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Magma
magma: G := TransitiveGroup(34, 39);
Group action invariants
| Degree $n$: | $34$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_{17}^2.Q_8$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,25,3,28,12,33,10,30)(2,18,16,22,11,23,14,19)(4,21,8,27,9,20,5,31)(6,24,17,32,7,34,13,26)(15,29), (1,33,10,22,12,29,3,23)(2,28,14,19,11,34,16,26)(4,18,5,30,9,27,8,32)(6,25,13,24,7,20,17,21)(15,31) | 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, $C_2^2$ $8$: $D_{4}$, $C_4\times C_2$, $Q_8$ $16$: $C_4:C_4$ $32$: 16T49 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 17: 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
| Label | 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 $ | $1$ | $1$ | $()$ | |
| $ 17, 17 $ | $32$ | $17$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,27,19,28,20,29,21,30, 22,31,23,32,24,33,25,34,26)$ | |
| $ 17, 17 $ | $32$ | $17$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,28,21,31,24,34,27,20, 30,23,33,26,19,29,22,32,25)$ | |
| $ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $16$ | $17$ | $(18,25,32,22,29,19,26,33,23,30,20,27,34,24,31,21,28)$ | |
| $ 17, 17 $ | $32$ | $17$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,34,33,32,31,30,29,28, 27,26,25,24,23,22,21,20,19)$ | |
| $ 17, 17 $ | $32$ | $17$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,21,24,27,30,33,19,22, 25,28,31,34,20,23,26,29,32)$ | |
| $ 17, 17 $ | $32$ | $17$ | $( 1,13, 8, 3,15,10, 5,17,12, 7, 2,14, 9, 4,16,11, 6)(18,23,28,33,21,26,31,19, 24,29,34,22,27,32,20,25,30)$ | |
| $ 17, 17 $ | $32$ | $17$ | $( 1, 7,13, 2, 8,14, 3, 9,15, 4,10,16, 5,11,17, 6,12)(18,24,30,19,25,31,20,26, 32,21,27,33,22,28,34,23,29)$ | |
| $ 17, 17 $ | $32$ | $17$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17)(18,22,26,30,34,21,25,29, 33,20,24,28,32,19,23,27,31)$ | |
| $ 17, 17 $ | $32$ | $17$ | $( 1,10, 2,11, 3,12, 4,13, 5,14, 6,15, 7,16, 8,17, 9)(18,32,29,26,23,20,34,31, 28,25,22,19,33,30,27,24,21)$ | |
| $ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $16$ | $17$ | $(18,22,26,30,34,21,25,29,33,20,24,28,32,19,23,27,31)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $289$ | $2$ | $( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,34)(20,33)(21,32) (22,31)(23,30)(24,29)(25,28)(26,27)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ | $289$ | $4$ | $( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)(19,31,34,22)(20,27,33,26) (21,23,32,30)(24,28,29,25)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ | $289$ | $4$ | $( 2, 5,17,14)( 3, 9,16,10)( 4,13,15, 6)( 7, 8,12,11)(19,22,34,31)(20,26,33,27) (21,30,32,23)(24,25,29,28)$ | |
| $ 8, 8, 8, 8, 2 $ | $1156$ | $8$ | $( 1,25, 3,28,12,33,10,30)( 2,18,16,22,11,23,14,19)( 4,21, 8,27, 9,20, 5,31) ( 6,24,17,32, 7,34,13,26)(15,29)$ | |
| $ 8, 8, 8, 8, 2 $ | $1156$ | $8$ | $( 1,24, 5,34, 4,23,17,30)( 2,18, 9,27, 3,29,13,20)( 6,28, 8,33,16,19,14,31) ( 7,22,12,26,15,25,10,21)(11,32)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $34$ | $2$ | $(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)$ | |
| $ 17, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $272$ | $34$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,27)(19,26)(20,25) (21,24)(22,23)(28,34)(29,33)(30,32)$ | |
| $ 17, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $272$ | $34$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,28)(19,27)(20,26) (21,25)(22,24)(29,34)(30,33)(31,32)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ | $578$ | $4$ | $( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)(19,22,34,31)(20,26,33,27) (21,30,32,23)(24,25,29,28)$ | |
| $ 8, 8, 8, 8, 1, 1 $ | $578$ | $8$ | $( 2, 3, 5, 9,17,16,14,10)( 4, 7,13, 8,15,12, 6,11)(19,27,31,33,34,26,22,20) (21,28,23,29,32,25,30,24)$ | |
| $ 8, 8, 8, 8, 1, 1 $ | $578$ | $8$ | $( 2,16, 5,10,17, 3,14, 9)( 4,12,13,11,15, 7, 6, 8)(19,26,31,20,34,27,22,33) (21,25,23,24,32,28,30,29)$ | |
| $ 8, 8, 8, 8, 1, 1 $ | $578$ | $8$ | $( 2,10,14,16,17, 9, 5, 3)( 4,11, 6,12,15, 8,13, 7)(19,33,22,27,34,20,31,26) (21,29,30,28,32,24,23,25)$ | |
| $ 8, 8, 8, 8, 1, 1 $ | $578$ | $8$ | $( 2, 9,14, 3,17,10, 5,16)( 4, 8, 6, 7,15,11,13,12)(19,20,22,26,34,33,31,27) (21,24,30,25,32,29,23,28)$ | |
| $ 8, 8, 8, 8, 2 $ | $1156$ | $8$ | $( 1,25)( 2,28,14,30,17,22, 5,20)( 3,31,10,18,16,19, 9,32)( 4,34, 6,23,15,33, 13,27)( 7,26,11,21,12,24, 8,29)$ | |
| $ 8, 8, 8, 8, 2 $ | $1156$ | $8$ | $( 1,24, 7,20,14,21, 8,25)( 2,29,11,23,13,33, 4,22)( 3,34,15,26,12,28,17,19) ( 5,27, 6,32,10,18, 9,30)(16,31)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $9248=2^{5} \cdot 17^{2}$ | 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: | 9248.y | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 17 P | |
| Type |
magma: CharacterTable(G);