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Magma
magma: G := TransitiveGroup(34, 40);
Group action invariants
| Degree $n$: | $34$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_{17}^2.D_4$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,13,8,3,15,10,5,17,12,7,2,14,9,4,16,11,6)(18,33,25,27)(19,20,24,23)(21,28,22,32)(26,31,34,29), (1,18,10,21,2,24,11,27,3,30,12,33,4,19,13,22,5,25,14,28,6,31,15,34,7,20,16,23,8,26,17,29,9,32) | 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, $C_2^2$ $8$: $D_{4}$ x 2, $C_4\times C_2$ $16$: $C_2^2:C_4$ $32$: $C_4\wr C_2$ 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
| 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $17$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)$ |
| $ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $17$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)$ |
| $ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $17$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)$ |
| $ 17, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $17$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)$ |
| $ 17, 17 $ | $16$ | $17$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(18,34,33,32,31,30,29,28, 27,26,25,24,23,22,21,20,19)$ |
| $ 17, 17 $ | $32$ | $17$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,34,33,32,31,30,29,28, 27,26,25,24,23,22,21,20,19)$ |
| $ 17, 17 $ | $32$ | $17$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,34,33,32,31,30,29,28, 27,26,25,24,23,22,21,20,19)$ |
| $ 17, 17 $ | $32$ | $17$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,34,33,32,31,30,29,28, 27,26,25,24,23,22,21,20,19)$ |
| $ 17, 17 $ | $16$ | $17$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(18,32,29,26,23,20,34,31, 28,25,22,19,33,30,27,24,21)$ |
| $ 17, 17 $ | $32$ | $17$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,32,29,26,23,20,34,31, 28,25,22,19,33,30,27,24,21)$ |
| $ 17, 17 $ | $32$ | $17$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,32,29,26,23,20,34,31, 28,25,22,19,33,30,27,24,21)$ |
| $ 17, 17 $ | $16$ | $17$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$ |
| $ 17, 17 $ | $32$ | $17$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,26,34,25,33,24,32,23, 31,22,30,21,29,20,28,19,27)$ |
| $ 17, 17 $ | $16$ | $17$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(18,25,32,22,29,19,26,33, 23,30,20,27,34,24,31,21,28)$ |
| $ 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 $ | $136$ | $34$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(19,34)(20,33)(21,32) (22,31)(23,30)(24,29)(25,28)(26,27)$ |
| $ 17, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $136$ | $34$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(19,34)(20,33)(21,32) (22,31)(23,30)(24,29)(25,28)(26,27)$ |
| $ 17, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $136$ | $34$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(19,34)(20,33)(21,32) (22,31)(23,30)(24,29)(25,28)(26,27)$ |
| $ 17, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $136$ | $34$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(19,34)(20,33)(21,32) (22,31)(23,30)(24,29)(25,28)(26,27)$ |
| $ 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $34$ | $4$ | $(19,22,34,31)(20,26,33,27)(21,30,32,23)(24,25,29,28)$ |
| $ 17, 4, 4, 4, 4, 1 $ | $136$ | $68$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(19,22,34,31)(20,26,33,27) (21,30,32,23)(24,25,29,28)$ |
