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Magma
magma: G := TransitiveGroup(34, 28);
Group action invariants
Degree $n$: | $34$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $28$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $\SOMinus(4,4)$ | ||
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: | (1,5,29,2,6,30)(3,23,33,15,20,9)(4,24,34,16,19,10)(7,21,25,13,32,17)(8,22,26,14,31,18)(11,12)(27,28), (1,25,18,32,7,16,29,14,10,6,3,33,22,20,24,28,12)(2,26,17,31,8,15,30,13,9,5,4,34,21,19,23,27,11) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: $\PSL(2,16):C_2$
Low degree siblings
17T7Siblings 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 $ | $1$ | $1$ | $()$ |
$ 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ | $272$ | $5$ | $( 1,25,29,16,32)( 2,26,30,15,31)( 3,10,14,22,20)( 4, 9,13,21,19) ( 5,34,17, 8,27)( 6,33,18, 7,28)$ |
$ 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ | $272$ | $5$ | $( 1,16,25,32,29)( 2,15,26,31,30)( 3,22,10,20,14)( 4,21, 9,19,13) ( 5, 8,34,27,17)( 6, 7,33,28,18)$ |
$ 17, 17 $ | $480$ | $17$ | $( 1, 3,18, 7,33,12,22,32, 6,10,24,16,25,29,20,28,14)( 2, 4,17, 8,34,11,21,31, 5, 9,23,15,26,30,19,27,13)$ |
$ 17, 17 $ | $480$ | $17$ | $( 1,18,33,22, 6,24,25,20,14, 3, 7,12,32,10,16,29,28)( 2,17,34,21, 5,23,26,19, 13, 4, 8,11,31, 9,15,30,27)$ |
$ 17, 17 $ | $480$ | $17$ | $( 1,22,25, 3,32,29,18, 6,20, 7,10,28,33,24,14,12,16)( 2,21,26, 4,31,30,17, 5, 19, 8, 9,27,34,23,13,11,15)$ |
$ 17, 17 $ | $480$ | $17$ | $( 1,25,32,18,20,10,33,14,16,22, 3,29, 6, 7,28,24,12)( 2,26,31,17,19, 9,34,13, 15,21, 4,30, 5, 8,27,23,11)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $68$ | $2$ | $( 1, 2)( 3, 8)( 4, 7)( 5,14)( 6,13)( 9,28)(10,27)(11,24)(12,23)(15,16)(17,20) (18,19)(21,33)(22,34)(25,26)(29,30)(31,32)$ |
$ 10, 10, 10, 2, 2 $ | $816$ | $10$ | $( 1,26,29,15,32, 2,25,30,16,31)( 3,27,14,34,20, 8,10, 5,22,17)( 4,28,13,33,19, 7, 9, 6,21,18)(11,24)(12,23)$ |
$ 10, 10, 10, 2, 2 $ | $816$ | $10$ | $( 1,30,32,26,16, 2,29,31,25,15)( 3, 5,20,27,22, 8,14,17,10,34)( 4, 6,19,28,21, 7,13,18, 9,33)(11,24)(12,23)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ | $272$ | $3$ | $( 1,33,22)( 2,34,21)( 3,29, 7)( 4,30, 8)( 5,13,31)( 6,14,32)( 9,15,27) (10,16,28)(17,19,26)(18,20,25)$ |
$ 15, 15, 1, 1, 1, 1 $ | $544$ | $15$ | $( 1,28, 7,20,12,10,24,22,32,16,18,29, 3,14, 6)( 2,27, 8,19,11, 9,23,21,31,15, 17,30, 4,13, 5)$ |
$ 15, 15, 1, 1, 1, 1 $ | $544$ | $15$ | $( 1,22, 6,24,14,10, 3,12,29,20,18, 7,16,28,32)( 2,21, 5,23,13, 9, 4,11,30,19, 17, 8,15,27,31)$ |
$ 6, 6, 6, 6, 6, 2, 2 $ | $1360$ | $6$ | $( 1,31,18,21, 7, 4)( 2,32,17,22, 8, 3)( 5,24,26,10,34,28)( 6,23,25, 9,33,27) (11,12)(13,20,30,14,19,29)(15,16)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $255$ | $2$ | $( 1,20)( 2,19)( 3,12)( 4,11)( 5,30)( 6,29)( 7,18)( 8,17)( 9,23)(10,24)(13,34) (14,33)(15,31)(16,32)(25,28)(26,27)$ |
$ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $1020$ | $4$ | $( 1,23,20, 9)( 2,24,19,10)( 3, 5,12,30)( 4, 6,11,29)( 7,26,18,27)( 8,25,17,28) (13,16,34,32)(14,15,33,31)(21,22)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $8160=2^{5} \cdot 3 \cdot 5 \cdot 17$ | 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: | no | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 8160.a | magma: IdentifyGroup(G);
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Character table: |
2 5 3 1 1 1 1 1 1 5 3 . . . . . . 3 1 1 1 1 . . 1 1 . . 1 1 . . . . 5 1 1 1 1 1 1 1 . . . 1 1 . . . . 17 1 . . . . . . . . . . . 1 1 1 1 1a 2a 5a 5b 10a 10b 3a 6a 2b 4a 15a 15b 17a 17b 17c 17d 2P 1a 1a 5b 5a 5a 5b 3a 3a 1a 2b 15b 15a 17b 17a 17d 17c 3P 1a 2a 5b 5a 10b 10a 1a 2a 2b 4a 5a 5b 17d 17c 17a 17b 5P 1a 2a 1a 1a 2a 2a 3a 6a 2b 4a 3a 3a 17d 17c 17a 17b 7P 1a 2a 5b 5a 10b 10a 3a 6a 2b 4a 15b 15a 17c 17d 17b 17a 11P 1a 2a 5a 5b 10a 10b 3a 6a 2b 4a 15a 15b 17c 17d 17b 17a 13P 1a 2a 5b 5a 10b 10a 3a 6a 2b 4a 15b 15a 17a 17b 17c 17d 17P 1a 2a 5b 5a 10b 10a 3a 6a 2b 4a 15b 15a 1a 1a 1a 1a X.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 X.3 16 -4 1 1 1 1 1 -1 . . 1 1 -1 -1 -1 -1 X.4 16 4 1 1 -1 -1 1 1 . . 1 1 -1 -1 -1 -1 X.5 17 5 2 2 . . -1 -1 1 1 -1 -1 . . . . X.6 17 -5 2 2 . . -1 1 1 -1 -1 -1 . . . . X.7 17 -3 A *A *A A 2 . 1 1 *A A . . . . X.8 17 -3 *A A A *A 2 . 1 1 A *A . . . . X.9 17 3 A *A -*A -A 2 . 1 -1 *A A . . . . X.10 17 3 *A A -A -*A 2 . 1 -1 A *A . . . . X.11 30 . . . . . . . -2 . . . C D F E X.12 30 . . . . . . . -2 . . . D C E F X.13 30 . . . . . . . -2 . . . E F C D X.14 30 . . . . . . . -2 . . . F E D C X.15 34 . B *B . . -2 . 2 . -*A -A . . . . X.16 34 . *B B . . -2 . 2 . -A -*A . . . . A = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 B = 2*E(5)^2+2*E(5)^3 = -1-Sqrt(5) = -1-r5 C = -E(17)^6-E(17)^7-E(17)^10-E(17)^11 D = -E(17)^3-E(17)^5-E(17)^12-E(17)^14 E = -E(17)-E(17)^4-E(17)^13-E(17)^16 F = -E(17)^2-E(17)^8-E(17)^9-E(17)^15 |
magma: CharacterTable(G);