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Magma
magma: G := TransitiveGroup(35, 6);
Group action invariants
| Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $6$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_7\times F_5$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $7$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,9,11,19,21,29,31,4,6,14,16,24,26,34)(2,8,12,18,22,28,32,3,7,13,17,23,27,33)(5,10,15,20,25,30,35), (1,5,3,4)(6,10,8,9)(11,15,13,14)(16,20,18,19)(21,25,23,24)(26,30,28,29)(31,35,33,34) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $7$: $C_7$ $14$: $C_{14}$ $20$: $F_5$ $28$: $C_{28}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $F_5$
Degree 7: $C_7$
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 $ | $1$ | $1$ | $()$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $4$ | $( 2, 3, 5, 4)( 7, 8,10, 9)(12,13,15,14)(17,18,20,19)(22,23,25,24)(27,28,30,29) (32,33,35,34)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $4$ | $( 2, 4, 5, 3)( 7, 9,10, 8)(12,14,15,13)(17,19,20,18)(22,24,25,23)(27,29,30,28) (32,34,35,33)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $2$ | $( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30) (28,29)(32,35)(33,34)$ | |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$ | |
| $ 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33) ( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$ | |
| $ 28, 7 $ | $5$ | $28$ | $( 1, 6,11,16,21,26,31)( 2, 8,15,19,22,28,35, 4, 7,13,20,24,27,33, 5, 9,12,18, 25,29,32, 3,10,14,17,23,30,34)$ | |
| $ 28, 7 $ | $5$ | $28$ | $( 1, 6,11,16,21,26,31)( 2, 9,15,18,22,29,35, 3, 7,14,20,23,27,34, 5, 8,12,19, 25,28,32, 4,10,13,17,24,30,33)$ | |
| $ 14, 14, 7 $ | $5$ | $14$ | $( 1, 6,11,16,21,26,31)( 2,10,12,20,22,30,32, 5, 7,15,17,25,27,35) ( 3, 9,13,19,23,29,33, 4, 8,14,18,24,28,34)$ | |
| $ 35 $ | $4$ | $35$ | $( 1, 7,13,19,25,26,32, 3, 9,15,16,22,28,34, 5, 6,12,18,24,30,31, 2, 8,14,20, 21,27,33, 4,10,11,17,23,29,35)$ | |
| $ 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,11,21,31, 6,16,26)( 2,12,22,32, 7,17,27)( 3,13,23,33, 8,18,28) ( 4,14,24,34, 9,19,29)( 5,15,25,35,10,20,30)$ | |
| $ 28, 7 $ | $5$ | $28$ | $( 1,11,21,31, 6,16,26)( 2,13,25,34, 7,18,30, 4,12,23,35, 9,17,28, 5,14,22,33, 10,19,27, 3,15,24,32, 8,20,29)$ | |
| $ 28, 7 $ | $5$ | $28$ | $( 1,11,21,31, 6,16,26)( 2,14,25,33, 7,19,30, 3,12,24,35, 8,17,29, 5,13,22,34, 10,18,27, 4,15,23,32, 9,20,28)$ | |
| $ 14, 14, 7 $ | $5$ | $14$ | $( 1,11,21,31, 6,16,26)( 2,15,22,35, 7,20,27, 5,12,25,32,10,17,30) ( 3,14,23,34, 8,19,28, 4,13,24,33, 9,18,29)$ | |
| $ 35 $ | $4$ | $35$ | $( 1,12,23,34,10,16,27, 3,14,25,31, 7,18,29, 5,11,22,33, 9,20,26, 2,13,24,35, 6,17,28, 4,15,21,32, 8,19,30)$ | |
| $ 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,16,31,11,26, 6,21)( 2,17,32,12,27, 7,22)( 3,18,33,13,28, 8,23) ( 4,19,34,14,29, 9,24)( 5,20,35,15,30,10,25)$ | |
| $ 28, 7 $ | $5$ | $28$ | $( 1,16,31,11,26, 6,21)( 2,18,35,14,27, 8,25, 4,17,33,15,29, 7,23, 5,19,32,13, 30, 9,22, 3,20,34,12,28,10,24)$ | |