| $ 17, 4, 4, 4, 4, 1 $ | $136$ | $68$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(19,22,34,31)(20,26,33,27) (21,30,32,23)(24,25,29,28)$ |
| $ 17, 4, 4, 4, 4, 1 $ | $136$ | $68$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(19,22,34,31)(20,26,33,27) (21,30,32,23)(24,25,29,28)$ |
| $ 17, 4, 4, 4, 4, 1 $ | $136$ | $68$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(19,22,34,31)(20,26,33,27) (21,30,32,23)(24,25,29,28)$ |
| $ 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $34$ | $4$ | $(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)$ |
| $ 17, 4, 4, 4, 4, 1 $ | $136$ | $68$ | $( 1,15,12, 9, 6, 3,17,14,11, 8, 5, 2,16,13,10, 7, 4)(19,31,34,22)(20,27,33,26) (21,23,32,30)(24,28,29,25)$ |
| $ 17, 4, 4, 4, 4, 1 $ | $136$ | $68$ | $( 1, 9,17, 8,16, 7,15, 6,14, 5,13, 4,12, 3,11, 2,10)(19,31,34,22)(20,27,33,26) (21,23,32,30)(24,28,29,25)$ |
| $ 17, 4, 4, 4, 4, 1 $ | $136$ | $68$ | $( 1, 8,15, 5,12, 2, 9,16, 6,13, 3,10,17, 7,14, 4,11)(19,31,34,22)(20,27,33,26) (21,23,32,30)(24,28,29,25)$ |
| $ 17, 4, 4, 4, 4, 1 $ | $136$ | $68$ | $( 1, 5, 9,13,17, 4, 8,12,16, 3, 7,11,15, 2, 6,10,14)(19,31,34,22)(20,27,33,26) (21,23,32,30)(24,28,29,25)$ |
| $ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $578$ | $4$ | $( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,22,34,31) (20,26,33,27)(21,30,32,23)(24,25,29,28)$ |
| $ 4, 4, 4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $578$ | $4$ | $( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,31,34,22) (20,27,33,26)(21,23,32,30)(24,28,29,25)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ | $289$ | $4$ | $( 2, 5,17,14)( 3, 9,16,10)( 4,13,15, 6)( 7, 8,12,11)(19,22,34,31)(20,26,33,27) (21,30,32,23)(24,25,29,28)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ | $578$ | $4$ | $( 2, 5,17,14)( 3, 9,16,10)( 4,13,15, 6)( 7, 8,12,11)(19,31,34,22)(20,27,33,26) (21,23,32,30)(24,28,29,25)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ | $289$ | $4$ | $( 2,14,17, 5)( 3,10,16, 9)( 4, 6,15,13)( 7,11,12, 8)(19,31,34,22)(20,27,33,26) (21,23,32,30)(24,28,29,25)$ |
| $ 34 $ | $272$ | $34$ | $( 1,18,10,21, 2,24,11,27, 3,30,12,33, 4,19,13,22, 5,25,14,28, 6,31,15,34, 7, 20,16,23, 8,26,17,29, 9,32)$ |
| $ 34 $ | $272$ | $34$ | $( 1,18, 7,20,13,22, 2,24, 8,26,14,28, 3,30, 9,32,15,34, 4,19,10,21,16,23, 5, 25,11,27,17,29, 6,31,12,33)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $68$ | $2$ | $( 1,18)( 2,24)( 3,30)( 4,19)( 5,25)( 6,31)( 7,20)( 8,26)( 9,32)(10,21)(11,27) (12,33)(13,22)(14,28)(15,34)(16,23)(17,29)$ |
| $ 34 $ | $272$ | $34$ | $( 1,18,17,29,16,23,15,34,14,28,13,22,12,33,11,27,10,21, 9,32, 8,26, 7,20, 6, 31, 5,25, 4,19, 3,30, 2,24)$ |
| $ 34 $ | $272$ | $34$ | $( 1,18,13,22, 8,26, 3,30,15,34,10,21, 5,25,17,29,12,33, 7,20, 2,24,14,28, 9, 32, 4,19,16,23,11,27, 6,31)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $1156$ | $4$ | $( 1,18,10,32)( 2,29, 9,21)( 3,23, 8,27)( 4,34, 7,33)( 5,28, 6,22)(11,26,17,24) (12,20,16,30)(13,31,15,19)(14,25)$ |
| $ 8, 8, 8, 8, 2 $ | $1156$ | $8$ | $( 1,18,10,30,12,27, 3,32)( 2,25,14,24,11,20,16,21)( 4,22, 5,29, 9,23, 8,33) ( 6,19,13,34, 7,26,17,28)(15,31)$ |
| $ 8, 8, 8, 8, 2 $ | $1156$ | $8$ | $( 1,18,10,23, 8,20,16,32)( 2,28, 6,34, 7,27, 3,21)( 4,31,15,22, 5,24,11,33) ( 9,30,12,26,17,25,14,29)(13,19)$ |
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.z | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);