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Magma
magma: G := TransitiveGroup(35, 13);
Group action invariants
| Degree $n$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $13$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_7\times A_5$ | ||
| 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: | (1,14,23,32,7,18,29,3,13,25,34,10,17,26)(2,15,22,33,8,19,30,4,12,21,35,6,16,28)(5,11,24,31,9,20,27), (1,29,17,7,34,23,13)(2,27,18,8,31,25,12,5,26,16,9,32,22,11,3,30,20,10,35,24,14)(4,28,19,6,33,21,15) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $7$: $C_7$ $60$: $A_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 5: $A_5$
Degree 7: $C_7$
Low degree siblings
42T86Siblings are shown 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 $ | $1$ | $1$ | $()$ | |
| $ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $20$ | $3$ | $( 3, 4, 5)( 6, 9,10)(11,14,15)(18,19,20)(21,24,25)(26,28,27)(31,32,33)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $15$ | $2$ | $( 2, 3)( 4, 5)( 6, 9)( 8,10)(11,15)(12,14)(16,18)(19,20)(21,24)(22,25)(26,30) (27,28)(31,33)(32,35)$ | |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1, 2, 3, 4, 5)( 6, 9, 7, 8,10)(11,13,12,14,15)(16,18,19,20,17) (21,24,23,22,25)(26,28,27,29,30)(31,34,35,32,33)$ | |
| $ 5, 5, 5, 5, 5, 5, 5 $ | $12$ | $5$ | $( 1, 2, 3, 5, 4)( 6, 7, 8,10, 9)(11,15,13,12,14)(16,18,20,19,17) (21,23,22,25,24)(26,27,28,29,30)(31,33,34,35,32)$ | |
| $ 35 $ | $12$ | $35$ | $( 1, 6,11,18,22,29,33, 5,10,12,17,21,27,32, 2, 7,15,20,25,30,34, 4, 9,14,16, 23,28,31, 3, 8,13,19,24,26,35)$ | |
| $ 35 $ | $12$ | $35$ | $( 1, 6,14,20,22,29,33, 3, 9,12,17,21,26,31, 2, 7,15,18,24,30,34, 4,10,11,16, 23,28,32, 5, 8,13,19,25,27,35)$ | |
| $ 21, 7, 7 $ | $20$ | $21$ | $( 1, 6,12,17,21,30,34, 4, 8,13,19,22,29,33, 2, 7,15,16,23,28,35) ( 3,10,14,18,25,26,32)( 5, 9,11,20,24,27,31)$ | |
| $ 14, 14, 7 $ | $15$ | $14$ | $( 1, 6,13,19,23,28,34, 4, 7,15,17,21,29,33)( 2, 8,12,16,22,30,35) ( 3, 9,14,20,25,27,32, 5,10,11,18,24,26,31)$ | |
| $ 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1, 7,13,17,23,29,34)( 2, 8,12,16,22,30,35)( 3,10,14,18,25,26,32) ( 4, 6,15,19,21,28,33)( 5, 9,11,20,24,27,31)$ | |
| $ 21, 7, 7 $ | $20$ | $21$ | $( 1,11,25,34, 9,18,29, 5,14,23,31,10,17,27, 3,13,24,32, 7,20,26) ( 2,12,22,35, 8,16,30)( 4,15,21,33, 6,19,28)$ | |
| $ 14, 14, 7 $ | $15$ | $14$ | $( 1,11,23,31, 7,20,29, 5,13,24,34, 9,17,27)( 2,12,22,35, 8,16,30) ( 3,15,25,33,10,19,26, 4,14,21,32, 6,18,28)$ | |
| $ 35 $ | $12$ | $35$ | $( 1,11,21,32, 8,17,27, 4,14,22,34, 9,19,26, 2,13,24,33,10,16,29, 5,15,25,35, 7,20,28, 3,12,23,31, 6,18,30)$ | |
| $ 35 $ | $12$ | $35$ | $( 1,11,25,33, 8,17,27, 3,15,22,34, 9,18,28, 2,13,24,32, 6,16,29, 5,14,21,35, 7,20,26, 4,12,23,31,10,19,30)$ | |
| $ 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,13,23,34, 7,17,29)( 2,12,22,35, 8,16,30)( 3,14,25,32,10,18,26) ( 4,15,21,33, 6,19,28)( 5,11,24,31, 9,20,27)$ | |
| $ 14, 14, 7 $ | $15$ | $14$ | $( 1,16,34,12,29, 8,23, 2,17,35,13,30, 7,22)( 3,18,32,14,26,10,25) ( 4,20,33,11,28, 9,21, 5,19,31,15,27, 6,24)$ | |
