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Magma
magma: G := TransitiveGroup(17, 3);
Group action invariants
Degree $n$: | $17$ | 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: | $C_{17}:C_{4}$ | ||
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: | (2,14,17,5)(3,10,16,9)(4,6,15,13)(7,11,12,8), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ 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 $ | $1$ | $1$ | $()$ |
$ 4, 4, 4, 4, 1 $ | $17$ | $4$ | $( 2, 5,17,14)( 3, 9,16,10)( 4,13,15, 6)( 7, 8,12,11)$ |
$ 4, 4, 4, 4, 1 $ | $17$ | $4$ | $( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $17$ | $2$ | $( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)$ |
$ 17 $ | $4$ | $17$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)$ |
$ 17 $ | $4$ | $17$ | $( 1, 3, 5, 7, 9,11,13,15,17, 2, 4, 6, 8,10,12,14,16)$ |
$ 17 $ | $4$ | $17$ | $( 1, 4, 7,10,13,16, 2, 5, 8,11,14,17, 3, 6, 9,12,15)$ |
$ 17 $ | $4$ | $17$ | $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $68=2^{2} \cdot 17$ | 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: | 68.3 | magma: IdentifyGroup(G);
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Character table: |
2 2 2 2 2 . . . . 17 1 . . . 1 1 1 1 1a 4a 4b 2a 17a 17b 17c 17d 2P 1a 2a 2a 1a 17b 17a 17d 17c 3P 1a 4b 4a 2a 17c 17d 17b 17a 5P 1a 4a 4b 2a 17c 17d 17b 17a 7P 1a 4b 4a 2a 17d 17c 17a 17b 11P 1a 4b 4a 2a 17d 17c 17a 17b 13P 1a 4a 4b 2a 17a 17b 17c 17d 17P 1a 4a 4b 2a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 1 1 1 X.3 1 A -A -1 1 1 1 1 X.4 1 -A A -1 1 1 1 1 X.5 4 . . . B E C D X.6 4 . . . C D E B X.7 4 . . . D C B E X.8 4 . . . E B D C A = -E(4) = -Sqrt(-1) = -i B = E(17)^3+E(17)^5+E(17)^12+E(17)^14 C = E(17)^2+E(17)^8+E(17)^9+E(17)^15 D = E(17)+E(17)^4+E(17)^13+E(17)^16 E = E(17)^6+E(17)^7+E(17)^10+E(17)^11 |
magma: CharacterTable(G);