| $ 28, 7 $ | $5$ | $28$ | $( 1,16,31,11,26, 6,21)( 2,19,35,13,27, 9,25, 3,17,34,15,28, 7,24, 5,18,32,14, 30, 8,22, 4,20,33,12,29,10,23)$ | |
| $ 14, 14, 7 $ | $5$ | $14$ | $( 1,16,31,11,26, 6,21)( 2,20,32,15,27,10,22, 5,17,35,12,30, 7,25) ( 3,19,33,14,28, 9,23, 4,18,34,13,29, 8,24)$ | |
| $ 35 $ | $4$ | $35$ | $( 1,17,33,14,30, 6,22, 3,19,35,11,27, 8,24, 5,16,32,13,29,10,21, 2,18,34,15, 26, 7,23, 4,20,31,12,28, 9,25)$ | |
| $ 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,21, 6,26,11,31,16)( 2,22, 7,27,12,32,17)( 3,23, 8,28,13,33,18) ( 4,24, 9,29,14,34,19)( 5,25,10,30,15,35,20)$ | |
| $ 28, 7 $ | $5$ | $28$ | $( 1,21, 6,26,11,31,16)( 2,23,10,29,12,33,20, 4,22, 8,30,14,32,18, 5,24, 7,28, 15,34,17, 3,25, 9,27,13,35,19)$ | |
| $ 28, 7 $ | $5$ | $28$ | $( 1,21, 6,26,11,31,16)( 2,24,10,28,12,34,20, 3,22, 9,30,13,32,19, 5,23, 7,29, 15,33,17, 4,25, 8,27,14,35,18)$ | |
| $ 14, 14, 7 $ | $5$ | $14$ | $( 1,21, 6,26,11,31,16)( 2,25, 7,30,12,35,17, 5,22,10,27,15,32,20) ( 3,24, 8,29,13,34,18, 4,23, 9,28,14,33,19)$ | |
| $ 35 $ | $4$ | $35$ | $( 1,22, 8,29,15,31,17, 3,24,10,26,12,33,19, 5,21, 7,28,14,35,16, 2,23, 9,30, 11,32,18, 4,25, 6,27,13,34,20)$ | |
| $ 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,26,16, 6,31,21,11)( 2,27,17, 7,32,22,12)( 3,28,18, 8,33,23,13) ( 4,29,19, 9,34,24,14)( 5,30,20,10,35,25,15)$ | |
| $ 28, 7 $ | $5$ | $28$ | $( 1,26,16, 6,31,21,11)( 2,28,20, 9,32,23,15, 4,27,18,10,34,22,13, 5,29,17, 8, 35,24,12, 3,30,19, 7,33,25,14)$ | |
| $ 28, 7 $ | $5$ | $28$ | $( 1,26,16, 6,31,21,11)( 2,29,20, 8,32,24,15, 3,27,19,10,33,22,14, 5,28,17, 9, 35,23,12, 4,30,18, 7,34,25,13)$ | |
| $ 14, 14, 7 $ | $5$ | $14$ | $( 1,26,16, 6,31,21,11)( 2,30,17,10,32,25,12, 5,27,20, 7,35,22,15) ( 3,29,18, 9,33,24,13, 4,28,19, 8,34,23,14)$ | |
| $ 35 $ | $4$ | $35$ | $( 1,27,18, 9,35,21,12, 3,29,20, 6,32,23,14, 5,26,17, 8,34,25,11, 2,28,19,10, 31,22,13, 4,30,16, 7,33,24,15)$ | |
| $ 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,31,26,21,16,11, 6)( 2,32,27,22,17,12, 7)( 3,33,28,23,18,13, 8) ( 4,34,29,24,19,14, 9)( 5,35,30,25,20,15,10)$ | |
| $ 28, 7 $ | $5$ | $28$ | $( 1,31,26,21,16,11, 6)( 2,33,30,24,17,13,10, 4,32,28,25,19,12, 8, 5,34,27,23, 20,14, 7, 3,35,29,22,18,15, 9)$ | |
| $ 28, 7 $ | $5$ | $28$ | $( 1,31,26,21,16,11, 6)( 2,34,30,23,17,14,10, 3,32,29,25,18,12, 9, 5,33,27,24, 20,13, 7, 4,35,28,22,19,15, 8)$ | |
| $ 14, 14, 7 $ | $5$ | $14$ | $( 1,31,26,21,16,11, 6)( 2,35,27,25,17,15, 7, 5,32,30,22,20,12,10) ( 3,34,28,24,18,14, 8, 4,33,29,23,19,13, 9)$ | |
| $ 35 $ | $4$ | $35$ | $( 1,32,28,24,20,11, 7, 3,34,30,21,17,13, 9, 5,31,27,23,19,15, 6, 2,33,29,25, 16,12, 8, 4,35,26,22,18,14,10)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $140=2^{2} \cdot 5 \cdot 7$ | 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: | 140.5 | magma: IdentifyGroup(G);
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| Character table: | 35 x 35 character table |
magma: CharacterTable(G);