = -i B = -E(8) C = -E(16)^3 D = -E(16) |
magma: CharacterTable(G);