| $ 21, 7, 7 $ | $20$ | $21$ | $( 1,16,32,13,30,10,23, 2,18,34,12,26, 7,22, 3,17,35,14,29, 8,25) ( 4,19,33,15,28, 6,21)( 5,20,31,11,27, 9,24)$ | |
| $ 35 $ | $12$ | $35$ | $( 1,16,32,15,27, 7,22, 3,19,31,13,30,10,21, 5,17,35,14,28, 9,23, 2,18,33,11, 29, 8,25, 4,20,34,12,26, 6,24)$ | |
| $ 35 $ | $12$ | $35$ | $( 1,16,32,11,28, 7,22, 3,20,33,13,30,10,24, 4,17,35,14,27, 6,23, 2,18,31,15, 29, 8,25, 5,19,34,12,26, 9,21)$ | |
| $ 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,17,34,13,29, 7,23)( 2,16,35,12,30, 8,22)( 3,18,32,14,26,10,25) ( 4,19,33,15,28, 6,21)( 5,20,31,11,27, 9,24)$ | |
| $ 21, 7, 7 $ | $20$ | $21$ | $( 1,21,10,29,15,32,17, 4,25, 7,28,14,34,19, 3,23, 6,26,13,33,18) ( 2,22, 8,30,12,35,16)( 5,24, 9,27,11,31,20)$ | |
| $ 14, 14, 7 $ | $15$ | $14$ | $( 1,21, 7,28,13,33,17, 4,23, 6,29,15,34,19)( 2,22, 8,30,12,35,16) ( 3,24,10,27,14,31,18, 5,25, 9,26,11,32,20)$ | |
| $ 35 $ | $12$ | $35$ | $( 1,21, 9,26,12,34,19, 5,25, 8,29,15,31,18, 2,23, 6,27,14,35,17, 4,24,10,30, 13,33,20, 3,22, 7,28,11,32,16)$ | |
| $ 35 $ | $12$ | $35$ | $( 1,21,10,27,12,34,19, 3,24, 8,29,15,32,20, 2,23, 6,26,11,35,17, 4,25, 9,30, 13,33,18, 5,22, 7,28,14,31,16)$ | |
| $ 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,23, 7,29,13,34,17)( 2,22, 8,30,12,35,16)( 3,25,10,26,14,32,18) ( 4,21, 6,28,15,33,19)( 5,24, 9,27,11,31,20)$ | |
| $ 35 $ | $12$ | $35$ | $( 1,26,19, 8,31,23,14, 4,30,20, 7,32,21,12, 5,29,18, 6,35,24,13, 3,28,16, 9, 34,25,15, 2,27,17,10,33,22,11)$ | |
| $ 14, 14, 7 $ | $15$ | $14$ | $( 1,26,17,10,34,25,13, 3,29,18, 7,32,23,14)( 2,27,16, 9,35,24,12, 5,30,20, 8, 31,22,11)( 4,28,19, 6,33,21,15)$ | |
| $ 35 $ | $12$ | $35$ | $( 1,26,20, 8,33,23,14, 5,30,19, 7,32,24,12, 4,29,18, 9,35,21,13, 3,27,16, 6, 34,25,11, 2,28,17,10,31,22,15)$ | |
| $ 21, 7, 7 $ | $20$ | $21$ | $( 1,26,16, 7,32,22,13, 3,30,17,10,35,23,14, 2,29,18, 8,34,25,12) ( 4,28,19, 6,33,21,15)( 5,27,20, 9,31,24,11)$ | |
| $ 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,29,17, 7,34,23,13)( 2,30,16, 8,35,22,12)( 3,26,18,10,32,25,14) ( 4,28,19, 6,33,21,15)( 5,27,20, 9,31,24,11)$ | |
| $ 35 $ | $12$ | $35$ | $( 1,31,30,25,19,13, 9, 2,32,28,23,20,12,10, 4,34,27,22,18,15, 7, 5,35,26,21, 17,11, 8, 3,33,29,24,16,14, 6)$ | |
| $ 14, 14, 7 $ | $15$ | $14$ | $( 1,31,29,24,17,11, 7, 5,34,27,23,20,13, 9)( 2,32,30,25,16,14, 8, 3,35,26,22, 18,12,10)( 4,33,28,21,19,15, 6)$ | |
| $ 35 $ | $12$ | $35$ | $( 1,31,30,21,18,13, 9, 2,33,26,23,20,12, 6, 3,34,27,22,19,14, 7, 5,35,28,25, 17,11, 8, 4,32,29,24,16,15,10)$ | |
| $ 21, 7, 7 $ | $20$ | $21$ | $( 1,31,30,23,20,12, 7, 5,35,29,24,16,13, 9, 2,34,27,22,17,11, 8) ( 3,32,26,25,18,14,10)( 4,33,28,21,19,15, 6)$ | |
| $ 7, 7, 7, 7, 7 $ | $1$ | $7$ | $( 1,34,29,23,17,13, 7)( 2,35,30,22,16,12, 8)( 3,32,26,25,18,14,10) ( 4,33,28,21,19,15, 6)( 5,31,27,24,20,11, 9)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $420=2^{2} \cdot 3 \cdot 5 \cdot 7$ | 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: | no | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 420.13 | magma: IdentifyGroup(G);
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| Character table: | 35 x 35 character table |
magma: CharacterTable(G);